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The inertia operator and Hall algebra of algebraic stacks Ronagh, Pooya 2016

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The inertia operator and Hall algebra of algebraic stacksbyPooya RonaghB.Sc. of Mathematics, Sharif University of Technology, 2009B.Sc. of Computer Science, Sharif University of Technology, 2009M.Sc. of Mathematics, The University of British Columbia, 2011a thesis submitted in partial fulfillment ofthe requirements for the degree ofDoctor of Philosophyinthe faculty of graduate and postdoctoral studies(Mathematics)The University of British Columbia(Vancouver)May 2016c© Pooya Ronagh, 2016AbstractWe view the inertia construction of algebraic stacks as an operator on the Grothendieckgroups of various categories of algebraic stacks. We are interested in showing that theinertia operator is (locally finite and) diagonalizable over for instance the field of ra-tional functions of the motivic class of the affine line q = [A1]. This is proved for theGrothendieck group of Deligne-Mumford stacks and the category of quasi-split Artinstacks.Motivated by the quasi-splitness condition we then develop a theory of linear al-gebraic stacks and algebroids, and define a space of stack functions over a linearalgebraic stack. We prove diagonalization of the semisimple inertia for the space ofstack functions. A different family of operators is then defined that are closely relatedto the semisimple inertia. These operators are diagonalizable on the Grothendieckring itself (i.e. without inverting polynomials in q) and their corresponding eigenvaluedecompositions are used to define a graded structure on the Grothendieck ring.We then define the structure of a Hall algebra on the space of stack functions. Thecommutative and non-commutative products of the Hall algebra respect the gradedstructure defined above. Moreover, the two multiplications coincide on the associatedgraded algebra.This result provides a geometric way of defining a Lie subalgebra of virtually in-decomposables. Finally, for any algebroid, an ε-element is defined and shown to becontained in the space of virtually indecomposables. This is a new approach to thetheory of generalized Donaldson-Thomas invariants.iiPrefaceThis dissertation is the original work of the author, P. Ronagh, in collaboration withhis PhD supervisor, Prof. Kai Behrend.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Overview of the Donaldson-Thomas invariants . . . . . . . . . . . . . . . . 11.2 A break-down of our program . . . . . . . . . . . . . . . . . . . . . . . . . . 4I Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Stratification of group schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1 Stratification of group spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Groups of multiplicative type . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.1 Quasi-split tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 Maximal tori of group schemes . . . . . . . . . . . . . . . . . . . . 122.3 Spreading out arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Tori and unipotent group schemes . . . . . . . . . . . . . . . . . . . . . . . 143 Inertia operator of K-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.1 K-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Inertia stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 Central band of a gerbe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.4 The semisimple and unipotent inertia . . . . . . . . . . . . . . . . . . . . . 19ivII Diagonalization of the inertia . . . . . . . . . . . . . . . . . . . . . . 214 Groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.1 Filtration by central number . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Local finiteness and diagonalization . . . . . . . . . . . . . . . . . . . . . . 234.3 The operators Ir and eigenprojections . . . . . . . . . . . . . . . . . . . . . 245 Inertia endomorphism on Deligne-Mumford stacks . . . . . . . . . . . . . . . . 275.1 Stratification of Deligne-Mumford stacks . . . . . . . . . . . . . . . . . . . 275.2 Filtration by split central number . . . . . . . . . . . . . . . . . . . . . . . . 285.3 Local finiteness and diagonalization . . . . . . . . . . . . . . . . . . . . . . 295.4 The operators Ir and eigenprojections . . . . . . . . . . . . . . . . . . . . . 316 Inertia endomorphism of algebraic stacks . . . . . . . . . . . . . . . . . . . . . . 356.1 Stratification of stacks in characteristic zero . . . . . . . . . . . . . . . . . 356.2 Filtration by central rank and split central number . . . . . . . . . . . . . 376.3 An ascending filtration and local finiteness . . . . . . . . . . . . . . . . . . 406.4 Spectrum of the unipotent inertia . . . . . . . . . . . . . . . . . . . . . . . . 427 Quasi-split stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447.1 Motivic classes of quasi-split tori . . . . . . . . . . . . . . . . . . . . . . . . 447.2 Quasi-split stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467.3 An ascending filtration and local finiteness . . . . . . . . . . . . . . . . . . 477.4 A descending filtration and diagonalization . . . . . . . . . . . . . . . . . . 487.5 Spectrum of the semisimple inertia of quasi-split stacks . . . . . . . . . . 498 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53III Algebroids and their Hall algebras . . . . . . . . . . . . . . . . . . 609 Linear Stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619.1 Algebraic stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619.2 Linear algebraic stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6210 Algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6710.1 Finite type algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6710.2 Algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7310.2.1 Algebroid inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7510.2.2 Clear algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76v11 K-algebra of stack functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7711.1 Stack functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7711.2 The filtration by split central rank . . . . . . . . . . . . . . . . . . . . . . . . 7811.3 The idempotent operators Er . . . . . . . . . . . . . . . . . . . . . . . . . . . 7811.4 The spectrum of semisimple inertia . . . . . . . . . . . . . . . . . . . . . . . 8311.5 Graded structure of multiplication . . . . . . . . . . . . . . . . . . . . . . . 8712 Hall Algebra of algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8812.1 The Hall algebra of a linear stack . . . . . . . . . . . . . . . . . . . . . . . . 8812.1.1 Filtered structure of the Hall algebra . . . . . . . . . . . . . . . . . 9012.1.2 Proof of Theorem 12.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 9012.2 Epsilon functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9412.3 The semi-classical Hall algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 96Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98A Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102A.1 Mobius numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102A.2 Identities involving Stirling numbers . . . . . . . . . . . . . . . . . . . . . . 103A.3 Labelled partitions and integer partitions . . . . . . . . . . . . . . . . . . . 104B Splitting covers of gerbes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105viList of TablesTable 8.1 Spectrum of the inertia endomorphism on a 4-dimensional K(Var)-submodule of K(St) containing [BGL2] . . . . . . . . . . . . . . . . . . . 55Table 8.2 Eigenprojections of [BGL2] . . . . . . . . . . . . . . . . . . . . . . . . . . 55Table 8.3 Stratification of GL3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Table 8.4 Stratification of R1(2,1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Table 8.5 Spectrum of the inertia endomorphism of a 9-dimensional K(Var)-submodule of K(St) containing [BGL3] . . . . . . . . . . . . . . . . . . . 59viiAcknowledgmentsMy deepest gratitude goes to my supervisor, Kai Behrend, for involving me in an amaz-ing research project and for his instructive guidance, patient mentorship and the manybrilliant ideas he has shared with me. I am grateful to Tom Bridgeland who played thewizard behind the many initial ideas that started this entire project.I would like to thank Jim Bryan and Zinovy Reichstein for their encouragementand for creating an amazing research environment here in UBC. Jim and Zinovy aregreat teachers of mine and have always been very helpful and enthusiastic in all myconversations with them.My thanks go to Sheida, Yousef and Maryam, my parents and my sister who havealways given me the warmth of choosing my path in life and for the support andsacrifices they have done for me, and to Madi who made sure I have a life outside ofmath.viiiChapter 1IntroductionDonaldson-Thomas invariants of a Calabi-Yau 3-fold M should count the number ofsemistable coherent sheaves on it. They are mathematically interesting because theyare invariant under continuous deformations of M and physically interesting becausethey count the BPS states of D-branes systems as predicted by string theory.The conventional Donaldson-Thomas invariants [5, 49] are defined only whenthe semistable and stable coherent sheaves coincide. The generalized theories ofDonaldson-Thomas invariants [29, 33] define these invariants without the mentionedrestriction.In [12], Bridgeland proposes a more conceptual and easy to understand approachto Joyce’s motivic Hall Algebras [27] and in [11], he shows how the machinery can beapplied to produce results on Donaldson-Thomas invariants of Calabi-Yau varieties.In our correspondence with Bridgeland, he suggests the idea of viewing the con-struction of the inertia stack as an operator on a Grothendieck group of algebraicstacks and replace Joyce’s virtual projections [27] with eigenprojections of this opera-tor. Our goal is to reproduce Joyce’s project [24–29] of generalized Donaldson-Thomasinvariants in a more geometric framework.1.1 Overview of the Donaldson-Thomas invariantsIn this section, we will be working over the base field C. Throughout, M will denotea fixed smooth complex projective Calabi-Yau threefold and by this we mean that thecanonical bundle KM is trivial and H1(M,OM) = 0. Coh(M) will denote the abeliancategory of coherent sheaves on M and M the moduli stack of objects of Coh(M).This means that the objects of M over a scheme S are coherent sheaves on S×M thatare flat over S. We will use the same notation for a scheme (if any) and the stack itrepresents.1Grothendieck group of the moduliLet K(Coh(M)) be the Grothendieck group of the category Coh(M). The bilinear formχ(E,F)=∑i(−1)idimCExti(E,F)is called the Euler form on K(Coh(M)). By Serre duality, the sets of left and rightorthogonal objects to Coh(M) with respect to the Euler form, i.e.{E : χ(E,F)= 0 for all objects F in Coh(M)}and{F : χ(E,F)= 0 for all objects E in Coh(M)}are the same subgroup K(Coh(M))⊥ and therefore the quotientN(M)=K(Coh(M))/K(Coh(M))⊥called the numerical Grothendieck group carries a well-defined non-degenerate bilinearform. There is a monoid Γ ⊂N(M) consisting of classes of sheaves. Fixing a class γ ∈ Γyields an open and closed substack Mγ ⊆M of objects of class γ.Stability conditionHere we will fix our notion of stability condition to be that of Gieseker, although thetheory should presumably work in more generality. Let OM(1) be a fixed very ampleline bundle onM . For a family E ∈Coh(M) of coherent sheaves, the Hilbert polynomialPE is the unique polynomial in Q[t] such thatPE(n)= dimH0(E(n)) for all n 0 .This polynomial only depends on the class of E in Γ . Thus we may write Pγ for γ ∈ Γ .We define τ(γ) = Pγ/rγ where rγ is the leading coefficient of Pγ . This means that τassociates to any class in Γ a monic polynomial of degree at most 3. If p(t) and p′(t)are two such polynomials we sayp ≤ p′, if degp > degp′ and in case of equality p(t)≤ p′(t) for all t 0.1This defines a notion of stability; E is stable if for all nonzero subobjects S ⊆ E, wehave τ([S]) < τ([E/S]) and it is semistable if for all nonzero subobjects S ⊆ E, wehave τ([S])≤ τ([E/S]).1The first condition on degrees serves to assure all semistable sheaves, E, are pure of dimensiondim(suppE).2The category Coh(M) equipped with this stability condition, has the propertiesone would expect from a stability condition (such as Harder-Narasimhan filtrations,Jordan-Holder filtrations, etc.)[23]. We write SS(τ) for the open substack of M consist-ing of semistable objects for τ . This itself consists of connected components SSγ(τ)of semistable objects of class γ. The latter stacks are all of finite type.Conventional Donaldson-Thomas invariantsFor a proper Deligne-Mumford stack X over C, with a symmetric obstruction theoryand virtual fundamental class [X]vir ∈ A0(X), the virtual count of points on X isdefined as the rational number#vir =∫[X]vir1∈Q .On the other hand, by Behrend’s theorem [6] this virtual count coincides with the Eulercharacteristic of X, weighted by the Behrend function νX of X:∫[X]vir1= χ(X,νX).Here is a few remarks on the concepts that appear in the above definition.• The virtual fundamental class: Behrend and Fantechi [7], show that for anyDeligne-Mumford stack with a perfect obstruction theory, the virtual class [X]viris a well-defined class in Adim(X)(X).• The virtual count: If moreover, X is proper, the push-forward of this class overa point is well-defined and produces a rational number,∫[X]vir 1. This numberis an integer if X is a scheme or an algebraic space.Let X → C be a family Xt of smooth projective varieties parametrized by t ∈ C ,and consider the family of moduli spaces Mt of stable sheaves on the fiber Xtthen the class [Mt]vir is independent of t (provided that the compact modulispaces and virtual fundamental classes we are talking about exist. This is forexample the case if Xt are smooth projective 3-folds andMt are moduli spacesof stable sheaves of fixed class in the numerical Grothendieck group and nostrictly semistable object occurs [49, Cor. 3.53].• Behrend function: This is a Z-valued locally constructible functions are definedover schemes (as done by Behrend [6]), they can be generalized to all algebraicspace, and algebraic stacks over C, locally of finite type according to the prop-erty that for any smooth morphism ϕ :W →X of relative dimension n, we haveϕ∗(νX)= (−1)nνW .3For Calabi-Yau 3-folds, Thomas [49] constructs a symmetric (in particular perfect)obstruction theory on Stγ(τ). If SSγ(τ) = Stγ(τ) then Stγ(τ) is proper as well whichis the conventional case in which Donaldson-Thomas invariants are defined:DTγ(τ)=∫[Stγ(τ)]vir1= χ(Stγ(τ),νStγ(τ))=∑n∈Zχ(ν−1Stγ(τ)(n)).By the simplicity of the Behrend function, it is therefore suggestive to take theweighted Euler characteristic as the definition of Donaldson-Thomas invariants formoduli stacks. The problem with this naive approach is that (1) the Euler characteristiccannot be defined even for simplest Artin stacks such as BXGm, and (2) the intersec-tion theory tools that result deformation invariance are not in our disposal anymore.The goals of projects of Kontsevich and Soibelman [33], Joyce and Song [29] and thatof ours is to find numbers DTγ(τ)∈Q that correctly count the semistable objects inSSγ(τ) and are (1) invariant under deformations of M ; and, (2) when SSγ(τ)= Stγ(τ)these numbers coincides with the conventional Donaldson-Thomas invariants above.Dependence on Stability conditionAnother caveat of these invariants is that they depend on the stability condition τ ,which in turn depends on the choice of a very ample line bundle OM(1). Explanationof how these numbers change subordinate to change of the stability condition is thecontent of the wall-crossing formulae in [33] and [29]. So another goal of ours wouldbe to prove some wall-crossing formulae, giving DTγ(τ) in terms of DTγ(τ′).1.2 A break-down of our programLocal-finiteness and diagonalization of inertia operatorLet S be a base category of schemes. For the case of Deligne-Mumford stacks in §5this would be the category of finite type schemes over a noetherian base scheme. Forthe cases of Artin stacks in §6 and §7 and all the rest of this work, S would be thecategory of varieties over the spectrum of an algebraically closed field in characteristiczero. The Grothendieck ring of S, will be denoted by K(S).We then consider a category of algebraic stacks over S. In §5 this category isthat of the Deligne-Mumford stacks DM and in §6 it will be the category St of alge-braic stacks with affine diagonals. In §7 we work with the category QS of quasi-splitalgebraic stacks and finally in §11 we consider the space of stack functions from alge-broids to a fixed base linear stacks.The Grothendieck ring of this category is defined as the commutative ring, K, gen-erated by isomorphism classes of algebraic stacks modulo season relations. K is a uni-tal associative K(S)-algebra in the obvious way. We will use the terms Grothendieck4ring, Grothendieck module and Grothendieck algebra to stress the structure of oursubject matter. For an object X of K, the construction of the inertia stack IX respectsthe equivalence and scissor relations and is K(S)-linear, and it hence induces a well-defined inertia operator on the K(S)-algebra K.Other important variants of this operator are the semisimple and unipotent inertiaoperators, respectively Iss and Iu, which are defined carefully in §3.4 but are roughlythe semisimple and unipotent loci of IX when viewed as a group object over X.We implicitly always assume that K is tensored with Q. Our main results on localfiniteness and diagonalization of the endormophism I : K → K are as follows. Hereq = [A1] is our notation for the class of the affine line.1. Corollary 5.11: In the case of Deligne-Mumford stacks the operator I is diago-nalizable as a K(Sch/B)-linear endomorphism, and the eigenvalue spectrum ofit is equal to N, the set of positive integers.2. Corollary 6.19: In the case of Artin stacks, the unipotent inertia Iu is diagonal-izable on K(St)[q−1,{(qk−1)−1 : k ≥ 1}] and the eigenvalue spectrum of it isthe set {qk : k≥ 0} of all power of q.3. Theorem 7.10: In the case of quasi-split stacks, the operator I is diagonalizableas a Q(q)-linear endomorphism and the eigenvalue spectrum of it is the set ofall polynomials of the formnquk∏i=1(qri −1).4. Theorem 7.14: In the case of quasi-split stacks the endormorphism Iss is diago-nalizable as a Q(q)-linear operator and the eigenvalue spectrum of it is the setof all polynomials of the formnk∏i=1(qri −1).5. Theorem 11.10: In the case of stacks functions of algebroids over a linear stackM, the operator Iss : K(M)(q)→ K(M)(q) is diagonalizable and the eigenvaluespectrum of it onsists of the set of all polynomials of the formk∏i=1(qri −1).5Eigenprojections and eigendecompositionsThe fact that in Theorem 11.10, the semisimple inertia is diagonalizable only afterinverting q is not convenient. However a technique that is represented in simpler casesin §4.3 and §5.4 for computing eigenprojections of the inertia of Deligne-Mumfordstacks, hints to definition of a different set of operators, {En}, that are diagonalizableon K(M).These operators are defined in terms of complete sets of mutually orthogonalidempotents in §11.3. They respect only a coarser filtration than the usual semisimpleinertia (explained in §11.2) but they are simultaneously diagonalizable (Corollary 11.5).This creates an eigenvalue decompositionK(M)=⊕k≥0Kk(M).and a filtration by the order of vanishing of the inertia at q = 1,K≤n(M)= kerEn+1 =⊕k≤nKk(M),also called the order filtration. Details are explained in §12.1.Hall algebraA noncommutative product denoted by ∗, is then defined in §12.1 on K(M) in theusual way of defining Hall algebras. The main result of §12 is that the commutativeproduct (defined by the cartesian product of stacks) and the new convolution productrespect the order filtration and on the associated graded they coincide (Theorem 12.3).The Hall product is known to be associative (cf. [12]) and induces the structure ofa Lie algebra on K(M). In particular Kind =K≤1(M) is a Lie subalgebra of K(M) whichis our counterpart for the space of virtually indecomposables of Joyce and Song [29].In §12.2 we introduce the elements εk associated to a stack function and showin Corollary 12.9 that it lives in K≤k(M). In particular, ε1 SSγ(τ) is a virtually in-decomposable, and can be viewed as a logarithm. This is the key point in defininginvariants similar to that of [29], details of which are to be published by the authorand K. Behrend.6Part IPreliminaries7Chapter 2Stratification of group schemesThroughout, Sch/B, denotes the big étale site of schemes of finite type over a fixednoetherian base scheme B. St/B denotes the category of algebraic stacks over Sch/Bwith affine diagonals. This in particular means that for an S-point s : S → X of analgebraic stack X, the sheaf of automorphisms Aut(s)→ S is an affine group scheme.2.1 Stratification of group spacesBy a group space G→ X, we mean a group object in the category St/B. In this sectionwe will see that we can always stratify such objects by nicely-behaved group schemes.The results of this section will be used in §6 and §7.Let G be a finitely presented group scheme over a base scheme X. Recall that theconnected component of unity, G0, is a priori defined in [4, Exp. VI(B), Def. 3.1] as thesubfunctor of G that assigns to any morphism T →X, the setG0(T)= {u∈G(T) :∀x ∈X,ux(Tx)⊂G0x}.Here G0x is the connected component of unity of the algebraic group Gx =G⊗X κ(x).By [4, Exp. VI(B), Thm. 3.10], if G is smooth over X, this functor is representableby a unique open subgroup scheme of G. Also note that in this case, G0 is smoothand finitely presented and is preserved by base change [4, Exp. VI(B), Prop. 3.3]. Thefollowing lemma is essentially stated through [8, §5], but we will reframe it for furtherreference:Lemma 2.1. Let G → X be a smooth group space of finite type, and assume X is anoetherian scheme. Then X can be written as a disjoint union of a finite family {Xα}α∈Aof reduced, locally closed subschemes, such that for each α ∈ A, G|Xα → Xα is a groupscheme and the functor of connected component of the identity is representable.8Remark 2.2. The above result holds without the smoothness condition if X is a schemeof finite type over a field of characteristic zero. This is a consequence of combing thisresult with generic smoothness theorem [50, Thm. 25.3.1] in characteristic zero.When G is finitely presented and smooth, the quotient space G/G0 exists as afinitely presented and étale algebraic space over X. For sheaf theoretic reasons, theformation of this quotient is also preserved by base change.G0 is not closed in general (interesting examples can be found in [40, §7.3 (iii)] and[4, Exp. XIX, §5]), however we have the followingLemma 2.3. Let G→ X be a smooth group scheme and assume that G =G/G0 is finiteand étale. Then G0 is a closed subscheme of G.Proof. G→X is finite, hence proper and consequently universally closed. Thus in thecartesian diagramG0×X Gϕ //G×X GG // G×X Gthe morphism ϕ : (h,g) , (hg,g) is a closed immersion. The property of being aclosed immersion is local in the fppf topology and thus by the cartesian diagramG0×X G//G×X GG0 // Gthe embedding of G0 in G is also a closed immersion. The vertical right hand arrow isdescribed by (g,h), gh−1. Corollary 2.4. Let G be a smooth finitely presented group scheme over X. There existsa stratification of X by finitely many locally closed subschemes X = {Xα}α∈A such thatfor all α∈ A and all group schemes Gα =G|Xα , (1) G0α is closed and (2) Gα/G0α is finiteand étale over Xα.Proof. Let G0 be the connected component of identity. Then G/G0 is a group objectin the category of algebraic spaces over X and by Lemma 2.1 we may without loss ofgenerality assume that G/G0 is a group scheme over X. By [46, Lem. 03I1] we canfurther assume that each (G/G0)|Xα is finite over Xα. By the above remarks on thebase change this means that with the notation Gα =G|Xα , each Gα =Gα/G0α is a finiteétale group scheme over Xα. The assertion now follows from Lemma 2.3. 9For a group scheme G → X and a closed subscheme Y of it, the functorial cen-tralizer ZG(Y) is defined as in [13, §2.2]. It is not generally true that this functor isrepresentable by a scheme. However by what we have proved so far, we may derivethe followingCorollary 2.5. Let G be a smooth group scheme over an integral base scheme X. Thereexists a stratification of the base X = {Xα}α∈A such that each restricted group schemeG|Xα has a closed subscheme representing its centre.Before stating a proof, we recall a Galois theoretic fact about finite étale covers ofschemes:Remark 2.6. If C → X is a connected degree d étale cover and L/K is a separableextension of the residue field of the generic point of X, then CXL →XL is the union of ddegree one coverings. More generally let {Ci}i=1,...,k be all the connected componentsof C with corresponding generic points {ηi} and residue fields {κ(ηi)}. Let L be acommon separable closure of the latter. Then XL→X is a finite étale covering and CXLis a union of connected components all of which are isomorphically mapped to XL.Proof. By Corollary 2.4 we may assume that G0 is a closed and open connected sub-scheme and G =G/G0 is a finite étale group scheme over X. We will find a finite étalecover X˜ →X such that G|X˜ has a scheme theoretic centre and then use affine descent.By the above remark we may take a covering X˜→X such that G×X X˜→ X˜ is a unionof connected components all of which are isomorphically mapped to X˜. Moreover Gis a torsor for a connected group over G, and thus so is G|X˜ over G|X˜ .G0|X˜ //// G|X˜ //G|X˜//X˜G0 //// G // G // XTherefore the connected components of the source and target correspond bijectively.But every connected component of G|X˜ maps isomorphically to X˜, thus each con-nected component of G|X˜ is isomorphic to G0. Now by [13, Lem. 2.2.4], the centralizerof each connected component exists over X˜ and their intersection is the centre of G|X˜ .Finally X˜→X is étale and in particular an fpqc covering and the centre of G|X˜ is affineover X˜, so affine descent finishes the proof. 102.2 Groups of multiplicative type2.2.1 Quasi-split toriA commutative group scheme T → X is said to be of multiplicative type if it is locallydiagonalizable over X in the fppf topology (and therefore in the étale topology [4, Exp.X, Cor. 4.5]). For the general theory of group schemes of multiplicative type we referthe reader to [4, Ch. IIIV–X], however we recall a few preliminary facts here. Associatedto T there exists [4, Exp. X, Prop. 1.1] a locally constant étale abelian sheafT ,M =HomX−gp(T ,Gm), (2.1)and T is the scheme representing the sheaf HomX−gp(M,Gm). This is an anti-equivalenceof the categories of X-group schemes of multiplicative type and locally constant étalesheaves on X whose geometric fibers are finitely generated abelian groups.We will restrict ourselves to isotrivial group schemes, i.e. ones that trivialize bya finite étale cover of X (which can be assumed to be connected and Galois by [39,Proposition 6.18]). The reason this does not limit us is the following lemma:Lemma 2.7. Let T be a group of multiplicative type over an integral base scheme X.There exists an open subset U ⊆ X such that after pulling back TU along a finite étalemorphisms U˜ → U , TU˜ is isomorphic to HU˜ ×Grm,U˜ where H is a finite commutativegroup.Proof. This immediately follows from the fact that every étale morphism is Zariskilocally quasi-finite and every quasi-finite morphism is Zariski locally finite [46, Lem.03I1]. So we may associate to T a Galois cover X′ → X with group Γ such that TX′ isisomorphic to HomX′−gp(MX′ ,Gm,X′) where M is a finitely generated abelian group.The action of Γ on X′ induces an action of it on MX′ .An X-torus, T , is an X-group scheme which is fppf locally isomorphic to Grm,X .This is equivalent to asking for M to be torsion-free. In this case we say T splits overX′ if TX′ = Gm,X′ and we denote MX′ by χT . Our anti-equivalence of categories isnow between isotrivial X-tori that split over X′ and Γ -lattices (i.e. a finitely generatedtorsion free abelian groups equipped with the structure of some Γ -module of finitetype) given byT ,H0(X′,HomX′−gp(X′×X T ,Gm))and byA,HomX−gp(X′×A/Γ ,T )in the reserve direction where A is a Γ -lattice.11In view of 2.1, the Γ -module MX′ is called the character lattice of T and also de-noted by χT . Note that, if ρ is the Galois action of Γ on TX′ , the induced action on χTis via pre-composition: ργ(m)=m◦ργ .Definition 2.8. Let T be an isotrivial X-torus splitting on the Galois cover X′→X withGalois group Γ . T is called a quasi-split torus if χT is a permutation Γ -lattice (i.e. theaction of Γ on χT is by permutation of the elements of a Z-basis).2.2.2 Maximal tori of group schemesWe recall [4, Exposé IXX, Def. 1.5] that the reductive rank, of algebraic k-group G, isthe rank of a maximal torus T of Gk where k is the algebraic closure of k:ρr (G)= dimkT .Likewise, the unipotent rank of G is the dimension of the unipotent radical U of Gkand denoted byρu(G)= dimkU.In this section we show that the structure theory of commutative algebraic groupsextend to a non-empty Zariski open neighborhood of the generic point of X.For an affine smooth group scheme G over a base scheme X, the above integerscan be more generally considered as functions on X, that assign to every point x ∈X,the correspondingρr (x)= ρr (Gx), and ρu(x)= ρu(Gx).The function ρr is lower semi-continuous in the Zariski topology [4, Exposé XII,Thm. 1.7]. Moreover, the condition of ρr being a locally constant function in theZariski topology, is equivalent to existence of a global maximal X-torus for G in theétale topology by [4, Exposé XII, Thm. 1.7]. If G is commutative, then this is further-more equivalent to existence of a global maximal X-torus for G in the Zariski topology[4, Exposé XII, Cor. 1.15]. This immediately implies the followingProposition 2.9. Let G be an affine smooth group scheme over a noetherian basescheme X. Then there exists a stratification of X by finitely many locally closed Zariskisubschemes {Xi} such that each group scheme G|Xi admits an isotrivial maximal torus.If G is moreover commutative, G|Xi admits a maximal torus in Zariski topology.Proof. Since ρr is lower semi-continuous and integer valued there exists a stratifica-tion by locally closed subspaces on which ρr is constant. We can further refine sucha stratification by Lemma 2.7 for the global tori to be trivial after finite étale basechange. 122.3 Spreading out argumentsWe say a group scheme Hη → X (or a property of it with respect to another groupscheme Gη) spreads out to a neighborhood of the generic point η∈X if there exists adense open subset U ⊂ X over which there is a U -group scheme H|U pulling back tothe prior one (and satisfying the same property with respect to a spreading out G|U ofGη).Lemma 2.10. Let X be an integral scheme and G a finitely presented affine groupscheme over X. Then closed subgroups of the generic fiber spread out: i.e. let η bethe generic point of X and Hη a closed subgroup of Gη. Then there exists a non-emptyopen set U ⊆X such that G|U contains a subgroup schemeHU fitting in the commutativediagramHη// HUGη // GUwhere the horizontal arrows are pull-back morphisms and the vertical arrows aremonomorphisms of group schemes.Proof. It suffices to consider the case where X is an affine scheme X = SpecR. Let Kbe the function field of R with the canonical homomorphism R→ K corresponding tothe inclusion of the generic point η→ X. Then G = SpecS, where S = R[x1, . . . ,xk]/Ifor some finitely generated ideal I = 〈p1, . . . ,p`〉 ⊂ R[x1, . . . ,xk] and therefore Gη =SpecK⊗R S. With this notation, Hη is cut out as a subscheme by a finitely generatedideal p ⊆ K⊗R S = K[x1, . . . ,xk]/IK. Thus each generator of p can be considered as apolynomial with coefficients in K. Since K is the inverse limit of localizations of R inits elements, there exists f ∈R such that all elements of p are defined with coefficientsin Rf . This defines a subscheme HU of GU satisfying the commutativity of the abovediagram, if we set U = SpecRf .Now we put a group scheme structure on HU by shrinking U further. Let i :G→Gandm :G×XG→G, respectively be the inversion and multiplication morphisms on G.Considering the inversion morphism, existence of group structure on Hη means thatin level of coordinate rings, we are given a commutative diagramRf [x1, . . . ,xk]/IRfi# //Rf [x1, . . . ,xk]/IRfK[x1, . . . ,xk]/IKi˜# // K[x1, . . . ,xk]/IKq // K[x1, . . . ,xk]/pIKand the composition of the induced morphism i˜# and the quotient map q has precisely13pIK as its kernel. Hence by a similar argument, there exists some g ∈ Rf lifting thiscomposition as in the cartesian diagramRfg[x1, . . . ,xk]/pIRfg //Rfg[x1, . . . ,xk]/pIRfgK[x1, . . . ,xk]/pIK // K[x1, . . . ,xk]/pIK.The case of multiplication morphism is similar. So by shrinking further we may as-sume that HU is a U -group scheme. Commutativity of the diagramHU ×HU // _HU _GU ×GU // GUwhere the horizontal arrows are morphisms (x,y), xy−1 is now obvious. Thus HUis the desired subgroup scheme of GU . Lemma 2.11. Group homomorphisms (repectively isomorphisms) spread out. Let G→Xand η∈X be as in previous lemma. If G′→X is another group scheme andϕη :Gη→G′ηis a group scheme homomorphism (resp. isomorphism), then there exists a non-emptyU ⊆X and a homomorphism (resp. isomorphism) ϕ :G′U →GU such that ϕ|η =ϕη.Proof. The proof is by similar arguments as in previous lemma. 2.4 Tori and unipotent group schemesLemma 2.12. The property of being a quasi-split torus spreads out. Let G → X andη ∈ X be as in the previous lemmas. If Gη is a quasi-split torus, then there exists anon-empty U ⊆X such that GU is a quasi-split torus.Proof. Since Gη is isotrivial, by Lemma 2.7 we may assume that G is also isotrivial.We recall that the character lattice of a torus is expressed in terms of the (étale lo-cally constant) sheaf χ(G) = HomU−gp(G,Gm,U ). Restriction from U to η induces ahomomorphism of finitely generated Z-modulesHomU−gp(G,Gm,U )→HomK−gp(Gη,Gm,η)and by Lemma 2.11 we may assume that this is an isomorphism of Z-modules. We alsonote that if U˜ →U is a finite étale covering that trivializes GU and restricts to the finite14separable extension L/κ(η), then Γ = Gal(L/K) is at the same time the fundamentalgroup of this covering and the associated action of Γ on χ(Gη) induces same action ofthis group on χ(G). Now we analyze how unipotency behaves with respect to stratifications. We firstclarify what we mean by a unipotent group scheme over a scheme X and then showthat unipotency spreads out from the generic fiber to a Zariski open neighborhood.Definition 2.13. An affine group scheme Z→X is said to be a unipotent group schemeif it is unipotent over each geometric fiber.Lemma 2.14. Let G be unipotent group scheme over an integral base scheme X. Thenthere exists a non-empty Zariski open set U ⊂ X such that GU has a filtration in sub-groups 1⊂G1⊂ . . .⊂Gr−1⊂Gr =G with all factors G`/G`−1 isomorphic to the constantU -group scheme Ga,U .Proof. Let Hη be a subgroup of the generic fiber Gη. By Lemma 2.10 the propertyof being a subgroup spreads out to a non-empty open in X. We also observe thatthe property of being isomorphic to Ga spreads out. That is, if Hη is isomorphic toGa,η, then there exists a non-empty open U ⊆X such that Hη spreads out over it to theconstant group scheme Ga,U . This is another straightforward spreading out argument:let K be the field of fraction of an integral domain R. Let S be a finitely presented R-algebra generated by x1, . . . ,x` and that there exists an R-algebra isomorphism ϕ :K⊗R S → K[t]. It is easy to check that there exists a localization Rf of R that extendsϕ to an isomorphism ϕ˜ : Rf ⊗R S → Rf [t]. The claim now follows by induction on thequotient scheme GU/Ga,U .1 Corollary 2.15. Let X be an integral scheme and G a finitely presented smooth affinecommutative group scheme over X. Let η be the generic point of X. Then the decom-position of Gη as Tη×Uη to a maximal torus and the unipotent radical spreads out;i.e. there exists a non-empty Zariski open V ⊆ X such that GV is isomorphic to TV ×UVwhere TV is a maximal torus with Tη as generic fiber and UV is a unipotent V -groupscheme with Uη as generic fiber. Moreover, if Tη is quasi-split we may assume TV is alsoquasi-split.Proof. This is now obvious by spreading the maximal torus of the generic fiber out byProposition 2.9, and spreading the unipotent radical out by Lemma 2.14, and observingthat the group structure of Tη×Uη also spreads out by Lemma 2.11. 1A relevant note is that A1-fibrations are always Zariski locally trivial (cf. [30]).15Chapter 3Inertia operator of K-groups3.1 K-groupsRecall Sch/B, the big étale site of schemes of finite type over a fixed noetherian basescheme B. The Grothendieck ring of Sch/B, denoted by K(Sch/B) is the free abeliangroup of isomorphism classes of such schemes, modulo the scissor relations,[X]= [Z]+ [X Z], for Z ⊂X a closed subscheme,equipped with structure of a commutative unital ring according to the fiber productin Sch/B,[X] · [Y]= [X×Y].We will always tensor this ring with Q. Hence throughout, any group, ring or algebradenoted as K is assumed to be a Q-vector space.The Grothendieck ring of the category St/B of algebraic stacks over Sch/B, is de-fined similar to above. As an abelian group, K(St/B) is generated by isomorphismclasses of algebraic stacks modulo similar relations; i.e. for any closed immersionZ↩ X of algebraic stacks we have [X] = [Z]+ [X Z]. And the fiber product over thebase category turns K(St/B) into a commutative ring. Hence for any algebraic stack Y,isomorphic to a fiber product X×Z, we have [Y]= [X][Z].Moreover, K(St/B) is a unital associative K(Sch/B)-algebra in the obvious way. Wewill use the terms Grothendieck ring, Grothendieck module and Grothendieck algebrato stress the structure of our subject matter.163.2 Inertia stacksFor any algebraic stack, X, the inertia stack IX is the fiber productIX// X∆X ∆ // X×Xwhere ∆ is the diagonal morphism. IX is isomorphic to the stack of objects (x,f ),where x is an object of X and f : x → x is an automorphism of it. Here a morphismh : (x,f )→ (y,g) is an arrow h : x→y in X satisfying g ◦h= h◦f .This construction respects the equivalence and scissor relations and is K(Sch/B)-linear, hence inducing a well-defined inertia operator on the K(Sch/B)-algebra K(St/B),and in particular an inertia endomorphism on the K(Sch/B)-module K(St/B).We use the notation I(k)X for k-times application of the inertia construction on thestack X. We may think of the objects of I(k)X as tuples (x,f1,··· ,fk) of an object x inX and pairwise commuting automorphisms f1,··· ,fk. A morphism (x,f1,··· ,fk)→(y,g1,··· ,gk) is an arrow h : x→y of X satisfying h◦fi = gi ◦h for all i= 1,··· ,k.3.3 Central band of a gerbeWe recall that to any algebraic stack X we can associate an fppf coarse moduli sheafX of isomorphism classes of objects of X [34, Rmk. 3.19]. The morphism of stacksX→X is always an fppf (in particular, étale) gerbe.Note that X is not generally represented by a scheme. For example, working overC, consider the quotient stack D= [A1/Gm], of the affine line A1 by the natural actionof the multiplicative group Gm on it. Over any scheme S, the objects of this stack arepairs (L→ S,s) of a line bundle L on S with a section s : S→L. This quotient stack playsa key role in logarithmic geometry and the coarse moduli sheaf of it is the classifyingspace of generalized cartier divisors (cf. [1] and [46, Tag 02T7]). We now show that Ddoes not admit a coarse moduli scheme. Suppose to the contrary that D is a coarsemoduli scheme with the universal map m :D→D. The quotient morphism p :A1→Dmaps every morphism f : S→A1, to the pair (S×A1, id×f). Restricting p to Gm ⊂A1,we the get a morphism p :Gm → [Gm/Gm] where the latter quotient stack is a point.We conclude that m◦p : A1 →D is a constant map from the affine line to a point. Byuniversality of m, this map has to be surjective therefore D = SpecC. However, for Dto be a moduli space, the isomorphism classes of C-points of D and D need to be inone-to-one bijection but D(C) consists of two isomorphic classes of objects and thisis a contradiction.Other examples of quotient stacks which do not admit coarse moduli spaces are17studied for instance in [2], [3] and [44]. We also refer the reader to [31] for a proof ofexistence of coarse moduli spaces for separated Deligne-Mumford stacks.Proposition 3.1. Let X → X be an étale gerbe. Then there exists a sheaf of abeliangroups Z →X and a morphism of sheaves of groups ϕ : Z×X X→ IX such that1. For every s : S → X, the induced morphism of sheaves of groups s∗ϕ : Z|S →Aut(s) identifies Z|S with the centre of the sheaf of groups Aut(s); and,2. The pair (Z,ϕ) is unique, up to isomorphism of sheaves of groups over X.Proof. This is explained in [18, Ch. IV, §1.5], specifically refer to [18, Ch. IV, §]for existence of the sheaf and to [18, Ch. IV, Cor. 1.5.5] for the properties of it. Definition 3.2. In the above setting, Z is called the central band associated to X andif it is a scheme we call it the central group scheme. The central inertia of X is definedto be the fiber productIzX//XZ // X.Lemma 3.3. The map ϕ : IzX→ IX as in Proposition 3.1 is representable and identifiesIzX with a closed (resp. open and closed) substack of IX, if the following condition issatisfied: For any object s ∈ X, Aut(s) is a group scheme and its functorial centralizeras defined in [13, §2.2] is representable by a closed (resp. open and closed) subscheme.Proof. For any scheme U , any U -point u∈ IX(U) is determined by a pair (u,σ) withu ∈ X(U) and σ ∈ Aut(u). Thus the S-points (for any scheme S) of the fiber productU ×IX IzX is given by objects〈f ∈U(S),s ∈X(S),τ ∈ Z(Aut(s)),ι : f∗u ›----→ s〉 such that ι◦f∗σ = τ ◦ ι.This stack is then 1-isomorphic to the stack of objectsf ∈U(S), such that f∗σ ∈ Z(Aut(f∗u)).The group sheaf Aut(u) is represented by an affine U -group scheme G and σ =σ(U) :U →G is a section of the structure morphism. Thus the stack of the objects above, isrepresented by the fiber product Z(G) j×G,σ U where j is the closed immersion of thecentre, Z(G)↩G. 18Discrete central inertiaSuppose we are in the case that Z → X is a group scheme and consider the opensubgroup scheme Z0 and the quotient algebraic space Z/Z0 over X. Pulling back to X,we define the connected component of the central inertia, Iz,0X as the sub-group space,and the discrete central inertia as the quotient group space, which are respectivelygiven by the following fiber productsIz,0X//XZ0 // XandIz/z,0X//XZ/Z0 // X.Here Z/Z0→X is called the discrete central band of X→X.3.4 The semisimple and unipotent inertiaRecall that if G is an affine group scheme of finite type on base scheme S, an elementg ∈G(S) is defined to be semisimple if for all scheme points s ∈ S given as spectrumof a field, gs is semisimple in fiber Gs .Definition 3.4. We define the semisimple inertia of an algebraic stack X to be thestrictly full subcategory IssX of the inertia stack IX consisting over a base S of thoseobjects (x,ϕ) such that ϕ ∈ Aut(x) is a semisimple element of the S-group schemeAut(x).According to [34, 3.5.1] in order to check that IssX is a substack of IX we only needto observe that if f : U → S is an étale surjection and ϕ ∈ Aut(x) is an automorphismof an object x over S where f∗ϕ∈Aut(f∗x) is semisimple then f is also semisimple.But this follows from the above definition and the fact that being semisimple is pre-served along field extensions. That is, given a group scheme G→ S, if g is a k-valuedpoint of Gk, and K/k is an algebraically closed extension, and g′ the K-valued pointof GK obtained by pullback, then g is semisimple if and only if g′ is [4, Exposé XII].Let u :U→X be an étale covering of X by an algebraic space. We note that Autss(u)may fail to be an algebraic space, however it is a locally constructible space by [4,Exposé XII, Proposition 8.1]. On the other hand, the diagonal of IssX is easily seento be representable, separated and quasi-compact. Therefore IssX can be written as awell-defined element in K(St) even though it is not necessarily an algebraic stack.19We use the notation Iss,zX for the locus IssX∩ IzX. We may also write this is afiber productIss,zX //XZss // Xwhich is a relative group scheme, since the set of semisimple elements Zss of thecentral band Z →X form a group scheme of multiplicative type. Equivalently,Iss,zX //XZ/U // Xwhere U is the unipotent radical of Z .Likewise the unipotent inertia of X is the strictly full subcategory IuX consistingof those objects (x,ϕ) such that ϕ is a unipotent element of Aut(x). It is easy tocheck that IuX⊂ IX is a closed substack of the inertia.20Part IIDiagonalization of the inertia21Chapter 4GroupoidsLet K(gpd) be the Q-vector space generated by finite groupoids, modulo equivalenceand scissor relations. It is easy to verify that the vector space K(gpd) is generated by[BG], for finite groups G.4.1 Filtration by central numberThe vector space K(gpd) has two natural gradings, which will be important for us.First, there is the grading by size of the automorphism group, denoted by upper in-dices, and then there is the grading by size of the centre of the automorphism group,denoted by lower indices. Thus [BG] is in Kni (gpd), if #G=n, and #Z(G)= i. We haveK(gpd)=∞⊕n=1∞⊕i=1Kni (gpd).Clearly, Kn(gpd) is finite-dimensional, for every n, but Ki(gpd) is infinite-dimensional,for all i. This grading defines an ascending filtration K≤n(gpd) and a descendingfiltration K≥i(gpd).Denote by I : K(gpd) → K(gpd) the endomorphism sending [X] to [IX], whereIX is the inertia groupoid of X. Note that inertia is compatible with equivalence andscissor relations, so that I is well-defined.Lemma 4.1. The endomorphism I preserves the associated filtrations K≤n(gpd) andK≥i(gpd). Moreover, on the associated graded, K≥i/K>i(gpd), the endomorphism I ismultiplication by i.Proof. Recall thatI(BG)›⊔g∈C(G)BZG(g),22where C(G) denotes the set of conjugacy classes of G, and ZG(g) is the centralizer ofg in G. Thus,I[BG]=∑g∈C(G)[BZG(g)]= #Z(G)[BG]+∑g∈C(G)∗[BZG(g)],where C(G)∗ denotes the set of non-central conjugacy classes. Now we note that fornon-central g we have strict inequalities#Z(G) < #ZG(g) < #G.This is enought to prove the claim. 4.2 Local finiteness and diagonalizationProposition 4.2. The endomorphism I : K(gpd)→ K(gpd) is diagonalizable, with spec-trum of eigenvalues equal to the positive integers.Proof. Every subspace K≤n(gpd) is finite dimensional, and preserved by I. On thisfinite dimensional subspace, I is triangular, with respect to the lower grading, andwith different eigenvalues on the diagonal. This proves that I is diagonalizable whenrestricted to K≤n(gpd) for all n. Thus K(gpd) has another natural grading, namely the grading induced by thedirect sum decomposition into eigenspaces under I, also called the grading by virtualsize of centre. Denote the corresponding projection operators by pin.Example 1. If A is a finite abelian group, then [BA] is an eigenvector for I, with eigen-value #A. Thus pin[BA]= [BA], if A had n elements, and pin[BA]= 0, otherwise.Example 2. We haveI[BS3]= [BS3]+ [BZ3]+ [BZ2],where we have commited the abuse of notation of writing Zn for Z/nZ. From this, andthe previous example, we can see that[BS3]− [BZ2]− 12 [BZ3]23is an eigenvector for I, with eigenvalue 1. Thuspin[BS3]=[BS3]− [BZ2]− 12 [BZ3] if n= 1[BZ2] if n= 212 [BZ3] if n= 30 otherwise .Example 3. For the dihedral group D4 with eight elements, we haveI[BD4]= 2[BD4]+ [BZ4]+2[BD2].Hence[BD4]− 12 [BZ4]− [BD2]is an eigenvalue of I with eigenvalue 2. It follows thatpin[BD4]=[BD4]− 12 [BZ4]− [BD2] if n= 212 [BZ4]+ [BD2] if n= 40 otherwise .4.3 The operators Ir and eigenprojectionsLet IrBG be the stack of tuples (s1, . . . ,sr ) where si are r distinct pairwise commutingelements of G:Ir (BG)= [(G×r )∗/G],where the brackets are used as the notation for quotient algebroids. In K(gpd) wewriteIr [BG]= [(G×r )∗/G],where bracket stands for the element in the K-group and the quotient notation isomitted.This defines another family of operators on K(gpd). For r = 0, I0 is identity on allBG and I1 is the usual inertia operator.Theorem 4.3. The operators Ir , for all r ≥ 0, preserve the filtration K≥k(gpd). On thequotient K≥k(gpd)/K>k(gpd), the operator Ir acts as multiplication by r !(kr).Proof. Let n be the size of the group G and k the size of its centre. Notice that thereare r !(kr)ways of choosing the r sections so that they are all in the centre. Thus, we24conclude,Ir [BG]= r !(kr)[BG]+∑S∈(G×r )∗S 6⊆Z(G)[BZG(S)].Corollary 4.4. The operators Ir , for r ≥ 0 are simultaneously diagonalizable. The com-mon eigenspaces form a family Πk(gpd) of subspaces of K(gpd) indexed by positiveintegers k≥ 0, andK(gpd)=⊕k≥0Πk(gpd).Let pik denote the projection onto Πk(gpd). We haveIrpik = r !(kr)pik ,for all r ,k≥ 0.Corollary 4.5. For r ≥ 0, we havekerIr =⊕k<rΠk(gpd).Corollary 4.6. For every k≥ 0, we havepik =∞∑r=k(−1)r+kr !(rk)Ir .In particular, pi0 =∑∞r=0(−1)rr ! Ir , and pi1 =∑∞r=1(−1)r−1(r−1)! Ir .Proof. We now use the “beautiful identity” [43]∑r(−1)r+k(`r)(rk)= δ`kto find the projections. We haveid=∑`≥0pi` ,and henceIr =∑`≥0Irpi` =∑`≥0r !(`r)pi` ,25and therefore∑r≥0(−1)r+kr !(rk)Ir =∑r≥0(−1)r+kr !(rk)∑`≥0r !(`r)pi`=∑`≥0∑r≥0(−1)r+k(rk)(`r)pi` = ∑`≥0δ`,kpi` =pik .Example 4. We previously showed that the projections for the inertia operator are asfollows:pin[BS3]=[BS3]− [BZ2]− 12 [BZ3] if n= 1[BZ2] if n= 212 [BZ3] if n= 30 otherwise .Notice that we haveIr [BS3]=[BS3]+ [BZ2]+ [BZ3] if r = 12[BZ2]+3[BZ3] if r = 23[BZ3] if r = 30 otherwise .This gives us a way of computingpin[BS3]=I1− I2+ 12 I3 = [BS3]− [BZ2]− 12 [BZ3] if n= 112 I2− 12 I3 = [BZ2] if n= 216 I3 = 12 [BZ3] if n= 30 otherwise .26Chapter 5Inertia endomorphism onDeligne-Mumford stacksFor the case of Deligne-Mumford stacks we fix a noetherian scheme B and work withcategories of objects defined over B. We let DM/B be the full subcategory of St/B ofall Deligne-Mumford stacks over B. We shortly write DM for this category and shortlywrite Sch for the category, Sch/B, of B-schemes of finite type. The inertia operator ofSt/B, induces an operator on DM. The Grothendieck ring of it, K(DM), also inherits thestructure of a K(Sch)-algebra. Therefore we have an induced inertia endomorphism onK(DM). In this chapter we prove our main results (local finiteness and diagonalization)for the K(Sch)-module K(DM).5.1 Stratification of Deligne-Mumford stacksDefinition 5.1. An irreducible gerbe X is a connected Deligne-Mumford stack, withfinite étale inertia, IX→X.Proposition 5.2. Every noetherian Deligne-Mumford stack can be stratified into finitelymany locally closed irreducible gerbes.Proof. Using [41, Prop. 5.7.6] we may assume that X is an integral Deligne-Mumfordstack. Flatness is an fppf-local property hence by generic flatness [21, Thm. 6.9.1]there exists an open substack of X such that IX→X is flat. Since IX→X is unramified,it is étale as well. A quasi-finite morphism of schemes is generically finite [46, Lem.03I1]. Therefore by fpqc-descent of finiteness on base [46, Lem. 02LA], IX→X is finiteon an open substack of X. This means that we can stratify X into finitely many locallyclosed substacks {Xi}i∈A such that each pullback morphism, IXi→Xi is finite étale. 27Lemma 5.3. If X is an irreducible gerbe, then IzX→X is finite étale.Proof. It suffices to show that the inclusion IzX→ IX is a closed immersion. Let U →Xbe an étale cover of X by a scheme U such that XU is the neutral gerbe BUG for anétale U -group scheme G. In fact according to Remark 2.6, we may assume that G splitsto finite union of copies of connected component of unity and hence has the structureof a constant U -group scheme: G = U ×H where H is a finite group. The centre of G˜jis obviously closed and open, and so is IzX→ IX by descent. Remark 5.4. If X is an irreducible gerbe, then each connected component Y of IX is anirreducible gerbe. This is clear since Y→X is finite étale and therefore so is IY→ IX bydefinition of the inertia stack. In other words, an inertia stack of an irreducible gerbehas a canonical stratification into irreducible gerbes by its connected components.5.2 Filtration by split central numberRecall [21, Cor. 17.9.3] that ifϕ :Y →X is a separated étale morphism over a connectedbase scheme X, there is a one-to-one correspondence between the sections of ϕ andthe number of connected components of Y isomorphic to X. Thus for a finite étalecovering, the number of such sections is an indication of how close Y is to being atrivial degree n covering,⊔nX →X.Definition 5.5. For an irreducible gerbe X we define the split central number, ν(X), tobe the number of sections of IzX→X.We define an ascending filtration of K(DM) by declaring [X], for an irreduciblegerbe X, to belong to K≥n if ν(X)≥n. Thus a linear combination of irreducible gerbesis in K≥n if the split central number is at least n for all terms.Proposition 5.6. The inertia endomorphism on K(DM) preserves the filtration K≥•.Furthermore, on the associated graded piece K≥n/K>n, the inertia endomorphism in-duces multiplication by the integer n.Proof. Consider an irreducible gerbe X, with split central number n and {Yα}α∈A bethe stratification of IX by connected components (hence irreducible gerbes). There areprecisely n of the Yα which are contained in IzX and map isomorphically to X (i.e. aredegree one connected étale covers of X). It suffices to show that any other strata Yhas split central number strictly larger than n.28There exists a diagramIz(IX)pi3&&IX×X IzX//pi2? _joo IzXpi1IX // X(5.1)where the square is cartesian. For any object x of X, elements of IX over x are pairs(x,ϕ) such that ϕ ∈ Aut(x) and objects of IzX over x are pairs (x,ψ) where ψ ∈Z(Aut(x)). The fibered product IX×X IzX is hence the stack of triples (x,ϕ,ψ) withx,ϕ and ψ as above. On the other hand, Iz(IX) is the stack of the objects (x,ϕ,ψ)such that ϕ ∈Aut(x),ψ∈ Z(ZAut(x)(ϕ)). Hence there is an embedding of the fiberedproduct into Iz(IX). Restricting to a substack Y⊂ IX we get the following diagram.IzYpi3$$Y×X IzX//pi2? _joo IzXpi1Y // X(5.2)In diagram 5.2 the embedding j is necessarily a union of connected components,because all vertical and diagonal maps in the diagram are representable finite étalecovering maps. Note also that there is a canonical section, δ, to pi3 : I(Y) → Y viathe diagonal Y→ Y×XY since any automorphism of an object x in X is in its owncentralizer. It is obvious that any section of pi1 pullback to a (distinct) section of pi2and gives a (distinct) section of pi3. This shows that inertia endomorphism preservesthe filtration K≥•.For the action of inertia on the graded piece K≥n/K>n we show that if Y is acomponent of IX which is not a section of pi1, then the associated section δ is notinduced by pulling back sections of pi1. In fact, if Y is not contained in IzX then δdoes not lift to pi2 and we are done. Otherwise, (when Y is completely contained inIzX), δ lifts to a section of pi2 but the image of this section in IzX is Y itself, which isnot a degree one cover of X. 5.3 Local finiteness and diagonalizationBefore we can deduce that I : K(DM) → K(DM) is diagonalizable, we need to provethat for every irreducible gerbe X the class [X] is contained in a finite dimensionalsubspace of K(DM), which is preserved by the inertia endomorphism. In this sectionwe use the notation∏kXY to denote the k-fold fiber product of a stack Y by itself overX.29Lemma 5.7. Let Y→ X be a finite étale representable morphism of algebraic stacks.Then the following family is a finite set up to isomorphism of stacks.C(Y→X)= {W :W is a connected component ofk∏XY for some k≥ 0}Proof. This is trivial since the Galois closure of Y with respect to X is a finite étale X-stack Y→ X. And every element in the above family is isomorphic to an intermediatecover, in between Y and Y. Corollary 5.8. Let Y1,··· ,Ys be finitely many algebraic stacks, finite étale over X. Thethe following family is finite up to isomorphism{W :W is a connected component ofk1∏XY1×X ···×Xks∏XYs for some k1,··· ,ks ≥ 0}Proof. There are s projection mapsp` :k1∏XY1×X ···×Xks∏XYs →k∏`XY`, ` = 1,··· ,swhich are all finite étale and in particular closed and open. The immersioni :k1∏XY1×X ···×Xks∏XYs →k1∏XY1×···×ks∏XYsis similarly closed and open. Hence W is isomorphic to its image i(W) which is aconnected component of pi1(W)×···×pis(W). However any fiber product W1×···×Ws where Wi ∈ C(Yi → X) has finitely many connected components and by Lemma5.7 there are only finitely many such fiber products. Corollary 5.9. Let X be an irreducible gerbe. Then the following family is finite up toisomophism.{W :W is a connected component of I(m)X for some m≥ 0}Proof. For an irreducible gerbe IX→X is finite étale, hence closed and open and there-fore the inertia stratifies to finitely many connected components Y1,··· ,Ys which are30finite étale over X. In the commutative diagramI(m)X  j //∏mX IXIXthe downward arrows are finite étale and hence so is the inclusion j. Consequently jis open and closed, and therefore any connected component of I(m)X is a stratum ofsome substackYi1 ×X ···×XYim ⊂m∏XIXfor a choice of i1,··· , im ∈ {1,··· ,s}. The claim now follows from Corollary 5.8. This completes the proof of our main results for Deligne-Mumford stacks:Theorem 5.10 (Local finiteness). Let X be a noetherian Deligne-Mumford B-stack and{Xi}i∈A, the stratification of it by irreducible gerbes. Then the K(Sch)-submodule ofK(DM) generated by the set of motivic classes of all Xi and all intermediate Galoiscovers between IXi → IXi is finitely generated, invariant under inertia endomorphism,and contains [X].Corollary 5.11 (Diagonalization). The endomorphism I : K(DM)→ K(DM) is diagonal-izable, with eigenvalue spectrum equal to N, the set of positive integers.5.4 The operators Ir and eigenprojectionsLet IrX be the stack of tuples (x,s1, . . . ,sr ) where si are distinct pairwise commutingautomorphisms of x. By this we mean that of x : X → X is an X-point of X, andG = Aut(x) is the X-group scheme of automorphisms of x, then si are sections ofG → X and not any two of them are identical sections. This definition applies alsoto r = 0. The stack I0X is just X. For r = 1, I1X is the usual inertia. Hence I1 isdiagonalizable with integer eigenvalues.Note that Ir is closely related to the k-fold inertia operators I(k) of §3.2. In fact itis easy to see that by an inclusion-exclusion argument that they satisfy the followingidentity,Ir =r∑k=1s(r ,k)I(k) ,where s(r ,k) are the signed Stirling number of the first kind.31We use the notation ZIrX for the substack of IrX consisting of objects (x,s1, . . . ,sr )such that all si are in the centre of Aut(x). The complement locus will be denoted byNZIrX.Let X be an irreducible gerbe with IX→ X an étale morphism of degree n. Thenthere exists a Galois covering X˜→X of X such that IzX|X˜ is a disjoint union of n copiesof X˜. So we have[ZIrX|X˜]= r !(nr)[X˜].We use the notation Inj(r ,n) for the set of injections from a set of cardinality rto a set of of cardinality n. So#Inj(r ,n)= r !(nr).Theorem 5.12. The operators Ir , for all r ≥ 0, preserve the filtration K≥k(DM) by splitcentral number. On the quotient K≥k(DM)/K>k(DM), the operator Ir acts as multipli-cation by r !(kr).Proof. Let X be an irreducible gerbe with split central number k and IzX→ X be ofdegree n. Let X˜→X be a splitting cover for IzX→X with Galois group Γ . Hence Γ actson n.IzX|X˜ι //IzXX˜ // XThen IzX|X˜ is the disjoint union of n copies of X˜ and Γ acts on it by permuting thesecopies. Let us rename the i-th copy to Y˜ and the image to Y. The integer i∈n is fixedunder the action of Γ precisely when Y › Y˜/Γ via the horizontal morphism. SinceX˜/Γ › X, the above happens precisely when Y is isomorphic to a copy of X by thevertical morphism. By Proposition 5.6 this only is the case if Y is one of the k copiesof X contributing to the split central number of X. Hence the set of fixed points of Γis of size k. Also,X˜× Inj(r ,n) '-→ ZIrX|X˜ ,and the action of Γ on n induces an action of it on Inj(r ,n) . A morphism ϕ : r ↩ nis invariant under this action if every element in the image of ϕ is so. Therefore the32number of fixed points of Inj(r ,n) is r !(kr). We may hence calculate as follows:ZIr [X]= [X˜×Γ Inj(r ,n)]=∑ϕ∈Inj(r ,n)/Γ[X˜/StabΓϕ]=∑ϕ∈Inj(r ,n)Γ[X]+∑ϕ∈Inj(r ,n)/ΓStabΓ ϕ 6=Γ[X˜/StabΓϕ]Thus, we conclude,ZIr [X]= r !(kr)[X]+∑ϕ∈Inj(r ,n)/ΓStabΓ ϕ 6=Γ[X˜/StabΓϕ].Finally note that each intermediate cover Y= X˜/StabΓϕ has a strictly larger split cen-tral number k. In fact, IzY = IzX|Y so every section of IzX→ X pulls back to a sec-tion of IzY→Y but also IzY→Y has sections induced by ϕ that do not descend toIzX→X.Finally for every irreducible gerbe Y ⊆ NZIrX, the split central rank is strictlylarger than n, because at least one of the sections si is noncentral. Corollary 5.13. The operators Ir , for r ≥ 0 are simultaneously diagonalizable. Thecommon eigenspaces form a family Πk(DM) of subspaces of K(DM) indexed by non-negative integers k≥ 0, andK(DM)=⊕k≥0Πk(DM).Let pik denote the projection onto Kk(DM). We haveIrpik = r !(kr)pik ,for all r ≥ 0, k≥ 0.Corollary 5.14. For r ≥ 1, we havekerIr =⊕k<rΠk(DM).Corollary 5.15. For every k≥ 0, we havepik =∞∑r=k(−1)r+kr !(rk)Ir .33In particular, pi0 =∑∞r=0(−1)rr ! Ir , and pi1 =∑∞r=1(−1)r−1(r−1)! Ir .The proof is similar to that of Corollary 4.6.34Chapter 6Inertia endomorphism of algebraicstacks6.1 Stratification of stacks in characteristic zeroFrom now on we need to work over a field k of characteristic zero and every stack isof finite-type. The categories of k-schemes, Sch/k would be shortened to Sch, and thatof k-algebraic stacks St/k would be written as St. We will show that by stratificationthe Grothendieck group of St is generated by nicely behaving algebraic stacks whichwill be named clear gerbes.Definition 6.1. Let X be a noetherian algebraic stack over the associated coarse modulispace X. We say X is a clear gerbe over the algebraic space X, if:(C1) X→X is an étale gerbe with faithfully flat structure morphism of finite type;(C2) the projection IX→X is a representable smooth morphism of finite type;(C3) X is a k-variety (i.e. a reduced, separated, k-scheme of finite type);(C4) the central inertia is a closed substack of the inertia stack;(C5) its central band is a smooth commutative X-group scheme;(C6) the discrete central band is an étale finite X-group scheme;(C7) the central band admits a maximal torus.Remark 6.2. Note that the above properties assure that IX is a noetherian group ob-ject in the category of algebraic spaces over X [46, Lem. 01T6]. The condition of affinediagonal implies further that IX is an affine group space over X.35We start with a modification of the stratification in [34, Thm. 11.5].Lemma 6.3. Let X be a noetherian algebraic stack of finite type. There exists a finitefamily {Xα}α∈A of locally closed substacks of X such that X is the disjoint union of Xα’sand for each index α∈A, the stack Xα satisfies (C1), (C2), (C3) over an associated coarsemoduli space Xα.Proof. By replacing generic smoothness instead of generic flatness in the proof of [34,Thm. 11.5] we may assume (C1) and (C2) are already satisfied and the coarse modulisheaf X, is a noetherian algebraic space of finite type and in particular quasi-compactand quasi-separated. Now we stratify X into k-varieties by [14, Thm. 3.1.1]. Theorem 6.4. Every noetherian algebraic stack of finite type with affine diagonal hasa stratification into a disjoint union of locally closed clear gerbes.Proof. By previous lemma we already assume that X→X satisfies (C1), (C2) and (C3).Now we stratify it such that for each stratum the central inertia descends to a smoothgroup scheme over the base coarse moduli space.First we stratify X→ X on the target so that the central band (which is a sheaf ofgroups) associated to each stratum is an algebraic space and its central inertia is aclosed substack of inertia stack.Let Z be the central band of X→X. It is easy to check that the diagonal morphismZ → Z×X Z is representable. So let U →X be an étale covering of X which trivializes Xto the trivial G-gerbe for some U -group scheme G. By generic smoothness we stratifyX to a finite family X = {Xi}i∈I such that each Gi =G|Ui is smooth over Xi. By descentof smoothness [21, Cor. 17.7.3], we conclude that Gi is a smooth Ui-group scheme. ByCorollary 2.5 we may further stratify each Xi by {Xij}β∈J such that each G|Xij admitsits centre Zij as a scheme. Then Zij is an étale cover of the restricted sheaf of groupsZ|Xij . The latter is hence an algebraic space.Notice also that the morphisms IzXij → IXij are now closed immersions accordingto Lemma 3.3. Moreover Z|Xij is noetherian, and smooth over Xij of finite type. Wenow use Lemma 2.1 to stratify each Xij further via {Xijk}k∈K such that each pullbackZ|ijk is a group scheme over Xijk.Finally, if X→X has smooth central group scheme Z→X, the quotient space Z/Z0is a finitely presented, étale algebraic space over X. By Lemma 2.1 we may assume thatZ/Z0 is a scheme. Then we stratify further using generic finiteness [46, Lem. 03I1].Recall that a smooth commutative algebraic group over a perfect field has a de-composition T ×U where U is the unipotent radical of the algebraic group and T is agroup of multiplicative type. If the group is connected then so is T , in which case T isa torus [37, XIV, Theorem 2.6].36Let η be a generic point of X. Then the mentioned decomposition holds for thegeneric fiber Zη of central band over X. We now use the machinery of §2.2 on how thestructure of a commutative algebraic groups spread out by Corollary 2.15. Definition 6.5. For a clear gerbe X→ X, the split central number of X is defined to bethe number of sections of Iz/z,0X→ X and is denoted by ν(X). The discrete degreeof the centre of X, is the degree of Iz/z,0X→ X as a finite étale map and is denotedby degz(X). The non-negative integer τ(X) = degz(X)− ν(X), is called the centraltwistedness of X.Definition 6.6. Let X → X be a clear gerbe. The relative dimensions dimX(IX) anddimX(IzX) are well-defined. The latter is called the central rank of X, and is denotedby ρ(X). This is also the rank of the associated commutative group scheme Z0 → X.The non-negative integer corank(X) = dimX(IX)−dimX(IzX) is called the co-centralrank of X. Let c(X) ≥ 1 will be the maximum number of geometrically connectedcomponents of the fibers of IX→ X. This number is well-defined and finite accordingto [46, Lem. 055Q].Definition 6.7. For a clear gerbe X→ X the unipotent and reductive ranks of the cen-tral band are constant over the coarse moduli space. We will denote these integersrespectively by ρu(X) and ρr (X).We will define two types of filtration on K(St) now. One is an descending doublefiltration K≥(·,·) and the other one is a ascending double filtration K≤(·,·,·). We willshow that the inertia operator preserves both filtrations. The latter filtration is used toprove local finiteness whereas the former filtration serves in proving diagonalizabilityresults in the next chapter.6.2 Filtration by central rank and split central numberLet K≥(r ,n) be the sub-module of K(St) spanned by clear gerbes X→ X, for which thecentral rank, is at least r and if this rank is exactly r then the split central number isat least n.First, we prove a property of monomorphisms of group schemes of same dimen-sion. If u :G→H is any homomorphism of group schemes, it follows from the defini-tion of the functor of connectedness component of identity [4, Exp. VI, Def. 3.1] thatG0 maps toH0. The following lemma shows a stronger fact in case of monomorphismsof group schemes of same dimension:37Lemma 6.8. Let u : G → H be a monomorphism of smooth group schemes, locally offinite type over base scheme S. If G and H are of same dimension over S, the inducedmap u0 :G0→H0 is surjective.1Proof. First suppose S is the spectrum of a local artin ring, A. Then G0 is of finite typeover A [4, Exp. VI(A), Prop. 2.4] hence noetherian, and therefore u0 is quasi-compactand consequently u0(G0) is closed in H0 [4, VI(B), Prop. 1.2]. But H0 is irreducible[4, VI(A), Cor. 2.4.1] and u0(G0) is of same dimension as H0 hence it has to be all ofH0. For a general base scheme, S, we now have that for any point s ∈ S, the inducedmorphism u0s :G0s →H0s is surjective, hence u0 is surjective. Let X is a clear gerbe. We fix a stratification of IX by clear gerbes and let Y be onesuch a stratum. We prove that if Y is not contained in IzX then either ρ(Y) > ρ(X) orν(Y) > ν(X). And if Y is contained in IzX, and maps to a component of Iz/z,0X, notof degree 1 over X, then ρ(X)= ρ(Y) but ν(Y) > ν(X).Proposition 6.9. Let Y be a stratum of IX not (completely) contained in IzX. Thenρ(Y)≥ ρ(X) and if ρ(Y)= ρ(X) then ν(Y) > ν(X).Proof. We use diagram 5.2 here again.IzYpi3$$Y×X IzX//pi2? _joo IzXpi1Y // XFrom this diagram, it is obvious that ρ(Y)≥ ρ(X).Now suppose ρ(Y) = ρ(X). Here j is a monomorphism of commutative groupspaces over Y thus by the previous lemma we have an induced monomorphism ofdiscrete central group schemesι : (IzX)Y/(IzX)0Y↩ IzY/(IzY)0.Since the connectedness component of identity and the quotient by it, are preservedby base change, we have the commutative diagramIz/z,0Ypi3&&(Iz/z,0X)Y?_joopi2// Iz/z,0Xpi1Y // X(6.1)1Also cf. [4, Exp. VI(B), Cor. 1.3.2].38in level of stacks. The morphism pi3 has a canonical section induced by the diagonalmorphism Y→Y×XY (explicitly (x,ϕ), (x,ϕ,[ϕ]), where x is an object of X, ϕ ∈Aut(x) and [ϕ] is the orbit of ϕ by the action of connectedness component of unity).Since Y is ultra-central, this section is not in the image of ι˜. Therefore ν(Y) > ν(X). When X is a clear gerbe, the connectedness components of the central inertiaIzX→X are also clear gerbes; this yields a canonical stratification of the central inertiaby clears gerbes. The next two propositions pertain to these strata.Proposition 6.10. The connectedness components of the central inertia IzX→X satisfythe following properties: (1) they all have the same central rank as that of X; and (2)there is a one-to-one correspondence between connectedness components of IzX andthat of Iz/z,0X and each compotent Y ⊆ IzX is an (Iz,0X)-torsor over the associatedconnected component Y′ ⊂ Iz/z,0X.Proof. It is easy to verify the isomorphism Iz(IzX)› IzX×X IzX of stacks which alsofit in the commutative diagramIz(IzX)//IzXZ×X Z // ZThus for any locally closed substack Y is IzX which descends to a locally closed sub-space Y of Z , we haveIz(Y)//YZ×X Y // Y .(6.2)In particular the central Y -group scheme associated to Y is the pull-back Z|Y . (1) isnow obvious form diagram 6.2. For (2) notice that the morphism IzX -→ Iz/z,0X is aprincipal bundle for the connected group scheme Iz,0X over X, and therefore there isa bijection between connectedness components of the source and the target of thismorphism. Passing to a component gives us the cartesian diagramY // _Y′ _IzX // Iz/z,0Xtogether with a finite étale mapping Y′→X proving the lemma. 39Proposition 6.11. Let Y be a connected component of IzX. We always have ν(Y) ≥ν(X), with equality happening if and only if the image of Y in Iz/z,0X maps downisomorphically to X.Proof. By last lemma, Y sits over a connectedness component Y′⊆ Iz/z,0X. The homo-morphism of commutative group schemes (IzX)Y → IzY over Y is an isomorphismgiving the left cartesian square of homomorphisms of commutative group schemesand inducing the right hand one:IzY//pi2IzXpi1Y // XIz/z,0Y //pi2Iz/z,0Xpi1Y // X(6.3)So distinct sections of pi1 pull back to distinct sections of pi2, therefore ν(Y)≥ ν(X).Now suppose Y′ is not isomorphic to X. Then, as the structure map Y→X factorsthrough Y′, and yields a section of pi2 that is not induced by pi1. In this case we haveν(Y) > ν(X). If Y′ maps isomorphically to X, the structure map Y→ X is an Iz,0X-torsor over X. Hence, the upper horizontal map in the left hand diagram of (6.3) is atorsor for a connected group scheme, and therefore we can push the sections forward.In this case, ν(Y)= ν(X). Remark 6.12. The Iz,0X-torsors Y in Proposition 6.10 and 6.11 all come from schemetorsors. In fact, the Iz,0X-principal bundle IzX→ Iz/z,0X is the pull-back of the Z0-principal bundle Z → Z/Z0. Passing to a strata Y, we likewise observe that Y→Y′ isthe pull back of a Z0-torsor.6.3 An ascending filtration and local finitenessLet K≤(d,c,t) be the sub-module of K(St) spanned by clear gerbes X→X, for which thecentral corank, is at most d, and if this number is exactly d then the maximum numberof geometrically connected components of IX→ X is at most c, and if this number isexactly c, then the central twistedness of X is at most t.Proposition 6.13. Let Y be a stratum of IX not completely contained in IzX. Thencorank(Y)≤ corank(X) and equality happens only if c(Y) < c(X).Proof. We consider a different commutative diagramIY""  // IX|Y// IXY // X(6.4)40The injective morphism is given on the fiber of any point x ∈ X by mapping of sub-group ZG(g)↩G where G=Aut(x) and g ∈G Z(G). Thus we always have dimX IX≥dimY IY. This together with Proposition 6.9 show that corank(Y)≤ corank(X).Now suppose corank(Y) = corank(X). In particular this means that dimY(IY) =dimX(IX). The inclusion in diagram 6.4 is given by ZG(g)↩ G for any object x ofX with G = Aut(x), where g is not contained in the centre. Therefore ZG(g) is asubgroup of G not equal to it. By [4, VI(A), Cor. 2.4.1], the closed immersion ZG(g)↩Gmaps connected components of ZG(g) to that of G. Thus the number of connectedcomponents of ZG(g) in strictly less than that of G. Proposition 6.14. LetY be a connected component of IzX. Then corank(Y)= corank(X)and c(X)= c(Y). Also τ(Y)≤ τ(X) with equality happening if and only if Y is isomor-phic to Iz,0X.Proof. It is easy to verify the isomorphism I(IzX) › IX×X IzX. But from Proposi-tion 6.10 we already know that ρ(Y) = ρ(X). So we also conclude that corank(Y) =corank(X). It is also obvious from the same isomorphism that c(X) = c(Y). Thelast claim follows from the fact that the commutative diagrams 6.3 also show thatdegz(Y)= degz(X). The claim is then clear from Proposition 6.11.Corollary 6.15. The filtration K≤(·,·,·) is preserved by the inertia operator of K(St).We see that even proving local finiteness in the general case of algebraic stacks isa challenge. However the above ascending filtration suggests sufficient conditions forlocal finiteness.Recall that a smooth commutative algebraic group Z over a perfect field has adecomposition Z =Zss×U where U is the unipotent radical of Z and Zss is a group ofmultiplicative type. If Z is connected then so is Zss , in which case the latter is a torus[37, XIV].Corollary 6.16. Let Z → X be a smooth connected commutative group scheme of di-mension r with constant reductive rank t and maximal torus T . Then [Z]= qr−t[T].Proof. Let η∈X be the generic point with residue field K. Over the algebraic closure,we have a decomposition ZK =TK×UK where UK is the unipotent radical and descentson K to the quotient group scheme Zη/Tη. By [36, Cor. 15.10] the latter is a unipotentalgebraic group as well. Now we use Lemma 2.14 and Corollary 2.15 to spread out theshort exact sequence1→ Tη→ Zη→ Zη/Tη→ 141to an open subscheme U of X. This implies that [ZU ] = qr−t[TU ] since the quotientZU/TU has a composition series with r − t line bundle factors. The claim now followsby noetherian induction. Let X be a clear gerbe in K≤(d,c,t). Then IX can be stratified by finitely many cleargerbes. Let Y be such an strata. If Y is not contained in IzX or is contained in IzX butis not an Iz,0X-torsor then it is in F<(d,c,t). If Y is an Iz,0X-torsor then [Y]= [Z0×XX]by Remark 6.12. Alternatively by condition (C7) we may write[Y]= qρu(X)[T ×X X]where T is the maximal torus of the central band of X. So by an inductive argument toprove a local finiteness result, we only need to deal with the computation of motivicclasses of such tori over X in terms of X.6.4 Spectrum of the unipotent inertiaLet K≥r be the sub-module of K(St) spanned by clear gerbes X→ X, for which theunipotent central rank, is at least r . This defines an descending filtration K≥· on K(St)and we show that it is preserved by the unipotent inertia. We also define an ascendingfiltration K≤· by declaring a clear gerbe X→ X to be in K≤s if the unipotent co-rankdefined as the difference dimX IX−ρu(X) is at most s.Proposition 6.17. Let X is a clear gerbe which unipotent rank r and counipotent ranks. Let Y be a clear gerbe contained in IuX Iu,zX. Then Y is contained in K>r and inK<s .Proof. The diagram 5.2 maps unipotent automorphisms to unipotent ones. Thereforewe haveIz,uYpi3&&Y×X Iu,zX//pi2? _joo Iz,uXpi1Y // XFrom this diagram, it is obvious that ρ(Y)≥ ρ(X). But if Y is not central then Iu,zY→Y has a trivial section that does not lift to pi2. So the unipotent part of the centralband of Y is strictly larger than that of X. Hence ρu(Y) > ρu(X).Now the diagram 6.4 showed that dimY(IY)≤ dimX(IX). Therefore the unipotentcorank of Y is strictly less that that of X, proving the second claim. Corollary 6.18. The endomorphism Iu :K(St)→K(St) is locally finite and triangulariz-able and the eigenvalue spectrum of it is the set of all monomials qu for u≥ 0.42Proof. By Remark 6.12 we have Iu,z[X]= qρu(X)[X]. The rest is clear. Corollary 6.19. The endomorphism Iu is diagonalizable on K(St)[q−1,{qk−1 :k≥ 1}].43Chapter 7Quasi-split stacksIn this section we present a criteria that if satisfied guarantees the inertia endomor-phism is locally finite and diagonalizable. This criteria restricts the central groupschemes of the clear gerbes to fit in particular exact sequences of nicely-behaved com-mutative group schemes in lines with Chevalley’s structure theorem. For some pre-liminaries on group schemes of multiplicative type and unipotent group schemes werefer the reader to § Motivic classes of quasi-split toriLet Γ be a finite group acting on the finite set r with orbit spacer/Γ = {O1,··· ,O`} .The polynomial∏`i=1(q|Oi|−1) depends only on the sizes of the orbits. In fact, to Γwe may associate a partition λ = (λi)i≥1 of the integer r with declaring λi to be thenumber of elements of r/Γ of size i. The the above polynomial is identical toQλ =∏`i=1(qi−1)λi .The reason quasi-split tori are interesting to us is the following computation oftheir motivic classes.Proposition 7.1. Let T be an isotrivial quasi-split torus over the integral scheme X andX be the minimal splitting Galois cover of it, with Galois group Γ . Let Λ= {b1,··· ,br }be a choice of basis for χT permuted by the Γ -action. Then the motivic class of T is given44by[T]=Qλ(q)[X]+∑I•∈F(r)/ΓStabΓ (I)⊊Γ(−1)`(I•)q|Imax|[X/StabΓ (I)] . (7.1)where λ` r is the partition of integer r induced by the action of Γ on Λ.Here, for a subset I ⊂ r , we denote by AI ⊂ Ar the subset of all (x1, . . . ,xr ) suchthat xi = 0, for i 6∈ I, and by GIm ⊂AI , the set of all (x1, . . . ,xr )∈AI , such that xi 6= 0,for all i∈ I. For every subset I ⊂ r , we have a Γ -equivariant splittingAI =⊔J⊂IGJm ,and hence,[GIm]= [AI]−∑J⊊I[GJm].By induction, we get an equivariant inclusion-exclusion principle[Grm]=∑k≥0(−1)k∑I•∈Fk(r)[AIk].Here Fk(r) is the set of all flags I• = Ik ⊊ . . .⊊ I1 ⊊ I0 = r in r . We denote the length ofa flag I• by k= `(I•), the maximal index by k=max, and the set of all flags, regardlessof length, by F(r).Proof. Γ acts on the split torus TX = Spec(OX[χT ]) and by [35, Prop. 5.21] we canrevive T from this pull-back as T › (X ×X T)/Γ . Then the surjection of Z-module,⊕ri=1biZ→ χT , induces a sheaf homomorphism OX[b1,··· ,br ]→OX[χT ] and con-sequently an open immersion Grm,X › TX → ArX , which is equivariant for the Γ -action.Hence we may pass to quotient schemes and getT =X×Γ Grm ↩X×Γ Ar .45Thus we have[T]= [X×Γ Grm]=∑k≥0(−1)k[X×Γ ⊔I•∈Fk(r)AIk]=∑k≥0(−1)k∑I•∈Fk(r)/Γ[X×StabΓ (I•)AIk]=∑I•∈F(r)/Γ(−1)`(I•)q|Imax|[X/StabΓ (I•)]=∑I•∈F(r)Γ(−1)`(I•)q|Imax|[X]+∑I•∈F(n)/ΓStabΓ (I)⊊Γ(−1)`(I•)q|Imax|[X/StabΓ (I)]=QΩ(q)[X]+ ∑I•∈F(r)/ΓStabΓ (I)⊊Γ(−1)`(I•)q|Imax|[X/StabΓ (I)] .Note that all forms of affine spaces occurring in this computation are vector bundlesover their base by Hilbert’s Theorem 90. This is the reason for the appearance of theterms q|Ik| in the calculation. 7.2 Quasi-split stacksDefinition 7.2. An algebraic stack X, is called quasi-split if for any point of its coarsemoduli space, x ∈X, the central band Z|x admits a quasi-split maximal torus.The category of quasi-split algebraic stacks is the full subcategory of St consistingof all quasi-split algebraic stacks is denoted by QS. It is easy to see that QS is closedunder ienrtia and contains all its products and open and closed substacks, thus thereis a well-defined induced K(Sch)-linear inertia endomorphism of the algebra K(QS).Remark 7.3. The results below are all about this subcategory of St however in §9, §10,and §11, we will be working with certain subcategories of QS. All results below holdtrue, for any full subcategory of QS that is (1) closed under inertia; and, (2) containsproducts and all closed and open subobjects of it.Quasi-split algebraic stacks, can be stratified further to nicely-behaving clear gerbesas far as motivic computations are concerned.Definition 7.4. A clear gerbe X→X with central group scheme Z →X, is called quasi-split if Z admits a maximal torus T which is quasi-split.46Theorem 7.5. A quasi-split stack can be stratified by finitely many quasi-split cleargerbes.Proof. In view of Theorem 6.4 we only need to prove the result for a quasi-split cleargerbe X→ X. Let η= SpecK be the generic point of X. By definition, the generic fiberXK , is then a quasi-split K-stack. Therefore the central group scheme Z → X, pullsback over η to the algebraic K-group Zη where the connected component of unity hasa decomposition as Z0 = Tη×Uη. Here Tη is a quasi-split maximal torus of Z and Uη isa unipotent algebraic K-group. It remains to observe that this decomposition spreadsout to an open neighborhood of η in X by Corollary 2.15. Let X be a quasi-split clear gerbe. The Iz,0X-torsors Y in Proposition 6.10 areessential in our motivic computations and therefore we finish this section by findingthe motivic class of such torsors in terms of the motivic class of X.Proposition 7.6. Let X be a quasi-split clear gerbe with coarse moduli space X andcentral X-group scheme Z of rank r and reductive rank t. Let Y → X be a Z0-torsorand Y be the pullback Iz,0X-torsor over X. Then[Y]= qr−tQλ`t(q)[X]+∑I•∈F(r)/ΓStabΓ (I)⊊Γ(−1)`(I•)qr−t+|Imax|[X/StabΓ (I)] , (7.2)for some finite étale covering X→X and an action of Γ =pi1(X/X) on the set {1,··· , t}.Here λ ` t is the integer partition type of the orbit space t/Γ in the sense of §7.1 andX=X|X .Proof. By [10, Prop 2.2] any torsor for a quasi-split torus is Zariski locally trivial andtherefore the assertion follows from Corollary 6.16 and Proposition 7.1 and pullingback along X→X. 7.3 An ascending filtration and local finitenessRecall the ascending filtration of §6.3 by central corank, maximum number of geomet-rically connected components, and then central twistedness. Recall that in the samesection we proved for any clear gerbe X→X, with central corank d, maximum numberof geometrically connected components c, and central twistedness t and a stratifica-tion of IX by clear gerbes, every stratum is Y⊆ IX is contained in K<(d,c,t) unless Y isan Iz,0X torsor. We denote the associated graded piece by K≤(d,c,t)/K<(d,c,t).Proposition 7.7. The endomorphism I : K(QS)→ K(QS) is locally finite. For every ele-ment X in QS there exists a finite-dimensional Z[q]-module of K(QS) that is invariantunder inertia and contains the motivic class [X].47Proof. Let X and Γ be defined as in Proposition 7.6. Consider the finite-dimensionalZ[q]-submodule, L generated by the finitely many intermediate covers X/StabΓ (I).Then the results of §6.3 and Proposition 7.6 show thatI(L)⊆ L modK<(d,c,t).The claim is now clear by induction. 7.4 A descending filtration and diagonalizationLet X→ X be a quasi-split clear gerbe with a central X-group scheme Z of reductiverank t. The Galois group Γ = pi1(X/X) of the minimal splitting Galois cover is then asubgroup of St , the group of permutations of t letters. This action induces an integerpartition λ` t as explained in §7.1.Definition 7.8. For a quasi-split clear gerbe X, the partition λ of integer t constructedas above is called the twist type of it.The double-filtration in section 6.2 induces a double filtration of K(QS) which willagain be denoted by K≥(·,·). We now introduce a refinement, K≥(·,·,·) of the formerfiltration and show that it is preserved by inertia endomorphism.We first impose a well-ordering on the set of all integer partitions: for a giveninteger t ≥ 0, we put the lexicographic ordering on the set of all partitions of t. Andfor any two integers t < s, we assume all partitions t are smaller than all partitions ofs. In fact any well-ordering that satisfies the following two conditions would work forus1. if λ` t and µ ` s and t < s then λ < µ; and2. if λ,µ ` t and b(λ) < b(µ) then λ < µ.Now, given integers r ,n≥0 and an integer partition λ` t, the submodule K≥(r ,n,λ)is generated by those quasi-split clear gerbes in K≥(r ,n) that have twist type is at leastλ.Lemma 7.9. The inertia endomorphism of K(QS) respects the filtration K≥(·,·,·).Proof. In view of results of section 6.2 it suffices to consider a tuple (r ,n,λ), and aquasi-split clear gerbe X→X with central rank r , discrete split central number n, andtwist type λ. Also from Propositions 6.9 and 6.10 we only need to consider a centralstratum Y of IX which is an Iz,0X-torsor over X. By Proposition 7.6, the motivicclass [Y] is a linear combination of [X] and the intermediate covers X/StabΓ (I) for aminimal splitting cover X→X of the maximal torus T →X of the central band of X. We48note that each of these stacks is a clear gerbe over the intermediate cover X/StabI andthe maximal torus of X pulls back to the maximal torus T |X/StabI of the central bandof X/StabΓ (I). Since Γ ′ = StabI is a proper subgroup of the Galois group Γ =pi1(X/X),the orbit space r/Γ ′ has strictly more elements than r/Γ . Therefore the twist type ofX/StabΓ (I) is strictly bigger than that of X in the well-ordering defined above.We can now finish the proof of our main result.Theorem 7.10. The operator I is diagonalizable on K(QS)(q)=Q(q)⊗Q[q]K(QS) as alinear endomorphism of a Q(q)-vector space. The eigenvalue spectrum of it is the setof all polynomials of the formnquk∏i=1(qri −1).Proof. In Lemma 7.9 we showed that I is triangularizable and by Proposition 7.6 theaction of it on each graded piece K≥(r ,n,γ)/K>(r ,n,γ) is multiplication by the polyno-mial nqr−tQγ`t(q). This polynomial uniquely determines the triply (r ,n,λ) resultingdistinct eigenvalues associated to each graded piece. Remark 7.11. In fact I is diagonalizable as an endomorphism of the Z[q]-moduleK(QS)[q−1,{(Qλ−Qµ)−1 :∀λ` t,µ ` s}).7.5 Spectrum of the semisimple inertia of quasi-splitstacksWe can now prove a semisimple version of Proposition 6.9 in terms of the reductiveranks of the quasi-split clear gerbes rather than their central ranks. Note that the finitegroup scheme Iz/Iz,0 is semisimple and product of semisimple commuting elementsis semisimple. Therefore there are ν(X) connected components of Iz,ss/Iz,ss,0 thatmap isomorphically to X. 1Proposition 7.12. Let Y be a stratum of IssX not completely contained in Iss,zX. Thenρ(Y)≥ ρ(X) and if ρ(Y)= ρ(X) then ν(Y) > ν(X).Proof. Let Y⊆ IssX be a strata of the semisimple inertia, in particular a locally closedsubstack of IX. Diagram 5.2 has a semisimple verison. The downward arrows pi1,pi2 and pi3 are all structure morphisms of relative commutative group schemes. Since1In fact, the morphism Zss/Zss,0 → Z/Z0 is an isomorphism of finite X-group schemes.49unipotency is preserved under the group homomorphisms we may divide each of thesecommutative group schemes with their unipotent radical.Iss,zYpi3&&Y×X Iss,zX//pi2? _joo Iss,zXpi1Y // X(7.3)From previous diagram it is obvious that ρ(Y)≥ ρ(X). Now suppose rankz(Y)=rankz(X). Similar to the case of Proposition 6.9 we now pass to the quotients byconnected components of unity to get a commutative diagramIss,z/ss,z,0Ypi3(((Iss,z/ss,z,0X)Y?_joopi2// Iss,z/ss,z,0Xpi1Y // X(7.4)in level of stacks. Same analysis as in case of Proposition 6.9 shows that pi3 has strictlymore sections that pi1. When X is a quasi-split clear gerbe, the connectedness components of the semisim-ple central inertia Iss,zX→ X are also quasi-split clear gerbes; this yields a canonicalstratification of Iss,zX. The analogue of Propositions 6.10 and 6.11 is stated below:Proposition 7.13. Let Y be a connectedness component of Iz,ssX→ X. Then ρ(Y) =ρ(X). We always have ν(Y)≥ ν(X), with equality happening if and only if the image ofY in Iss,z/ss,z,0X maps down isomorphically to X.Proof. The proof is similar to that of Propositions 6.10 and 6.11 by considering thecommutative diagramIss,z(Iss,zX)//Iss,zX// XZss ×X Zss // Zss // Xfor the first claim where by restricting to Y we getIss,z(Y)//YZss ×X Y // Y .50Note that Iz,ssX -→ Iss,z/ss,z,0X is a principal bundle for the connected group schemeIss,z,0X over X, and therefore there is a bijection between connectedness componentsof the source and the target of this morphism. So Y sits over a connectedness com-ponent Y′ ⊆ Iz/z,0X. Similar to the case in Proposition 6.11 we now can form thefollowing cartesian diagramIss,z/ss,z,0Y //pi2Iss,z/ss,z,0Xpi1Y // Xand the rest of the proof is now similar to Proposition 6.11. The proofs of local finiteness and diagonalization of Iss now follow similar to thecase of the full inertia operator. We skip the details and only carry the computation ofthe spectrum as follows. When X is a quasi-split clear gerbe,[IssX]= [Iss,zX]+ terms with larger reductive rank+ terms with same reductive rank but larger discrete-split central number.For central strata of the semisimple inertia has a simpler description than thecase of the full inertia. The relative group space Iss,z,0X→ X is the base change of aquasi-split torusIss,z,0X //XT // Xwhere X is the coarse moduli space of X and T is the maximal torus of the centralband of X. Computation of Proposition 7.6 therefore shows[Iz,ssX]= ν(X)[Iss,z,0X]+ terms of lower co-untwistedness= ν(X)Qλîr (q)[X]+ terms with finer twist types γ > λ.We may now consider a simpler filtration, K≥(·,·,·) than before: given integersr ,n≥0, and a partition λ` r the submodule K≥(r ,n,λ) is generated by those quasi-splitclear gerbes that have reductive rank at least r and if this rank is exactly r then theuntwistedness of the central group scheme is at least n and if this quantity is exactlyn, then the twist type is at least λ. This filtration is preserved by the semisimple51inertia endomorphism and leads toTheorem 7.14. The endomorphism Iss :K(QS)→K(QS) is locally finite and triangular-izable and the eigenvalue spectrum of it is the set of all polynomials of the formnk∏i=1(qri −1).Moreover Iss is a diagonalizableQ(q)-linear endomorphism of the vector space K(QS)(q).52Chapter 8ExamplesExample 5. A first simple example is the case of [BGL2]. Here, and in following exam-ples we are suppressing the notation [.] for quotient stacks; thus unless mentionedotherwise, all quotients (of schemes) are stack quotients. Note that we haveI BGL2 = GL2 /GL2 = (GL2)ss, eq/GL2unionsq(GL2)dist/GL2unionsq(GL2)ns/GL2 .The first stratum contains diagonalizable matrices with one eigenvalue, the secondstratum diagonalizable matrices with distinct eigenvalues, and the third stratum thenon-semisimple matrices. We study these three strata and their inertia:First stratum: Consider the mapping Gm → GL2 via x,(x 00 x). This is equivariantwith respect to the natural GL2-action, so we get an induced morphism of stacksGm×BGL2→ GL2 /GL2which is easily seen to be an isomorphism onto the first stratum.Second stratum: Let T be the standard maximal torus of GL2. Let ∆ be the centreof GL2, which is the diagonal subtorus of T . Let N be the normalizer of T . We have ashort exact sequence0 //T //N //Z2 //0where Z2 is the Weyl group of GL2. Note that N = G2m ÏZ2 is in fact a semi-directproduct, by taking(0 11 0)as the nontrivial element of Z2 ⊂ N . The induced action ofthe Weyl group Z2 on T is by swapping the two entries. The natural inclusion mapT ∆→ GL2 is equivariant for the inclusion of groups N ⊂ GL2, so we get an inducedmorphism of stacks(T ∆)/N -→ GL2 /GL253which is an isomorphism onto the second stratum. We will abbreviate this asX= (T ∆)/N.Third stratum: Let H be the (commutative) subgroup of all matrices of the form(λ µ0 λ). Note that H is the centralizer of every matrix of the form(a 10 a), with a 6= 0.Thus we see that the third stratum is isomorphic to Gm×BH.We conclude that in the level of motivic classes the inertia of the class [BGL2] isgiven byI[BGL2]= (q−1)[BGL2]+ [X]+ (q−1)[BH].Since H is commutative, we also have I[BH]= q(q−1)[BH]. We will now find theinertia of the second stratum X. Note that that the coarse moduli space of X is thesmooth variety X = T ∆/Z2. Note also that IX = IzX as the stabilizer of any pointin T ∆ is commutative. We will write X˜ = T ∆ to emphasize the fact that T ∆ is adegree 2 cover of X.Associated to the Z2-action on the group T , there exists a commutative X-groupschemeT ′ = X˜×Z2 T ,with fibre T . By Lemma 8.1, X is the neutral gerbe, X= BXT ′. And IzX= IX fits in thecartesian diagramIX //XT ′ // X.The representation of Z2 on A2 given by swapping entries, yields a canonicalclosed embedding T ′ ⊂ V , into a rank 2 vector bundle over X. As in Proposition 7.1,this leads toI[X]= (q2−1)[X]− (q−1)2(q−2)[BG2m]Thus, the 4-dimensional K(Var)-module, L, of motives generated by the 4 motives[BGL2],[BH],[X], and [BG2m]is preserved by the inertia endomorphism I. The first element in this set is of centralrank 1 and the other three are of central rank 2. BH has reductive rank 1, X hasreductive rank 2 with the nontrivial partition of 2 associated to it, and BG2m has arank 2 torus with the trivial partition of 2 associated to it. The eigenvalue spectrum ishence {q−1,q(q−1),q2−1,(q−1)2}. Inertia endomorphism is lower triangularizable54on L and we haveI =q−1 0 0 0q−1 q(q−1) 0 01 0 q2−1 00 0 −(q−1)2(q−2) (q−1)2with a set of eigenvectorsEigenvalues Eigenvectorsq−1 −q(q−1)[BGL2]+q[BH]+ [X]+ (q−1)[BG2m]q(q−1) [BH]q2−1 [X]− (q−1)(q−2)2 [BG2m](q−1)2 [BG2m]Table 8.1: Spectrum of the inertia endomorphism on a 4-dimensional K(Var)-submodule of K(St) containing [BGL2]Also I is diagonalizable on L[q−1,(q−1)−1] and the eigenprojections of [BGL2]areEigenvalues EigenvectorsΠq−1 [BGL2]− qq−1 [BH]− 1q(q−1) [X]− 1q [BG2m]Πq(q−1) qq−1 [BH]Πq2−1 1q(q−1) [X]− (q−2)2q [BG2m]Π(q−1)2 12 [BG2m]Table 8.2: Eigenprojections of [BGL2]Lemma 8.1. Let Γ be a group acting on the group T by automorphisms, Γ →Aut(T), andlet G=NÏH be the associated semi-direct product of groups. Let X be a variety, X˜→Xa principal Γ -bundle, and T ′→X is the associated form of T over X. The BXT ′ = X˜/G.Proof. Consider the diagramX˜×T //X˜×TX˜ // X˜55where Γ acts on the first column and G on the second column in the obvious way. Thenthe horizontal arrows are a morphism of T -bundles which is Γ →G equivariant. Thuswe get an induced cartesian diagram of stacks(X˜×T)/Γ //(X˜×T)/GX˜/H // X˜/Gwhich we may rewrite asT ′ //XX // X˜/G.Then the latter diagram induces a morphism X → X˜/G, which is then obviously anisomorphism. Example 6. In the previous example the central group schemes were always con-nected. We will now present an example that demonstrates how non-cennected centralgroup schemes contribute to non-monic eigenvalues. Let N = G2mÏZ2 be the groupscheme introduced in previous example. In this example we study BN . The inertia ofBN has two obvious connectedness components:I BN =N/N =G2m/NunionsqG2m×{σ}/Nwhere σ is the nontrivial element of Z2.First stratum: This stratum is not already a gerbe (as the stabilizer of points ondiagonal ∆ ⊂ G2m is not isomorphic to the stabilizer of other points). However thefollowing is a stratification of it into clear gerbes:G2m/N =∆/NunionsqX,where X = G2m ∆/N is the same quotient stack that appeared in previous example.The action of N on ∆ is trivial so we have[G2m/N]= (q−1)[BN]+ [X].Second stratum: This stratum is already a clear gerbe and we will denote it as Y.Any point 〈(µ,γ),σ〉 of G2m×{σ} is conjugate to 〈(µγ,1),σ〉 which is canonical forthe orbit. Thus the subscheme Y representing the points,{〈(x,1),σ〉} ⊂G2m×{σ},is a coarse moduli space for this gerbe. This is isomorphic to A1 {0}. The stabilizerof this subscheme is a subgroup scheme N′ ⊆N , the fiber of which over a geometric56point x of Y is the κ(x)-algebraic groupN′x = {〈(t,t),1〉 : t ∈ κ(s)×}∪{(xt,t),σ〉 : t ∈ κ(s)×}.We notice that the mapping Y/N′→Y is an isomorphism of stacks and also that N′ isa commutative Y -group scheme, acting trivially on Y . Therefore[Y]= [Y][BN′]= (q−1)[BN′]I[BN′]= [N′/N′]= [N′][BN′]Finally N′/(N′)0→ Y is a degree two covering of X. The image of (N′)0 in N′/(N′)0 isisomorphic to Y and therefore so is the image of the other connected component. SoN′ is Zariski locally the union of two Gm-torsors over Y . Pulling back along Y→ Y wehaveI[BN′]= 2(q−1)[BN′].We conclude that the K(Var)-submodule of K(St) generated by[BN],[BN′],[X], and [BG2m]is invariant under inertia endomorphism. The first two generators have central rankone, and [BN] has split central number one whereas [BN′] has split central numbertwo. The spectrum of I restricted to this submodule is the set{(q−1),2(q−1),q2−1,(q−1)2}as expected.Example 7. Another simple example that shows many features of this theory is thestack BGL3. As before, the inertia stack is isomorphic to the quotient stack [GL3 /GL3]via conjugation action of GL3 on itself. We first stratify this quotient according toJordan canonical forms: let Jkλ be the subscheme of all general linear matrices withk-distinct eigenvalues and λ î 3 is a partition of 3 indicating format of the Jordanblocks and Rkλ ⇒ Jkλ is the groupoid representation of restriction of [GL3 /GL3] to Jkλ .Then we have a stratification[GL3 /GL3]= [J1(3)/R1(3)]unionsq [J1(2,1)/R1(2,1)]unionsq [J1(1,1,1)/R1(1,1,1)]unionsq [J2(2,1)/R2(2,1)]unionsq [J2(1,1,1)/R2(1,1,1)]unionsq [J3(1,1,1)/R3(1,1,1)]The action of Rkλ on Jkλ by conjugation is always trivial unless in presence of Jordan57blocks of same dimension with distinct eigenvalues (which can then be permuted).Thus[GL3 /GL3]= J1(3)×BR1(3)unionsqJ1(2,1)×BR1(2,1)unionsqJ1(1,1,1)×BR1(1,1,1)unionsqJ2(2,1)×BR2(2,1)unionsqJ2(1,1,1)×BR2(1,1,1)unionsqJ3(1,1,1)×BR3(1,1,1)We recall the notation of Example 5 for the subgroup of upper-triangular 2×2 matriceswith a single eigenvalue of multiplicity two:H =a b0 a : a,b ∈Gm .This represents a commutative group scheme. Now, easy computations show that allRkλ’s are subgroup schemes of GL3 and in factGroupoid Group scheme structure Commutative?R1(3)a b c0 a b0 0 a : a∈Gm,b,c ∈A1 YesR1(2,1)a b c0 a 00 d e : a,e∈Gm,b,c,d∈A1 NoR1(1,1,1) GL3 NoR2(2,1) H×Gm YesR2(1,1,1) GL2×Gm NoR3(1,1,1) G3mÏS3 YesTable 8.3: Stratification of GL3[GL3 /GL3]= (q−1)[BR1(3)]+ (q−1)[BR1(2,1)]+ (q−1)[BGL3]+ (q−1)(q−2)[BH][BGm]+ (q−1)(q−2)[BGL2][BGm]+ [G3m/G3mÏS3]Since inertia respects the commutative algebra structure of K(St) we may use theprevious example to compute the effect of inertia on terms of the second line above.Since R1(3) is commutative we also haveI[BR1(3)]= [R1(3)][BR(3)1]= q2(q−1)[BR1(3)].58Strata Canonical form for an orbit Centralizer of the canonical forma b c0 a 00 d e : a≠ ea b+cd/(a−e) 00 a 00 0 e G1 =x y 00 x 00 0 z : x,z ≠ 0a b c0 a 00 d a : c,d≠ 0a 0 10 a 00 cd a G2 =x y z0 x 00 w x : x ≠ 0a b 00 a 00 d a : d≠ 0a b+d 00 a 00 d a G3 =x y 00 x 00 z x : x ≠ 0a b c0 a 00 0 a : c ≠ 0a 0 10 a 00 0 a G4 =x y z0 x 00 0 x : x ≠ 0a b 00 a 00 0 a itself GTable 8.4: Stratification of R1(2,1)The case of Y = [G3m/G3mÏS3] is similar to that of [G2m/G2mÏZ2]. It remains toanalyze the action of G=R1(2,1) on itself. We need to stratify G/G to several substackswhich is carried out in Table 8.4. It follows thatI[BG]= q(q−1)[BG]+q3(q−1)(q−2)[BG1]+q(q−1)3[BG2]+q(q−1)2[BG3]+q(q−1)2[BG4].We conclude that [BGL3] is contained in a 9-dimensional K(Var)-submodule of K(St)which is diagonalizable (Table 8.5).Central rank Reductive rank Twist type Pivot elements Eigenvalue1 1 (1) [BGL3] q−12 2 (2,0) [BGL2][BGm] (q−1)21 (1) [BG] q(q−1)3 3 (0,0,1) [Y] q3−1(1,1,0) [X][BGm] (q2−1)(q−1)(3,0,0) [BG3m] (q−1)32 (2,0) [BH][BGm],[BG1] q(q−1)21 (1) [BR1(3)],[BG3],[BG4] q2(q−1)4 1 (1) [BG2] q3(q−1)Table 8.5: Spectrum of the inertia endomorphism of a 9-dimensional K(Var)-submodule of K(St) containing [BGL3]59Part IIIAlgebroids and their Hall algebras60Chapter 9Linear Stacks9.1 Algebraic stacksLet us briefly summarize our conventions about algebraic stacks.We choose a noetherian base ring R, and we fix our base category S to be thecategory of R-schemes, endowed with the étale topology.Over S we have a canonicalsheaf of R-algebras OS, it is represented by A1=A1SpecR , and called the structure sheaf.We will assume our algebraic stacks to be locally of finite type. Thus, an algebraicstack, is a stack over the site S, which admits a presentation by a smooth groupoidX1 ⇒ X0, where X0 and X1 are algebraic spaces, locally of finite type, the source andtarget morphism s,t : X1 → X0 are smooth, and the diagonal X1 → X0×X0 is of finitetype.If G is an algebraic group acting on the algebraic space X, we will denote thequotients stack by X/G, because we fear the more common notation [X/G] wouldlead to confusion with the notation for elements of various K-groups of schemes andstacks.Coherent sheavesIn particular, an algebraic stack X is a fibered category X→S. The category X inheritsa topology from S, called the étale topology, and X endowed with this topology is thebig étale site of X. Sheaves over X are by definition sheaves on this big étale site. Forexample, OS induces a sheaf of R-algebras on X, which is denoted by OX , and calledthe structure sheaf of X. It is represented by A1X .A sheaf F over X induces for every object x of X lying over the object U of Sa sheaf on the usual (small) étale site of the scheme U , denoted FU . Moreover, forevery morphispm α : y → x lying over f : V → U , we obtain a morphism of sheavesα∗ :FU →f∗FV . (The α∗ satisfy an obvious cocycle condition.) The data of the sheaves61FU , together with the compatibility morphisms α∗, is equivalent to the data definingF. For example, the structure sheaf OX induces the structure sheaf on U , for everysuch x/U .A sheaf of OX -modules is coherent, if for all x/U the sheaf FU is a coherent sheafof OU -modules, and all compatibility morphisms α∗ : f∗FU →FV are isomorphisms ofsheaves of OV -modules.For example, a groupoid presentation X1 ⇒ X0 of X, and a coherent sheaf F0 onX0, together with an isomorphism s∗F0→ t∗F0, satisfying the usual cocycle conditionon X2 =X1×X0 X1, give rise to a coherent sheaf on X.Representable coherent sheavesA sheaf F over X is representable, if there exists an algebraic stack Y →X, with repre-sentable structure morphism Y →X, such thatF is isomorphic to the sheaf of sectionsof Y →X.A coherent sheaf F over the algebraic stack X is representable if and only if forevery x/U , the coherent OU -module FU is reflexive, i.e., isomorphic to F∨∨U . So F isrepresentable if and only if F is isomorphic to F∨∨ as an OX -module. If F is repre-sentable, it is represented by the finite type affine X-scheme Y = SpecX(SymOXF∨).9.2 Linear algebraic stacksWe will review the definition of linear algebraic stacks, and some basic constructions.For definitions and basic properties of fibered categories we refer the reader to [20,Exposé VI].Suppose X→S is a category over S. We write X(S) for the fiber of X over theobject S of S. If f : S′ → S is a morphism in S, and x′ ∈ X(S′) and x ∈ X(S) are X-objects lying over S′ and S, respectivly, we write Homf (x′,x) for the set of morphismsfrom x′ to x in X, lying over f . For S′ = S and f = idS , we write HomS(x′,x).Recall that a morphism α :x′→x lying over f : S′→ S is cartesian, if for every ob-ject x′′ of X(S), composition withα induces a bijection HomS(x′′,x′)'-→Homf (x′′,x).Recall further that X→S is a fibered category, if every composition of cartesian mor-phisms is cartesian, and if for every f : S′ → S in S, and every x over S, there exists acartesian morphism over f with target x. A cartesian functor between categories overS is one that preserves cartesian morphisms.If X is a fibered category over S, the subcategory of X, consisting of the sameobjects and all cartesian morphisms is a category fibered in groupoids over S. Wedenote it by Xcfg, and call it the underlying category fibered in groupoids.Definition 9.1. A category X over S is an O-linear category over S, if for every f :S′ → S in S and all x′ ∈ X(S′), x ∈ X(S), the set Homf (x′,x) is endowed with the62structure of an O(S′)-module, in such a way that for every pair of morphisms g :S′′ → S′, f : S′ → S, and every triple of objects x′′ ∈ X(S′′), x′ ∈ X(S′), x ∈ X(S), thecompositionHomf (x′,x)×Homg(x′′,x′) -→Homf◦g(x′′,x)is O(S′)-bilinear. Here the O(S′′)-modules, Homg(x′′,x′) and Homf◦g(x′′,x) inheritthe structure of O(S′)-modules via pullback along g.An O-linear functor F : X → Y between O-linear categories is a functor of cate-gories over S, such that for every f : S′ → S, and all x′ ∈ X(S′), x ∈ X(S) the mapHomf (x′,x)→Homf(F(x′),F(x))is O(S′)-linear.Let X be an O-linear fibered category over S. Pullback in X is O-linear, i.e., iff :S′→S is a morphism inS, and x,y ∈X(S) are objects with pullbacks x′,y′∈X(S′),the pullback map f∗ : HomS(x,y)→HomS′(x′,y′) is O(S)-linear. So if we fix objectsx,y ∈X(S), the presheaf HomS(x,y) over the usual (small) étale site of S, defined byHomS(x,y)(T)=HomT (x|T ,y|T ), for every étale T → S, is a presheaf of OS -modules.Moreover, for any morphism f : S′ → S in S, we have a natural homomorphism ofsheaves of OS′ -modules f∗HomS(x,y)→HomS′(x′,y′).Definition 9.2. A linear algebraic stack is an O-linear fibered category X over S, suchthat(i) for every object S ∈S, and every pair x,y ∈ X(S), the presheaf of OS -modulesHomS(x,y) is a coherent sheaf over S, which is representable by a finite typeaffine S-scheme;(ii) for every morphism f : S′ → S the pullback homomorphism f∗HomS(x,y)→HomS′(x′,y′) is an isomorphism of coherent OS′ -modules; and,(iii) the underlying category fibered in groupoids Xcfg→S is an algebraic stack overR (locally of finite type).A morphism of linear algebraic stacks is an O-linear cartesian functor over S.Remark 9.3. If X is a linear algebraic stack, with underling algebraic stack X = Xcfg,there exists a representable coherent sheaf H over X×X, which represents the sheafover X ×X, whose set of sections over the pair x,y ∈ X(S) is the O(S)-moduleHomS(x,y). The sheaf H is the universal sheaf of homomorphisms. The subsheafI⊂H representing isomorphisms is naturally identified with X, and the projection toX×X with the diagonal.Pulling back H via the diagonal to X, we obtain the universal sheaf of endomor-phisms, which represents the sheaf whose set of sections over x ∈ X(S) is the O(S)-algebra EndS(x).63The linear algebraic stack X can be reconstructed from its underlying algebraicstack X, and the sheaf of OX×X -algebras H. We leave it to the reader to write downaxioms for the pair (X,H), which assure that (X,H) comes from a linear algebraicstack.ExamplesExample 8. Let X be a projective R-scheme. The linear stack CohX has as objects ly-ing over the R-scheme S, the coherent sheaves on X ×S, which are flat over S. Fora morphism of R-schemes f : S′ → S, and F′ ∈ CohX(S′), and F ∈ CohX(S), we setHomf (F′,F)=HomOX×S′ (F′,f∗F). A morphismF′→F in CohX over f in S is carte-sian, if it induces an isomorphism F′ › f∗F.The linear stack CohX is algebraic. Let pi : X × S → S be the projection on thesecond component. The fact that HomS(F, G) is represented by a finite type affineS-scheme (whose formation commutes with base change) follows from the fact thatpi∗HomS(F, G) is reflexive, and equal to the dual of a coherent OS -module, whoseformation commutes with base change (see [21, EGA III 7.7.8, 7.7.9]).The fact that (CohX)cfg is algebraic and locally of finite type is proved in [34,].Example 9. As a special case of the previous example, consider the case X = SpecR.Then the linear algebraic stack CohSpecR is the linear stack of vector bundles, notationVect. The underlying algebraic stack Vectcfg is the disjoint union⊔n≥0BGLn. Thesheaf H over ⊔n≥0BGLn×⊔n≥0BGLn =⊔n,m≥0B(GLn×GLm)is given by the natural representation M(m×n) of GLn×GLm over the componentB(GLn×GLm).Example 10. A generalization of Vect in a different direction is given by quiver repre-sentations.Let Q be a quiver. The stack of representations of Q, notation RepQ, has asRepQ(S) the set of diagrams (F) in the shape of Q of locally free finite rank OS -modules. For a morphism f : S′ → S of R-schemes we have Homf (F′,F) is the O(S′)-module of homomorphisms F′→ f∗F of diagrams of locally free OS′ -modules.Example 11. As a toy example, let A be an R-algebra scheme of finite type, with groupscheme of units A×, also of finite type. Then we define the linear stack of A×-torsors64to have as objects over the R-scheme S the right A×-torsors over S, and for f : S′ →S and A×-torsors P ′ over S′ and P over S, we set Homf (P ′,P) = HomS′(P ′,f∗P) =P ′×A× A×A× f∗P . In this example, the underlying algebraic stack is BA× and we haveH =A×\A/A×.The case A = 0 is not excluded. The associated linear stack is id : S → S. AllHomf (x,y) are singletons, endowed with their unique module structure. This stackis represented by SpecR. It can also be thought of as the stack of zero-dimensionalvector bundles.SubstacksLet X be a linear algebraic stack with underlying algebraic stack X = Xcfg. If Y ⊂ X isa locally closed algebraic substack, there is a canonical linear algebraic stack Y, withunderlying algebraic stack Ycfg = Y . In fact, we can define Y to be the full subcategoryof X consisting of objects which are in X.Fibered productsLet F : X→ Z and G : Y→ Z be cartesian morphisms of O-linear fibered categories. Wedefine a new O-linear fibered category W as follows: objects of W over the object T ofS are triples (x,α,y), where x is an X-object over T , y is a Y-object over T , and αis an isomorphism α : F(x)→G(y), over T . A morphism from (x′,α′,y′) to (x,α,y)over T ′→ T is a pair of morphisms f : x′→ x over T ′→ T and g :y′→y over T ′→ T ,such that α◦F(f)=G(g)◦α′.In other words, we can write the set of morphisms from (x′,α′,y′) to (x,α,y)over ϕ : T ′→ T as the fibered productHomϕ(x′,x)×Homϕ(F(x′),G(y))Homϕ(y′,y),and as each of the sets in this fibered product is an O(T ′)-module, and the maps arelinear, this fibered product is also an O(T ′)-module. We leave it to the reader to verifythat composition is bilinear.Let us verify that W is a fibered category. Suppose that (x,α,y) is a triple over T ,and ϕ : T ′ → T a morphism in S. We construct a triple (x′,α′,y′) over T ′ by takingas x′ a pullback of x via ϕ, and for y′ a pullback of y via ϕ. Then, as G is cartesian,G(y′) is a pullback of G(y) via ϕ. Hence there exists a unique morphism α′ : F(x′)→G(y′) covering T ′, such that α ◦ F(x′ → x) = G(y′ → y) ◦α′. Then α′ is cartesian,because cartesian morphisms satisfy the necessary two out of three property. Thenα′ is invertible, because cartesion morphisms covering an identity are invertible. Thetriple (x′,α′,y′) comes with a given morphism to (x,α,y) which coversϕ. It is easilyverified that this morphism is cartesian.65Therefore, W is an O-linear fibered category. By construction, the two projectionsW→X and W→Y are cartesian. We call W the fibered product of X and Y over Z.Suppose X, Y and Z are linear algebraic stacks, with underlying algebraic stacksX, Y and Z , respectively. For triples (x′,α′,y′) and (x,α,y) over S, the presheafHomS((x′,α′,y′),(x,α,y))is equal to the fibered productHomS(x′,x)×HomS(Fx′,Gy)HomS(y′,y),and is therefore a representable coherent sheaf of OS -modules. We see that W is againa linear algebraic stack. Morover, the underlying algebraic stack of W is the fiberedproduct X×Z Y .Lack of localityRemark 9.4. Suppose X and Y are linear algebraic stacks, with underlying algebraicstacks X and Y . We can construct a disjoint union linear algebraic stack XunionsqY whoseunderlying algebraic stack is XunionsqY , by declaring all homomorphisms between objectsof X and objects of Y to be zero. This concept of disjoint union is not useful for ourpurposes. For the linear algebraic stacks we are interested in, the underlying algebraicstack often decomposes into a disjoint union, even though the linear algebraic stackdoes not. An example is given by the linear stack of vector bundles Vect, Example 9.Thus linear algebraic stacks exhibit less local behaviour than algebraic stacks, andare therefore less geometrical. This is one of the reasons we prefer to work withalgebroids, rather than linear algebraic stacks.66Chapter 10Algebroids10.1 Finite type algebrasDefinition 10.1. Let X be an algebraic stack. By an algebra over X, we mean a sheafof OX -algebras over X. If the algebra A over the algebraic stack X is an algebraic stackitself, i.e., if the structure morphism A→ X is a representable morphism of stacks,then we say that A is representable. If, in addition, the underlying sheaf of O-modulesof A coherent, we call A a finite type algebra over X.Inertia representationWhenever A→ X is an algebra over the algebraic stack X, we have a tautological mor-phism of sheaves of groups over XIX -→AutX(A). (10.1)Here IX is the inertia stack of X, i.e., the stack of pairs (x,ϕ), where x is an object of X,andϕ an automorphism of x, and AutX(A) is the sheaf of automorphisms of the sheafof algebras A over X. To construct (10.1), consider the stack of sheaves of algebrasAlg over S, which has as objects over the scheme S, the sheaves of OS -algebras on theusual (small) étale site of S. A morphism from the sheaf of OS′ -algebras A′ over S′,covering the morphism of schemes f : S′ → S, to the sheaf of OS -algebras A over S,is, by definition, an isomorphism of sheaves of OS′ -algebras A′ → f∗A. The sheaf ofalgebras A→ X gives rise to a morphism of S-stacks a : X → Alg. We get an inducedmorphism on inertia stacks IX → IAlg, and notice that a∗IAlg =AutX(A).With this definition, an automorphism ϕ of the object x of the stack X is mappedto the inverse of the restriction morphism ϕ∗ :A(x)→A(x).67Lemma 10.2. Suppose X is a gerbe over the algebraic space S, and A→X is an algebra.Then there exists a sheaf of OY -algebras B, and an isomorphism A› B|X if and only ifthe inertia representation IX →AutX(A) is trivial.If this is the case, then A is representable or of finite type if and only if B is. We can pull back the sheaf of algebras A over X, via the structure morphismIX →X, to obtain the sheaf of algebras A|IX . This sheaf of algebras is endowed with atautological automorphism, induced from (10.1). In fact, the morphism 10.1 inducesover each object x ∈ X an action of Aut(x) on Ax . Therefore the objects of AIX overan object x of X, are triples (x,ϕ,a) where ϕ is an automorphism of x, and a ∈ Axis an object of A lying over x. The tautological automorphism of AIX maps (x,ϕ,a)to (x,ϕ,ϕ(a)). The algebra of invariants for this automorphism consists of objects(ϕ,a) such that ϕ(a)= a. We shall denote this algebra by AfixIX .The following statement is somewhat tautological, and holds more generally thanfor algebras.Proposition 10.3. Suppose that A is a representable algebra over the algebraic stackX. Then the inertia stack of A is naturally identified with AfixIX . In particular, IA is arepresentable algebra over IX .Proof. We have a commutative diagram of algebraic stacksIA //AIX // Xwhich identifies IA with a substack of A|IX . In fact, the factorization IA → A|IX is amonomorphism if and only if A→X is representable [46, Tag 04YY]. The algebra A|IXis the stack of triples (x,ϕ,a), where x is an object of X, ϕ is an automorphism ofx, and a ∈ A(x) is an object of A lying over x. Such a triple is in IA, if and only ifϕ∈Aut(x) is in the subgroup Aut(a)⊂Aut(x). This is equivalent toϕ fixing a underthe action of Aut(x) on A(x). This is the claim. In fact, the fibre of IA over the objects x of X is equal toIA(x)= {(ϕ,a)∈Aut(x)×A(x) |ϕ∗(a)= a} .The fibre of IA(x) over ϕ ∈ Aut(x) is the subalgebra A(x)ϕ ⊂ A(x), and the fibre ofIA(x) over a∈A(x) is the subgroup StabAut(x)(a)⊂Aut(x).68Algebra bundlesDefinition 10.4. We call a finite type algebra A→X an algebra bundle, if the underly-ing OX -module is locally free (necessarily of finite rank).Lemma 10.5. Let A→ X be a finite type algebra over the algebraic stack X. There is anon-empty open substack U ⊂X, such that A|U is an algebra bundle.Proof. The claim is true for the scheme case, so by considering a smooth presentationof X, we can show that there exists a non-empty scheme V , together with a smoothmorphism V → X, such that A|V is an algebra bundle. The image U of V in X is anopen substack, and A|U is an algebra bundle, because local freeness is local in thesmooth topology. Remark 10.6. By considering the representation of A on itself by left multiplication,we see that every algebra bundle is a sheaf of subalgebras of the algebra End(V) ofendomorphisms of a vector bundle V over the stack X.Central idempotentsLemma 10.7. The centre of a finite type algebra is a finite type algebra.Proof. ByHom(E∨,Hom(E,O))=Hom(E∨⊗E,O) ,the endomorphism sheaf of a reflexive sheaf is reflexive, and therefore the algebra ofO-linear endomorphisms of a finite type algebra is of finite type.The centre of A is the kernel of the O-linear homomorphism of representablecoherent sheaves A → EndO(A), given by a , [a,·]. As such, it is a representablecoherent sheaf itself. For a representable algebra A over X, we denote the closed substack of idempo-tents in A by E(A).Lemma 10.8. Suppose A→X is a commutative finite type algebra. The stack E(A)→Xof idempotents in A is unramified over X.Proof. The claim is local in the smooth topology, so we may assume that X is ascheme. It is then sufficient to prove the claim over a stratification of X, so we mayassume that A is an algebra bundle, and hence a subalgebra of GL(V), for a vectorbundle V over X, which we may as well assume is trivial, of rank n.69We use the formal criterion. So let T ⊂ T ′ be a square zero extension of affineschemes, with ideal I, and e,f two n×n matrices with entries in O(T ′), which agreeon T . Hence the entries of e−f are in I, which implies that (e−f)2 = 0. Therefore, wehave0= e(e−f)2 = e(e−2ef +f)= e−2ef +ef = e−ef .So we have e = ef , and by symmetry also f = fe, and as e and f commute, we havee= f . Corollary 10.9. If A→ X is a commutative finite type algebra, and X is reduced, thereis a non-empty open substack U of X, such that E(A|U )→U if finite étale.Proof. First we use generic flatness, and the fact that a flat and unramified morphismsis necessarily étale, to prove that E(A)→ X is generically étale. Then we use Zariski’smain theorem to prove that E(A)→X is generically finite. By this corollary, when studying the centre of finite type algebras over the finitetype stack X, we may, after passing to a locally closed stratificationof X, assume thatthe stack of central idempotents is finite étale over X.Also note that a finite étale morphism to the stack X is locally trivial in the smoothtopology.Recall that a non-zero idempotent e is called primitive, if e= e1+e2, for orthogonalidempotents e1, e2, implies that e1 = 0 or e2 = 0.In a commutative algebra, the following is true(i) every idempotent is in a unique way (up to order of the summands) a sum ofprimitive idempotents, this is the primitive decomposition,(ii) orthogonal idempotents have disjoint primitive decompositions,(iii) distinct primitive idempotents are orthogonal to each other,(iv) the primitive idempotents add up to 1.Definition 10.10. Assume that the commutative finite type algebra A has finite étalestack of idempotents E(A)→X. An idempotent local section of A→X is primitive, ifit is a primitive idempotent locally in the smooth topology.Lemma 10.11. Let A→ X be a commutative finite type algebra, with finite étale stackof idempotents E(A)→ X. There is an open and closed substack PE(A) ⊂ E(A), suchthat an idempotent factors through PE(A) if and only if it is primitive.Proof. We may assume that I(A)→ X is constant. Then the multiplication operationand the partially defined addition operation on E(A) are also constant. The claimfollows. 70Definition 10.12. Let A→ X be a finite type algebra, with centre Z → X. Let ZE(A)be the stack of idempotents in Z , in other words the stack of central idempotents inA. Assume that ZE(A)→ X is finite étale. The substack of primitive idempotents inZE(A) is denoted by PZE(A), and called the stack of primitive central idempotentsof A. It is finite étale over X. The degree of PZE(A)→ X is called the central rank ofA.If X is smooth and connected, the number of connected components of PZE(A)is the split central rank of A. More precisely, the partition of the central rank givenby the connected components of PZE(A) is called the central type of A. (So the splitcentral rank is the length of the type.)Remark 10.13. Let X be smooth and connected, and let A→X be a commutative finitetype algebra, with finite étale stack of idempotents E(A)→ X. Then there is a one-to-one correspondence between the connected components of PE(A) and the primitiveidempotents in the algebra of global sections Γ(X,A).The semisimple centreFor a finite type algebra over a field, we have(i) the primitive idempotents are linearly independent,(ii) an element is semisimple if and only if it is a linear combination of primitiveidempotents.We need a version of this statement for finite type algebras over stacks.Let A→X be a commutative finite type algebra whose stack of idempotents E(A) isfinite étale. Consider the finite étale cover of primitive idempotents pi : PE(A)→X. Wehave a tautological global section e of A|PE(A), and a, ae defines a homomorphismof OPE(A)-modules OPE(A)→A|PE(A). Pushing forward with pi and composing with thetrace map pi∗(A|PE(A))→A defines the morphism of algebras over Xpi∗OPE(A) -→A. (10.2)In fact, (10.2) is an isomorphism onto Ass , the subalgebra of semisimple elements inA.If we drop the assumption that A is commutative, we get a canonical embeddingof algebraspi∗OPZE(A) -→A, (10.3)whose image is the semisimple centre Z(A)ss .71Permanence of rank and split rankProposition 10.14. LetA↩A′ be a monomorphism of commutative finite type algebraswith finite étale stacks of idemtpotents over the smooth and connected stack X. Denotethe ranks of A and A′ by n and n′, and the split ranks by k and k′, repsectively. Thenn≤n′ and k≤ k′. Morover,(i) if A′ admits a semisimple global section, which is not in A, then n<n′,(ii) if A′ admits an idempotent global section, which is not in A, then k < k′.Proof. The embedding A↩ A′ induces an embedding of finite étale X-stacks E(A)↩E(A′). Every idempotent e in A can be decomposed uniquely into a sum of orthogonalprimitive idempotents in A′. Let us call this the primitive decomposition of e in A′.Consider the correspondence Q⊂ PE(A)×X PE(A′) defined by(e,e′)∈Q ⇐⇒ e′ partakes in the primitive decomposition of e in A′ .One shows thatQ is a finite étale cover of X locally in the étale topology of X, reducingto the case where both E(A) and E(A′) are trivial covers. By properties of the primitivedecomposition, the projectionQ→ PE(A) is surjective, and the projectionQ→ PE(A′)is injective. Thus we haven= degPE(A)≤ degQ≤ degPE(A′)=n.If n = n′, then both Q → PE(A) and Q → PE(A′) are isomorphisms, showing thatPE(A)= PE(A′), and hence Ass = (A′)ss . This proves (i).We can repeat the argument for the algebras of global sections Γ(X,A)↩ Γ(X,A′).We deduce that k ≤ k′, and if k = k′, every primitive idempotent in Γ(X,A) remainsprimitive in Γ(X,A′), and every primitive idempotent of Γ(X,A′) is in Γ(X,A). Wededuce that Γ(X,A) and Γ(X,A′) have the same idempotents, which proves (ii). Families of idempotentsDefinition 10.15. For a finite type algebra A→ X, we denote by En(A)→ X the stackof n-tuples of non-zero idempotents in A, which are pairwise orthogonal, and add upto unity. We call sections of En(A) also complete sets of orthogonal idempotents.Note that the family members of sections of En(A) need not be central.The stack En(A) is algebraic, and of finite type over X. For n= 0, the stack E0(A)is empty, unless A= 0, in which case it is identified with X. For n= 1, the stack E1(A)contains exactly the unit in A (so is identified with X), unless A = 0, in which caseE1(A) is empty.7210.2 AlgebroidsDefinition 10.16. An algebroid is a triple (X,A,ι), where X is an algebraic stack, A isa finite type algebra over X, and ι : IX → A× is an isomorphism of sheaves of groupsover X, such that the natural diagramIXι //##A×AutX(A)(10.4)of sheaves of groups over X commutes.The vertical map A× → AutX(A) associates toa unit u of A the inner automorphism x , uxu−1. The diagonal map IX → AutX(A)is the inertia representation (10.1). We sometimes refer to the commutativity of (10.4)as the algebroid condition.Thus, if (X,A) is an algebroid, the inertia representation acts through inner automor-phisms on A.We will usually abbreviate the triple (X,A,ι) to X, and write AX for A, if we needto specify the algbra.Example 12. Let X be a linear algebraic stack with underlying algebraic stack X, andlet A→ X be the universal sheaf of endomorphisms of Remark 9.3. Then automor-phisms are invertible endomorphisms, so we use for ι : IX → A× the tautological em-bedding. The inertia representation being the inverse of the pullback action, it is,indeed, given by (left) inner automorphisms.We call (X,A) the algebroid underlying the linear algebraic stack X.Example 13. As toy example, consider an R-algebra scheme of finite type A, withgroup scheme of units A×, as in Example 11. Let A× act on A from the left by innerautomorphisms. Then A×\A is an a finite type algebra over the stack of right A×-torsors BA×. The inertia stack of BA× is the quotient stack A×\A×, where A× actson A× from the left by inner automorphisms. The morphism ι :A×\A×→A×\A is theinclusion map.Example 14. Every scheme Z is an algebroid via the definition AZ = 0Z . There is nonatural way to enhance the algebroid (Z,0Z) to a linear algebraic stack. This exhibitsone way in which algebroids are more flexible than linear algebraic stacks.73Example 15. Consider the linear stack of vector bundles Vect, Example 9. The under-lying algebroid consists of the disjoint union of the quotient stacks GLn \M(n×n),given by the adjoint representations M(n×n), for n ≥ 0. Thus, in passing from thelinear stack to the underlying algebroid, we discard all M(m×n), for m 6= n, and form = n, restrict the left-right bi-action of GLn on M(n×n) to the (left only) adjointaction. Thus we remove exactly the information which we consider non-local, see Re-mark 9.4.Example 16. If X→ SpecR is a gerbe, any algebroid over X can be promoted to a linearalgebraic stack, whose underlying algebraic stack is X.Definition 10.17. A morphism of algebroids X → Y is a pair (f ,ϕ), where f : X → Yis a morphism of algebraic stacks, and ϕ :AX →AY is a morphism of algebras over f ,such that the diagramIXι //IfA×XϕIYι // A×Ycommutes.Definition 10.18. A morphism (f ,ϕ) of algebroids is representable, if f : X → Y is arepresentable morphism of algebraic stacks, and ϕ :AX → f∗AY is a monomorphismof sheaves of algebras over X.There is a natural notion of 2-morphism of algebroid, which makes algebroids intoa 2-category. Representable algebroids over a fixed algebroid form a 1-category.Remark 10.19. Suppose (X,A) is an algebroid, f : Y →X is a representable morphismof algebraic stacks, B→ Y is a finite type algebra over Y , and ϕ : B→A is a morphismof algebras, covering f , such that B → f∗A is a monomorphism. If the image of thecomposition IY → IX |Y → A×|Y is equal to the image of B× → A×|Y , then the inducedisomorphism IY → B× makes (Y ,B) into an algebroid, and (f ,ϕ) into a representablemorphism of algebroids. (The algebroid condition on (X,B) is automatic.)Commutative algebroidsProposition 10.20. If X is a gerbe over the algebraic space S, and A→ X is a com-mutative algebroid over X, then the algebra A descends to a finite type algebra overS.Proof. This follows from the fact that the inertia representation is trivial, being givenby inner automorphisms, and Lemma 10.2. 74Fibered productsLet (X,AX)→ (Z,AZ) and (Y ,AY )→ (Z,AZ) be morphisms of algebroids. The fiberedproduct of (X,AX) and (Y ,AY ) over (Z,AZ) has as underlying algebraic stack thefibered productW =X×Z Y . The algebraAW is given by the fibered product of algebrasover WAW =AX |W ×AZ |W AY |W .This fibered product (W,AW ) has a natural universal mapping property. If (X,AX),(Y ,AY ) and (Z,AZ) are the algebroids underlying linear algebraic stacks X, Y and Z,respectively, then the algebroid (W,AW ) is the algebroid underlying the linear alge-braic stack W=X×ZY.Pullback algebroidA morphism of algebraic stacks f : Y → X is called inertia preserving if IY = IX |Y .Examples of inertia preserving morphispms include monomorphisms, and projectionsZ×X → X, for algebraic spaces Z , as well as pullback of gerbes via morphisms to thecoarse moduli space. Inertia preserving morphisms are necessarily representable.If (X,AX) is an algebroid, and f : Y → X is inertia preserving, then (Y ,f∗AX) isan algebroid, called the pullback of the algebroid AX via f . The pullback (Y ,f∗AX)comes wtih a representable structure morphism to (X,AX).Definition 10.21. A closed immersion of algebroids (Y ,AY )→ (X,AX) is a morphismwhere Y →X is a closed immersion of algebraic stacks, such that AY =AX |Y .10.2.1 Algebroid inertiaA key observation is that if (X,A) is an algebroid, then (IX , IA) is another algebroid.In fact, by Proposition 10.3, IA → IX makes IA into a finite type algebra over IX , andthe algebra IA(x,ϕ) lying over the object (x,ϕ) of IX is equal to the fixed algebraA(x)ϕ. The fibre of I(IX)→ IX over (x,ϕ) is the centralizer in Aut(x) of ϕ, whichwe shall denote by Aut(x)ϕ. The isomorphism ι(x) : Aut(x)→ A(x)× restricts to anisomorphismAut(x)ϕ -→ (A(x)×)ϕ = (A(x)ϕ)× ,and therefore we get an induced isomorphism I(IX)→ (IA)× of groups over IX . Onechecks that (IX , IA) inherits the commutativity of (10.4) from (X,A). The algebroid(IX , IA) comes with a natural representable morphism to (X,A).There is also a semisimple version.It is given by (IssX , IssA ), and it is isomorphic tothe pullback of (IX , IA) via the inertia preserving morphism IssX → IX . The fibre of IssAover the object x of X isIssA (x)= {(ϕ,a)∈Aut(x)×A(x) |ϕ∗(a)= a and ϕ is semisimple} .75Definition 10.22. We call (IX , IA) the inertia algebroid of the algebroid (X,A), nota-tion I(X,A). Also, (IssX , IssA ) is called the semisimple inertia of the algebroid (X,A),notation Iss(X,A).Proposition 10.23. Let (X,A) be an algebroid. Then En(A) is the base of a canonicalalgebroid, mapping to X with a representable morphism of algebroids.Proof. The pullback ofA via the structure morphism En(A)→X is an algebra endowedwith an n-tuple of idempotents e1, . . . ,en . We define Afix to be the subalgebra ofA|En(A) commuting with e1, . . . ,en. The inertia stack of En(A) is the subgroup stack ofIX |En(A) consisting of automorphisms which stabilize each of the ei, under the inertiarepresentation on A. If we map it via ι into A×|En(A), we obtain exactly (A×)fix, by thealgebroid property of A, applied to the idempotents e1, . . . ,en. By Remark 10.19, thisis enough to assure that(En(A),Afix)is an algebroid. 10.2.2 Clear algebroidsSuppose that (X,A) is an algebroid, and that X is a gerbe over the scheme X. Thenthe centre Z(A) descends to a commutative finite type algebra over X, by Proposi-tion 10.20. Note that this is analogue to Proposition 3.1 in the case of bands of gerbes.Definition 10.24. We call an algebroid (X,A) clear, if(i) X is a gerbe with coarse moduli space X,(ii) the space X is a non-singular variety,(iii) A and Z(A) are algebra bundles over X,(iv) ZE(A)→X is finite étale.For a clear algebroid, ZE(A) and PZE(A) descend to a finite étale X-schemes. Thedefinition of central rank, split central rank, and central type apply to clear algebroids.For every algebroid (X,A), over a finite type algebraic stack X, there exists a strat-ification of X, such that the restricted algebroids over the pieces of the stratificationare all clear. This follows from Lemma 10.5 and Corollary 10.9.76Chapter 11K-algebra of stack functions11.1 Stack functionsLet M be a linear stack, and A→M its universal endomorphism algebra. In Example 12we showed that (M,A) is an algebroid. A main example to have in mind is the linearstack CohX of Example 8.Definition 11.1. A stack function is a representable morphism of algebroids (X,A)→(M,A), such that X is of finite type.Recall Remark 10.19, which says that to prove that a given (X,A)→ (M,A) is a stackfunction, we do not need to check the algebroid condition on (X,A).The K-algebra of M, notation K(M), is the free Q-vector space on (isomorphismclasses of) stack functions, modulo the scissor relations relative M. The class in K(M)defined by a stack function X→M will be denoted [X→M]. A scissor relation relativeM is[X →M]= [Z →X →M]+ [X Z →X →M],for any closed immersion of algebroids Z ↩ X, and any stack function X →M. Thesubstacks Z and X Z are endowed with their respective pullback algebroids.Example 17. Consider the linear stack Vect. Stack functions are triples (X,V ,A),where X is a finite type algebraic stack, V is a vector bundle over X, and A⊂ EndX(V)is a subalgebra, such that the inertia representation IX → AutX(V) identifies IX withA× ⊂AutX(V).Example 18. As a trivial example, consider the case where M is the stack of zerodimensional vector bundles. In this case the underlying algebroid of M is SpecR,77endowed with the 0-algebra. Stack functions are the same as algebraic spaces over R.We denote the corresponding K-algebra by K(Var). Fibered product over SpecR makesK(Var) into a Q-agebra.The vector space K(M) is a commutative ring with the multiplication[X →M] · [Y →M]= [X×Y →M×M ⊕-→M].Also a K(Var)-module structure on K(M) is given by [Z]·[X→M]= [Z×X→X→M].This turns K(M) into a K(Var)-algebra. The additive zero in K(M) is given by theempty algebroid 0= [œ→M] and the multiplicative unit is 1= [SpecR 0-→M].11.2 The filtration by split central rankDefinition 11.2. We introduce the filtration by split central rank K≥k(M) on K(M),by declaring K≥k(M) to be generated as a Q-vector space by stack functions [X →M], where X is a clear algebroid (Definition 10.24) such that AX admits k orthogonalcentral non-zero idempotents (globally).Alternatively, K≥k(M) is generated by [X→M], where X is a clear algebraoid suchthat PZE(AX) has at least k components.Remark 11.3. Trying to define a direct sum decomposition of K(M) by split centralrank would not work, because a clear algebroid X of split central rank k may very welladmit a closed substack Z ⊂ X whose restricted algebroid is again clear, but of splitcentral rank larger than k.The 0-ring has no non-zero central idempotents, but any non-zero ring has at leastone. Therefore, K(Var)⊂ K(M) is a complement for K>0(M) in K(M)= K≥0(M), i.e.,K(M)=K(Var)⊕K>0(M). In particular, we haveK≥0(M)/K>0(M)=K(Var).11.3 The idempotent operators ErLet Er denote the operator on K(M) which maps a stack function [X→M] to [Er (X)→X →M], where Er (X)= Er (AX) is the stack of r -tuples of non-zero orthogonal idem-potents adding to unity in AX , see Definition 10.15. The algebroid structure on Er (X)is described in Proposition 10.23.This definition applies also to r = 0. The stack E0(X) is empty if AX 6= 0, andE0(X)= X, if X is a scheme. Hence E0 is diagonalizable, and has eigenvalues 0 and 1.78The kernel (0-eigenspaces) is K>0(M) ⊂ K(M), the image (1-eigenspaces) is K(Var) ⊂K(M).For r = 1, the operator E1 vanishes on stack functions [X →M], where X is analgebraic space, and acts as identity on stack functions for which AX 6= 0. Hence,E1 is also diagonalizable with eigenvalues 0, and 1. The kernel of E1 is K(Var), andthe image is K>0(M). Hence E0 and E1 are complementary idempotent operators onK(M), i.e., they are orthogonal to each other and add up to the identity.Recall the Stirling number of the second kind, S(k,r), which is defined in such away that r !S(k,r) is the number of surjections from k to r .Theorem 11.4. The operators Er , for all r ≥ 0, preserve the filtration K≥k(M) by splitcentral rank. On the quotient K≥k(M)/K>k(M), the operator Er acts as multiplicationby r !S(k,r).Proof. Consider a clear algebroid (X,A) with a morphism X →M defining the stackfunction [X→M] in K(M). Let n be the central rank of X, and k the split central rankof X. The filtered piece K≥k(M) is generated by such [X →M].Denote by X → X the coarse space of X, which is a non-singular variety, by as-sumption.Let X˜ → X be a connected Galois cover with Galois group Γ , which trivializesPZE(A) → X. As PZE(A) descends to X, this Galois cover can be constsructed asa pullback from the non-singular variety X. Therefore, the morphism X˜ →X is inertiapreserving and hence X˜ inherits, via pullback, the structure of an algebroid, and hence[X˜ →X →M] is a stack function.Recall that the degree of the cover PZE(A)→ X is n, and the number of compo-nents of PZE(A) is k.By labelling the components of the pullback of PZE(A) to X˜, we obtain an actionof Γ on the set n= {1, . . . ,n} and an isomorphism of finite étale covers of XX˜×Γ n '-→ PZE(A)[x,ν] 7 -→ e[x,ν] .Both source and target of this isomorphism support natural algebroids and the iso-morphism preserves them.The number of orbits of Γ on n is k.Then we also have an isomorphismX˜×Γ Epi(n,r) '-→ ZEr (A)[x,ϕ] 7 -→( ∑ϕ(ν)=ρe[x,ν])ρ=1,...,r ,where ZEr denotes the stack of labelled complete sets of r orthogonal central idem-79potents. Again, both stacks involved are in fact algebroids, and this isomorphism isan isomorphism of algebroids.Hence, we may calculate as follows (all stacks involved are endowed with theirnatural algebroid structures):ZEr [X →M]= [X˜×Γ Epi(n,r)→M]= [X˜×Γ unionsqϕ∈Epi(n,r)/Γ Γ/StabΓϕ→M]=∑ϕ∈Epi(n,r)/Γ[X˜/StabΓϕ→M]=∑ϕ∈Epi(n,r)Γ[X →M]+∑ϕ∈Epi(n,r)/ΓStabΓ ϕ 6=Γ[X˜/StabΓϕ→M].Now, we have Epi(n,r)Γ = Epi(n/Γ ,r ), and hence#Epi(n,r)Γ = r !S(|λ|,r )= r !S(k,r).Thus, we conclude,ZEr [X →M]= r !S(k,r)[X →M]+∑ϕ∈Epi(n,r)/ΓStabΓ ϕ 6=Γ[X˜/StabΓϕ→M].For any proper subgroup Γ ′ ⊂ Γ , the quotient X′ = X˜/Γ ′ is an intermediate cover X˜ →X′→X, such that X′ 6=X. The pullback of PZE(A) to X′ has more than k components,because the number of orbits of Γ ′ on n is larger than k. Thus we have proved thetheorem for ZEr , instead of Er .Now observe that ZEr (A) ⊂ Er (A) is a closed substack, because ZEr (A)→ X isproper and Er (A)→X is separated. So we can writeEr [X →M]= ZEr [X →M]+ [NZEr (A)→X →M],where NZEr (A) is the complement of ZEr (A) in Er (A). To prove that [NZEr (A)→M]∈K>k(M), let Y ↩NZEr (A) be a locally closed embedding, such that the algebroid(Er (A),Afix)|Y is clear.Consider the embedding of algebras Afix|Y ↩ A|Y . It induces an embedding ofcommutative algebras Z(A|Y )↩ Z(Afix|Y ), because Z(A|Y )⊂Afix|Y . The algebra A|Ycomes with r tautological idempotent sections, all of which are contained in Z(Afix|Y ),but at least one of which is not contained in Z(A|Y ). So by Proposition 10.14 (ii), thesplit central rank of Afix|Y is strictly larger than the split central rank of A|Y . The latteris at least as big as k, the split central rank of A, because the split central rank cannotdecrease under base extension. This shows that [Y →M] ∈ K>k(M) and finishes the80proof. Corollary 11.5. The operators Er , for r ≥ 0 are simultaneously diagonalizable. Thecommon eigenspaces form a family Kk(M) of subspaces of K(M) indexed by non-negative integers k≥ 0, andK(M)=⊕k≥0Kk(M). (11.1)Let pik denote the projection onto Kk(M). We haveErpik = r !S(k,r)pik ,for all r ≥ 0, k≥ 0.Proof. First remark that for given r , the numbers r !S(k,r) form a monotone increas-ing sequence of integers.Then note that the operators Er pairwise commute: the composition Er ◦Er ′ as-sociates to an algebroid (X,A) the stack of pairs (e,e′), where both e and e′ are com-plete families of non-zero orthogonal idempotents in A, the length of e being r andthe length of e′ being r ′, and the members of e commuting with the members of e′.Finally, let us prove that, for every k and every r , the Q-vector space K≥k(M) is aunion of finite-dimensional subspaces invariant by Er .For this, define K(M)≤N to be generated asQ-vector space by stack functions [X→M], where X is a clear algebroid, such that the rank of the vector bundle underlyingthe algebra AX → X is bounded above by N . This is an ascending filtration of K(M),which is preserved by Er . SetK≥k(M)∩K(M)≤N =K≥k(M)≤N .Suppose x = [X →M] is a stack function with X a clear algebroid of split centralrank k, and let N be the rank of the vector bundle underlying AX . Note that k ≤ N ,because for a commutative algebra, the number of primitive idempotents is boundedby the rank of the underlying vector bundle. We deduce that for k > N , we haveK≥k(M)≤N = 0.On the other hand, Theorem 11.4 implies by induction thatEir (x)∈Qx+QEr (x)+ . . .+QEi−1r (x)+K≥k+i(M).Applying this for i=N−k+1, we see thatEr(EN−kr (x))∈Qx+QEr (x)+ . . .+QEN−kr (x),and hence that Qx+QEr (x)+ . . .+QEN−kr (x) is invariant under Er .81This proves that any x ∈ K≥k(M) is contained in a finite-dimensional subspaceinvariant under Er . Standard techniques from finite-dimensional linear algebra nowimply the result. Remark 11.6. The proof of Theorem 11.4 and its corollary show that the central ver-sions ZEr of the Er are also diagonalizable. On the other hand, the ZEr do not com-mute with each other, and so are less useful.Corollary 11.7. For r ≥ 1, we havekerEr =⊕k<rKk(M).In particular, for any element x, we have Erx = 0, for r  0.Corollary 11.8. For every k≥ 0, we havepik =∞∑r=ks(r ,k)r !Er ,where the s(n,k) are the Stirling numbers of the first kind. In particular, pi0 = E0, andpi1 =∞∑r=1(−1)r+1rEr .Proof. We haveid=∑`≥0pi` ,and henceEr =∑`≥0Erpi` =∑`≥0r !S(`,r)pi` ,and therefore∑r≥0s(r ,k)r !Er =∑r≥0s(r ,k)r !∑`≥0r !S(`,r)pi`=∑`≥0( ∑r≥0S(`,r)s(r ,k))pi` =∑`≥0δ`,kpi` =pik ,by the inverse relationship between the Stirling numbers of the first and second kind. 82Remark 11.9. The Stirling numbers of the first kind appear in the Taylor expansionsof the powers of the logarithm:∞∑r=ks(r ,k)r !tr = log(1+ t)k .Example 19. The universal rank 2 vector bundle GL2\A2 → BGL2, and its classifyingmorphism to Vect define a Hall algebra element [BGL2 → Vect] ∈ K(Vect), which wewill abbreviate to [BGL2]. To decompose [BGL2] into its pieces according to (11.1),we consider the action of E2, as we have Er [BGL2]= 0, for all r > 2. In fact,E2[BGL2]= [BT], and E2[BT]= 2[BT],where T is a maximal torus in GL2. Thus Q[BGL2]+Q[BT] is a subspace of K(Vect)invariant under E2, and the matrix of E2 acting on this subspace is0 01 2 . (11.2)This matrix is lower triangular, with different numbers on the diagonal, hence di-agonalizable over Q. In fact, the diagonal entries are 2S(1,2) = 0 and 2S(2,2) = 2.Diagonalizing (11.2) gives the eigenvectors(i) v1 = [BGL2]− 12 [BT] with eigenvalue 0,(ii) v2 = 12 [BT] with eigenvalue 2.Therefore, we have v1 ∈ K1(Vect) and v2 ∈ K2(Vect), and since [BGL2]= v1+v2, wehave found the required decomposition of [BGL2].11.4 The spectrum of semisimple inertiaThe semisimple inertia operator on K(M) is the Q-linear endomorphismIss :K(M) -→K(M)[X →M] 7 -→ [IssX →X →M].Here IssX denotes the semisimple algebroid inertia of Definition 10.22. In fact, Iss islinear over K(Var). We will use as scalars the subring Q[q] ⊂ K(Var), and extendscalars to the quotient field Q(q). Thus, we will consider Iss as a Q(q)-linear operatorIss :K(M)(q) -→K(M)(q),where K(M)(q)=K(M)⊗Q[q]Q(q).83Recall the definition of Qλ for a partition λ`n as in §7.1. This is a polynomial inq, of degree n, which vanishes to order |λ| at q= 1. Recall the well-ordering on the setof all partitions of all integers introduced in §7.4.Theorem 11.10. The operator Iss : K(M)(q)→ K(M)(q) is diagonalizable. Its eigen-value spectrum consists of the Qλ ∈Q(q), for all paritions λ. Denote the eigenspacecorresponding to the eigenvalue Qλ by Kλ(M)(q). We haveKk(M)(q)=⊕|λ|=kKλ(M)(q),where Kk(M)(q)=Kk(M)⊗Q[q]Q(q). Thus, the decomposition of K(M)(q) accordingto eigenspaces of Iss refines the decomposition according to eigenspaces of the familyof operators (Er ).Proof. We define a clear stack function to be a stack function X →M for which X is aclear algebroid.Then define a decreasing filtration K≥λ(M)(q) indexed by partitions, by declaringK≥λ(M)(q) to be generated by clear stack functions whose central type is ≥ λ. We willprove(i) the operator Iss preserves the filtration by partitions,(ii) on the quotient K≥λ(M)(q)/K>λ(M)(q), the operator Iss acts as multiplicationby Qλ.(iii) the operator Iss is locally finite.This will prove the claims concerning diagonalizability of Iss .Consider a clear stack function X →M, of central rank n, central type λ` n, andsplit central rank k = |λ|. As in the proof of Theorem 11.4, let X˜ → X be a connectedGalois cover with Galois group Γ , acting on the set n, such thatX˜×Γ n '-→ PZE(AX).We get an induced isomorphismX˜×Γ An '-→ Z(AX)ssonto the semisimple centre of AX , by reformulating (10.3). It follows that we have anisomorphismX˜×Γ Gnm '-→ ZIss(X)onto the semisimple central inertia of X. A computation similar to that of §7.1 wouldnow show that84ZIss[X →M]=Q(λ)[X →M]+∑I•∈F(n)/ΓStabΓ (I)⊊Γ(−1)`(I•)q|Imax|[X˜/StabΓ (I)→M].Recall that F(n) is the set of all flags I• = Ik ⊊ . . . ⊊ I1 ⊊ I0 = n in n. The length of aflag I• is denoted by k= `(I•), and the maximal index is k=max.Note that the cover X˜→X, as well as all intermediate covers X˜→ X˜/Γ ′→X, for anysubgroup Γ ′ ⊂ Γ , come via base extension from covers of the variety X, and are there-fore endowed with canonical structures of algebroids over M, and define elements ofK(M)(q), as in the proof of Theorem 11.4.As in the proof of Theorem 11.4, all stack functions [X˜/Γ ′ →M], for Γ ′ ⊊ Γ arecontained in K>k(M)(q), hence in K>λ(M)(q) by the first property of our partitionordering.Let us now consider Iss , instead of ZIss . We have a closed immersion of algebroidsZIssX ↩ IssX. Thus we can writeIss[X →M]= ZIss[X →M]+ [NZIss(X)→X →M].Now consider a locally closed embedding Y ↩ NZIss(X), such that Y is a clear alge-broid. Over Y , we have the inclusion of algebras Afix|Y ↩A|Y , inducing an embeddingof the centres in the opposite direction Z(A)|Y ↩ Z(Afix)|Y . There is one semisimplesection, namely the tautological one, which is in A(Afix)|Y , but not in Z(A)|Y , and soby Proposotion 10.14 (i), we have that the central rank of Y is larger than n. By thesecond property of our partition ordering, we have therefore [Y →M]∈K>λ(M)This proves the first two claims we made about Iss . To prove local finiteness ofIss , proceed as in the proof of Corollary 11.5. Every time we apply Iss either the centralrank or the split central rank will increase, but their sum can be bounded in terms ofthe dimension.We have now proved that Iss is diagonalizable. Next, note that the Er are linearover K(Var), and hence also induceQ(q)-linear endomoprhism of K(M)(q). Moreover,Iss commutes with Er , for every r . Both compositions Er ◦ Iss and Iss ◦Er associate toan algebroid (X,A) the stack of pairs (a,e), where a is a semisimple unit in A, and e alabelled complete set of r orthogonal idempotents, all commuting with a.Therefore, Er preserves the eigenspace Kλ(M)(q) of the Iss . AsK≥|λ| ⊃K≥λ ⊃K>λ ⊃K>|λ|it follows from Theorem 11.4, that Er acts on Kλ(M)(q) by scalar multiplication by85r !S(|λ|,r ), and hence that Kλ(M)(q)⊂K|λ|(M)(q). Example 20. Let us continue with Example 19. The stack function [BGL2] is clear, itscentral rank is 1. We haveIss[BGL2]= [∆/GL2]+ [T∗/N]= (q−1)[GL2]+ [T∗/N],where ∆ is the central torus of GL2, and T∗ = T ∆. Also, N is the normalizer of Tin GL2. Moreover, [∆/GL2]=Gm× [BGL2] is a clear stack function of central rank 1,and [T∗/N] is a clear stack function of central rank 2, and split central rank 1.Applying Iss to [T∗/N], we getIss[T∗/N]= (q2−1)[T∗/N]− (q−1)[T∗/T]= (q2−1)[T∗/N]− (q−1)[T∗][BT]= (q2−1)[T∗/N]− (q−1)2(q−2)[BT].Finally, [BT] is a clear stack function of central rank 2 and split central rank 2. It is aneigenvector for Iss :Iss[BT]= [T/T]= [T][BT]= (q−1)2[BT].We see that Q(q)[BGL2]+Q(q)[T∗/N]+Q(q)[BT] is invariant under Iss , and thematrix of Iss on this subspace isq−1 0 01 q2−1 00 −(q−1)2(q−2) (q−1)2This matrix is lower trianguler, with distinct scalars on the diagonal, and is thereforediagonalizable over Q(q). Diagonalizing, we get the following eigenvectors(i) v(1) = [BGL2]− 1q(q−1) [T∗/N]− 1q [BT],(ii) v(2) = 1q(q−1) [T∗/N]− q−22q [BT],(iii) v(1,1) = 12 [BT],where v(1) ∈ K(1)(Vect)(q), v(2) ∈ K(2)(Vect)(q) and v(1,1) ∈ K(1,1)(Vect)(q). More-over, [BGL2] = v(1)+v(2)+v(1,1), and this is the spectral decomposition of [BGL2]into eigenvectors of Iss . Of course, v(1)+v(2) = v1 and v(1,1) = v2.8611.5 Graded structure of multiplicationProposition 11.11. For x,y ∈K(M), we haveIss(x ·y)= Iss(x) · Iss(y).Proof. This follows immediately from the fact that, for any two algebroids X, Y , wehave Iss(X×Y)= Iss(X)× Iss(Y), as algebroids over X×Y . Denote the disjoint union of two partitions λ and µ by λ+µ.Corollary 11.12. We have Kλ(M)(q) ·Kµ ⊂ Kλ+µ(M)(q), and hence also Kk(M)(q) ·K`(M)(q)⊂Kk+`(M)(q).Remark 11.13. So the Q(q)-vector spaceK(M)(q)=⊕k≥0Kk(M)(q)is a graded Q(q)-algebra, with respect to the commutative product on K(M)(q). Onecan show that this fact is true for K(M) itself, without tensoring with Q(q). In otherwords if x ∈Kk(M) and y ∈K`(M), then x.y ∈Kk+`(M).87Chapter 12Hall Algebra of algebroidsLet M be a linear stack, and A→M be its universal endomorphism algebra. Recallthat (M,A) forms an algebroid (c.f. Example 12). In this chapter, we define the Hallproduct of the category of stack functions over M following a similar treatment as tothat of [12]. This restricts the possible choices for M. For example we need existenceof linear algebraic stacks of flags of objects of M, denoted by M(n). We also requirethe projection M(n) →M onto largest element in the flag to be a morphism of linearalgebraic stacks of finite type. In particular, M(2) is the linear algebraic stack of shortexact sequences of objects of M and will be used to define a non-commutative butassociate Hall product on K(M).For sake of simplicity we focus on M= CohX of Examples 8, although some otherpossible choices are Vect and RepQ.12.1 The Hall algebra of a linear stackWe define the linear category M(n) by setting the objects of over every R-scheme Sbeing the flags 0=M0 ⊆M1 ⊆ ··· ⊆Mn =M of objects of M over the same scheme S,where each factor Mi/Mi−1 is also an object of M over S. Morphisms over f : T → Sare diagramsf∗(M1) //f∗(M2)f∗(Mn−1) //f∗(Mn)N1 // N2 Nn−1 // Nn(12.1)where each vertical arrow is a morphism in M and each square is a commutativediagram of objects of M. For all i = 1, . . . ,n there are morphisms of linear categoriesαi :M(n)→M, sending the above flag to its i-th factor Mi/Mi−1. And there is another88morphism b :M(n)→M sending the above flag to An =A.M(n)b //a1×···×anMM×···×M.In particular M(2) is the linear category of short exact sequences M′ →M →M′′ in Mand morphism between the short exact sequences.Proposition 12.1. For the linear algebraic stacks M = CohX , (1) the linear categoryM(n) is a linear algebraic stack; (2) b is a representable morphism of linear algebraicstacks; and (3) a1×···×an is a morphism of linear algebraic stacks of finite type.Proof. The underlying algebraic stacks of M(n) are all constructed by restricting thevertical morphisms in (12.1) to be isomorphisms of sheaves. The claims then followfrom results of [12, Section 4.1]. Recall the space K(M) of stack functions as defined in §11. We have the followingstructures on the Hall algebra K(M).1. Module structure. There is an action of K(Var) on K(M), given by [Z] · [X →M]= [Z×X →X →M]. This action turns K(M) into a K(Var)-module.2. Multiplication. We multiply two stack functions [X →M] and [Y →M] by theformula[X →M] · [Y →M]= [X×Y →M×M ⊕-→M].3. Hall product. The Hall product of the stack functions [X →M] and [Y →M] isdefined by first constructing the fibered productX∗Y// M(2)a1×a2X×Y // M×Mand then setting[X →M]∗ [Y →M]= [X∗Y -→M(2) b-→M].The additive zero in K(M) is given by the empty algebroid 0 = [œ→M]. The multi-plication is associative and commutative, and the Hall product is associative by [12,Theorem 4.3]. The unit with respect to both multiplications is 1 = [SpecR 0-→M], i.e.the 0-object of M.8912.1.1 Filtered structure of the Hall algebraDefinition 12.2. For n≥ 0, we defineK≤n(M)= kerEn+1 =⊕k≤nKk(M).This is an ascending filtration on K(M), called the filtration by the order of vanishingof inertia at q = 1, or simply the order filtration of K(M).This is a slight abuse of language, because we have to tensor with Q(q), before wecan state that K≤n(M)(q) is the direct sum of all eigenspaces of Iss whose coresspond-ing eigenvalue Q∈Q[q] has order of vanishing at q = 1 less than or equal to n.Theorem 12.3. The Hall product respects the order filtration: if ξ ∈ K≤m(M) and η∈K≤n(M), then ξ∗η∈K≤m+n(M) and ξ.η∈K≤m+n(M). On the associated graded theHall product coincides with the commutative product.12.1.2 Proof of Theorem 12.3Analysis of Ep(En∗Em)Suppose ξ = (X →M) and χ = (Y →M) are stack functions. The stack function ξ∗χis defined by the cartesian diagram:X∗Y// M(2)// MX×Y // M×MExplicitely, X∗Y is the stack of triples (x,M,y),xyM′ // M // M′′(12.2)where x and y are objects of X and Y , respectively, M is an object of M(2), i.e., ashort exact sequence M′ → M → M′′ of objects in M, and x → M′ and y → M′′ areisomorphisms from the images of x and y in M to M′ and M′′, respectively. (We omitthese isomorphisms from the triple to simplify the notation.)90The stack function En(ξ)∗Em(χ) is defined by the enlarged diagram:En(X)∗Em(Y) //X∗Y// M(2)// MEn(X)×Em(Y) // X×Y // M×MExplicitly, En(X)∗Em(Y) is the stack of 5-tuples(x,(eν),M,y,(fµ)), where (x,M,y)represents a diagram (12.2), and (eν) = (e1, . . . ,en) is a complete set of non-zero or-thogonal idempotents in A(x), and (fµ) = (f1, . . . ,fm) is a complete set of non-zeroorthogonal idempotents in A(y).Finally, the stack Ep(En(X)∗Em(Y))is the stack of objects of En(X)∗Em(Y),endowed with a complete set of p non-zero labelled idempotents. Explicitly, it consitsof 6-tuples (x,(eν,ρ),M,(gρ),y,(fµ,ρ)), (12.3)where (x,M,y) is as in (12.2), and (gρ)ρ∈p is a complete set of non-zero orthogonalidempotents for the short exact sequence M′ →M →M′′. Moreover, (eρ,ν)ρ∈p,ν∈n isa pn-tuple of orthogonal idempotents in A(x), and (fρ,µ)ρ∈p,µ∈m is a pm-tuple oforthogonal idempotents in A(y), such that for every ρ = 1, . . . ,p we have ∑nν=1 eρ,ν =gρ|E′ and∑mµ=1fρ,µ =gρ|E′′ . Finally, we require for all ν =1, . . . ,n that eν =∑pρ=1 eρ,ν 6=0 and for all µ = 1, . . . ,m that fµ =∑pρ=1fρ,µ 6= 0.Decomposing Ep(En∗Em)We use the notation σ î u for partitions of sets as opposed to σ î` r for labelledset partitions (where the order of blocks matter). For more details in labelled andunlabelled partitions we refer the reader to §A.3. If b(σ) = p then for an elementω ∈ u we say σ(ω) = ρ or ω, ρ if ρ ∈ p is the label of the partition ω belongs to.Also if σ and γ are both labelled set partitions of u, we write ω, (ρ,µ) if ω is in theset with label ρ in σ and in the set with label µ in γ.Given non-negative integers p, u, v , and labelled set partitions γ îuunionsqv such thatb(γ)= p, we define a new stack function (X∗Y)γ →M, denoted (ξ∗χ)γ , as follows.Let (X∗Y)γ be the algebraic stack of 6-tuples(x,(eω),M,(gρ),y,(fη)), (12.4)where (x,M,y) is as in (12.2), and (eω)ω∈u, (fη)η∈v and (gρ)ρ∈p are complete setsof non-zero orthogonal idempotents for x, y and the short exact sequence E, respec-91tively. Moreover, we require that for all ρ = 1, . . . ,p we havegρ|M′ =∑γ(ω)=ρeω and gρ|M′′ =∑γ(η)=ρfη . (12.5)There is a natural algebroid structure on (X∗Y)γ . The morphism to M given bymapping the 6-tuple (12.4) to the middle object b(M) of the short exact sequence M ,makes (X∗Y)γ into a stack function.There is a morphismEu(X)×Ev(Y) -→ (X∗Y)γ (12.6)which maps a quadruple(x,(eω),y,(fη))to the 6-tuple (12.4) where M =M′⊕M′′,with M′ denoting the image of x in M, and M′′ the image of y in M. The family ofidempotents (gρ) on M is defined by formulas (12.5).Lemma 12.4. If for every ρ = 1, . . . ,p exactly one of the two preimages γ−1(ρ)∩u andγ−1(ρ)∩v is empty, the morphism (12.6) is an isomorphism.Proof. Given an object (12.4) of (X∗Y)γ , the short exact sequence M is split into adirect sum of p short exact sequences. Each one of these sequences is canonicallysplit, because either the subobject or the quotient object vanishes, by the assumptionon ϕ and ψ. Therefore the sequence M is split, canonically, too. Now suppose given ϕ î` u with n labelled blocks and ψ î` v with m labelledblocks such that ϕ∩γ|u = 0 and ψ∩γ|v = 0. Then we can define a morphism ofstacks(X∗Y)γ -→ Ep(En(X)∗Em(Y)), (12.7)by mapping the 6-tuple (12.4) to the 6-tuple (12.3) by definingeρ,ν =∑ω,(ρ,ν)eω and fρ,µ =∑η,(ρ,µ)fη .By our assumptions, these sums are either empty or consist of a single summand, sothe eρ,ν and the fρ,µ are obtained from the eω and the fη essentially by relabelling.Lemma 12.5. The morphism (12.7) gives rise to a morphism of stack functions (ξ∗χ)γ → Ep(En(ξ)∗Em(χ)), which is both an open and a closed immersion. If we changeany of u, v or γ or ϕ or ψ, we get a morphism with disjoint image. The images of allmorphisms (12.7) cover Ep(En(X)∗Em(Y)).Proof. This follows from the fact that the source and target of (12.7) only differ in theway the idempotents in Ax and Ay are indexed. 92Corollary 12.6. In K(M) we have the equationEp(En(ξ)∗Em(χ))= ∑u,v,γ∑ϕ,ψ(ξ∗η)γ =∑u,v,γn!m![nγ|u][mγ|v](ξ∗η)γ ,where u,v run over all positive integers and γ runs over all labelled partitions of uunionsqvwith p blocks. The bracket notation is adapted from §A.2.The Hall algebra is filteredWe can now calculate as follows:Ep(pik(ξ)∗pi`(χ))= ∑n,ms(n,k)n!s(m,`)m!Ep(En(ξ)∗Em(χ))=∑n,ms(n,k)s(m,`)∑u,v,γ[nγ|u][mγ|u](ξ∗χ)γ=∑u,v,γ(∑ns(n,k)[nγ|u])(∑ns(m,`)[mγ|u])(ξ∗χ)γ .For both brackets to be non-zero, the number of blocks of γ|u must be at most k, andthe number of blocks of γ|v must be at most `, by Lemma A.3. We conclude that forall p>k+` we have Ep(pik(ξ)∗pi`(χ))= 0, which proves the first part of the theorem.If p = k+` then the only possible case is if the number of blocks of γ|u is exactlyk, and the number of blocks of γ|v is exactly `, by Lemma A.3. In this case we haveEp(pik(ξ)∗pi`(χ))= ∑u,v,γµ(0u,γ|u)µ(0v ,γ|v)(ξ∗χ)γ ,where γ runs over those partitions of uunionsqv such that for every ρ = 1, . . . ,p exactly oneof the two preimages γ−1(ρ)∩u and γ−1(ρ)∩v is non-empty. By Lemma 12.4, wehave thereforeEp(pik(ξ)∗pi`(χ))= ∑u,v,γµ(0u,γ|u)µ(0v ,γ|v)Euξ.Evχ=∑u,v(k+`)!u!v !Euξ.Evχ ∑γ1îub(γ1)=kµ(0,γ1) ∑γ2îvb(γ2)=`µ(0,γ2)= (k+`)!(∑us(u,k)u!Euξ)(∑vs(v,`)v !Evχ)= (k+`)!pik(ξ)pi`(χ).93The associated graded algebraBy what we just proved, we havepik+`(pik(ξ)∗pi`(χ))=∑ps(p,k+`)p!Ep(pik(ξ)∗pi`(χ))only nonzero if p = k+`= s(k+`,k+`)(k+`)! Ek+`(pik(ξ)∗pi`(χ))=pik(ξ).pi`(χ).The proof for the commutative product is similar and givespik+`(pik(ξ).pi`(χ))=pik(ξ).pi`(χ).This finishes the proof of the theorem.12.2 Epsilon functionsWe consider a stack function ξ = (X →M), and an idempotent operator Ek. Let usdenote by FnX the fibered productFnX //XM(n)b // MThen we will consider Ek(FnX). This is the stack of triples(x,(eκ),F)where x is an object of X, and F = (F1→ . . .→ Fn) is a flag in M, such Fn =M , the imageof x in M. Moreover, (eκ)κ=1,...k is a complete set of non-zero orthogonal idempotentsfor x, such that the induced idempotent operators on M respect the flag F . We getinduced idempotents fκ,ν = eκ|Fν/Fν−1 . They have the properties that∑κ fκ,ν = 1, forall ν , and for every κ, at least one of the fκ,ν does not vanish.We will decompose Ek(FnX) according to which of the fκ,ν vanish. For this, con-sider a non-negative integer p and a labelled partition αî` p with k blocks. Then wedefine FαX to be the stack of triples(x,(eκ),(Fκ)).Here x is an object of X, and (eκ) is a complete set of orthogonal non-zero idempo-94tents for x. If we denote the image of x in M by M , then these idempotents define adirect sum decomposition M =M1⊕ . . .⊕Mk. For every κ, we have a filtration Fκ of Eκindexed by α−1(κ).Now let us suppose given another labelled partition β î` p with n blocks suchthat α∩β= 0. Using β, we define a morphismFαX -→ Ek(FnX), (12.8)by defining the flag F in terms of the k-tuple of flags (Fκ) byFν =⊕κ∑α(ρ)=κβ(ρ)≤nFρ .Note that the sum for fixed κ is not really a sum, it is just the largest of the spaces Fρ ,such that α(ρ)= κ and β(ρ)≤n.Lemma 12.7. The morphism (12.8) is an isomorphism onto the locus in Ek(FnX), givenby fκ,ν 6= 0 if and only if α−1(κ)∩β−1(ν) is nonempty (i.e. is a singleton).Corollary 12.8. We haveEk(Fnξ)=∑p∑α:b(α)=k[nα]Fαξ .A special case of the following result is the main outcome of theories of general-ized Donaldson-Thomas invariants and is used in many applications (cf. for example[11, §6.3]).Corollary 12.9. We haveεk(ξ)=∑ns(n,k)n!Fn(ξ)∈K≤k(M).Proof. Let us do a calculation:Ek+1∑ns(n,k)n!Fn(ξ)=∑ns(n,k)n!Ek+1Fn(ξ)=∑ns(n,k)∑p∑α:b(α)=k+1[nα]Fαξ=∑p∑α:b(α)=k+1(∑ns(n,k)[nα])Fαξ= 0 .95The expression in the bracket vanishes, by Corollary A.3.In particular, in the case of k= 1, we haveε1(ξ)=∑n>0(−1)nnFn(ξ).For M = CohX the moduli stack of coherent sheaves on a projective C-scheme X (cf.Example 8) and τ a stability condition as in §1.1, let ξ = [SSγ(τ)↩ CohX] be the in-clusion of the substack of semistable sheaves of class γ. Applying the above result tothis case producesε1(ξ)=∑n(−1)nn!∑06=γ1,...,γn∈Γτ(γi)=τ,∑γi=γSSγ1(τ)∗ . . .∗SSγn(τ)∈K≤1(CohX).This is what is called the no-poles theorem in [29] or absence of poles in [33] andproving them is involved. In contrast, our framework provides a simple derivation ofthis result.12.3 The semi-classical Hall algebraBy Theorem 12.3, the submoduleK(M)=⊕n≥0tnK≤n(M)of K(M)JtK is a K(Var)[t]-subalgebra with respect to the Hall product. The algebraK(M) is a one-parameter flat family of algebras. The special fibre at t=0 is canonicallyisomorphic to the graded algebra associated to the filtered algebra(K(M),∗). Thequotient map K →K/tK is identified with the map ∑nxntn,∑npin(xn).The graded algebra associated to the filtered algebra(K(M),∗), is canonically iso-morphic to the commutative graded algebra(K(M), ·), by Theorem 12.3. The specialfibre inherits therefore a Poisson bracket, which encodes the Hall product to secondorder. This Poisson bracket has degree −1, and is given by the formula{x,y} =pik+`−1(x∗y−y∗x), for x ∈Kk(M), y ∈K`(M). (12.9)Corollary 12.10. The graded K(Var)-algebra(K(M), ·) is endowed with a Poisson bracketof degree −1, given by (12.9).96Corollary 12.11. In particular, K1(M) is a Lie algebra with respect to the Poissonbracket (12.9). In fact, for x,y ∈ K1(M), we have that x∗y −y ∗x ∈ K1(M), soin this case, the Poisson bracket is equal to the Lie bracket.Corollary 12.12. For every stack function ξ = (X→M), the epsilon element ε1ξ definesa virtual indecomposable.Thus, K1(M) is a Lie algebra over the ring of scalars K(Var). We call K1(M) theLie algebra of virtually indecomposable stack functions. This terminology is used inanalogy with that of [29]. It is a work in progress to show that K1(M) serves a similarpurpose as the virtual indecomposables of [29] in defining generalized Donaldson-Thomas invariants.97Bibliography[1] D. Abramovich, Q. Chen, D. Gillam, Y. Huang, M. Olsson, M. Satriano, and S. Sun.Logarithmic Geometry and Moduli. ArXiv e-prints, June 2010. → pages 17[2] J. Alper. Good moduli spaces for Artin stacks. ArXiv e-prints, Apr. 2008. →pages 18[3] J. Alper. On the local quotient structure of Artin stacks. ArXiv e-prints, Apr.2009. → pages 18[4] M. Artin, J. E. Bertin, M. Demazure, A. Grothendieck, P. Gabriel, M. Raynaud, andJ.-P. Serre. Schémas en groupes. Séminaire de Géométrie Algébrique de l’Institutdes Hautes Études Scientifiques. 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URLhttp://dx.doi.org/10.1007/978-1-4614-1099-7. Basic results and techniques. →pages101Appendix APartitionsA.1 Mobius numbersWe follow the notation of [9]. Let (S,≤) be a poset. The two elements 0,1 ∈ S arerespectively the smallest and the largest elements in the poset. The Mobius functionis an integer valued function of two variables on S that associates to a pair (x,y) ofelements of S the Mobius number of it, defined by µ(x,z) = 0 when x œ z and whenx ≤ z by ∑y :x≤y≤zµ(x,y)= δ(x,z). (A.1)Here δ is the Kronecker delta function, defined as δ(x,z) = 1 if x = z and is zerootherwise.The meet of two elements x,y ∈ S is defined by x∧y being the unique greatestlower bound for x and y . A poset with well-defined meet operator ∧ is called a meetsemilattices [15].For every two elements x,y ∈ S, the notation [x,y] stands for the subset of allnodes z such that z ≥ x and z ≤ y . If S is a met semi-lattice then so it [x,y]. S iscalled locally finite if for every two elements x,y ∈ S, the segment [x,y] has finitelymany nodes.Proposition A.1. Given three fixed elements x,z,w of a locally finite meet semilatticeS ∑y :y∧w=xµ(y,z)= µ(x,z) w ≥ z0 w 6≥ z.In the following proof we use the µw(x,z) for this type of summations.Proof. The case of w ≥ z is easy because if y ∧w = x, and y ≤ z ≤w then y = x.By locally finiteness of S the segment [x,1] is a finite poset. So we may proceed by102induction on x. If x = 1 then the claim is easy to check from the above definition.Suppose the claim is proved for every element x > x. The set of all y ∈ [x,z] is thedisjoint union of the following sets{y :y∧w = x} ∀x ∈ [x,z]which meansµw(x,z)=∑x:x≤x≤zµw(x,z)=∑y :x≤y≤zµ(x,y)= 0where the first identity follows from the induction hypothesis and the last identityfollows from (A.1) in the definition of µ. A.2 Identities involving Stirling numbersLet u be a positive integer. We use the notation u for the set of first u integers{1,2, . . . ,u} and Ωu for the set of all partitions λîu. Given two partitions λ,µ îu wewrite λ ≤ µ if every element of λ is a subset of an element of µ. In this case we sayλ is finer than µ and µ is coarser than λ. This way, (Ωu,≤) is a poset. We use thenotations 0u and 1u respectively for the finest and coarsest partitions of u. In placeswhere no confusion should arise we suppress the subscript u and simply write 0 and1 respectively for 0u and 1u.Given any other partition αîu the partition α∩λ is the coarsest partition that isfiner than both α and λ. This turns Ωu into a meet semilattice.We use the notation b(λ) for the number of blocks of a partition. In [47, Example3.10.4] the k-th Whitney number of the first kind associated to Ωu is defined aswk =∑τîub(τ)=u−kµ(0u,τ)and is it shown to be equal to a Stirling number of the first kind by wk = s(u,u−k).We like to rewrite this result ass(b(α),k)=∑τîub(τ)=kµ(α,τ) (A.2)by substituting Ωu with the segment [α,1u]. Note that if τ 6∈ [α,1u] then µ(α,τ)= 0hence the above identity holds without the need to restrict τ to be in [α,1u].Lemma A.2. We have∑α:α∩λ=0s(b(α),k)=0 if k < b(λ)µ(0,λ) if k= b(λ).103Here α runs over all partitions of u.Proof. We use the identity (A.2) and have∑α:α∩λ=0s(b(α),k)=∑α:α∩λ=0∑τ :b(τ)=kµ(α,τ)=∑τ :b(τ)=k∑α:α∩λ=0µ(α,τ)=∑τ :b(τ)=k:τ≤λµ(0,τ)where the last identity follows from Proposition A.1. The set of all such τ can only benonempty if k≥ b(λ) proving the lemma. Let u be a positive integer and λî u a set partition. The notation[nλ]means thenumber of partitions αîu with n blocks such that α∩λ= 0.Corollary A.3. We have∑ns(n,k)[nλ]=0 if k < b(λ)µ(0,λ) if k= b(λ).Proof. This follows from previous lemma and the following computation∑ns(n,k)[nλ]=∑n∑α,b(α)=n,α∩λ=0s(n,k)=∑α:α∩λ=0s(b(α),k).A.3 Labelled partitions and integer partitionsA labelled partition of u is a partition λîu where the order of elements of λ is impor-tant. We denote a labelled partition by λî` u. If we forget the labelling the resultingpartition is called the associated unlabelled partition to λ. Note that given a labelledpartition λ there are b(λ)! other labelled partitions with the same associated unla-belled partition as that of λ. If λ,µ î` u are two labelled partitions we adapt thenotation µ(λ,µ) to denote the Mobius number of the associated unlabelled partitionsof λ and µ.Recall that given an integer t, an integer partition is a sequence (si)i≥1 of integerssuch that∑i≥1 isi = t. To a set partition λ î t we may therefore associate an integerpartition λ ` t defined as λ = (λ1,λ2, . . .) if λ has λi elements of size i. We call λ theinteger partition type of λ. We define the length of λ as b(λ)=∑iλi. So the length ofthe integer type of λ is identical to the number of blocks of λ.104Appendix BSplitting covers of gerbesLet X→ X be a gerbe over a smooth connected scheme X and G an X-group scheme.Recall that triviality of X is obstructed [16]. However X is by definition étale locallyneutral. After passing to a dense open on X, we may assume there exists a finite étalecovering X →X, such that the pullback of X is a neutral gerbes over X, written asX|X › BXGfor some X-group schemeG. We say X is a trivializing cover for X. We may also assumethat X → X is a Galois cover [48, Proposition 5.3.9] with Galois group Γ = Aut(X/X).Then by classification of gerbes there exists a homomorphismϕ : Γ →Aut(BG)= [AutG/G]such that X›X×Γ ,ϕ BG. We may assume ϕ is injective by passing to an intermediatecover:Xkerϕ--------------------→ X˜ Γ/kerϕ-------------------------------→X.Note that X˜ is now the minimal trivializing Galois cover of X→X.Definition B.1. The above covering X˜ →X is called a splitting cover of X over X.Lemma B.2. Given a Galois covering X → X and Γ = Aut(X/X), an action of Γ on BGthat results X as a form of BG on X is unique up to 2-isomorphisms of stacks.Proof. Let ϕ : Γ →Aut(BG) and ϕ′ : Γ →Aut(BG) be two such actions. IfX˜×Γ ,ϕ BG › X˜×Γ ,ψ BG105as gerbes over X, then from classification of gerbes we can construct a natural trans-formation between the functors ϕ and ψ. Lemma B.3. Splitting covers are unique up to isomorphisms of stacks.Proof. Let X1 and X2 be two splitting covers with corresponding ϕ : Γ →Aut(BG) andϕ′ : Γ ′→Aut(BG). We can create a product trivializing cover X˜→X with ∆=Aut(X˜/X)a new group of automorphisms over X. Then by the previous lemma there is a naturaltransformation between the two composition functorsΓ  v))∆88 88&& &&Aut(BG)Γ ′ (  55both corresponding to trivializing covers of X. Therefore there exists a natural trans-formation between ϕ and ϕ′. This proves uniqueness. We finally show that under base change, each connected component of the pull-back of a splitting cover is a splitting cover.Corollary B.4. Let Y → X be a smooth morphism of schemes. If X˜ → X is the splittingcover for X, then Y˜ = Y ×X X˜ → Y is a union⊔Y˜i of splitting covers of X|Y → Y .BY˜G  // Y˜i×Γi BG""⊔Y˜i //YBX˜G//!!X##X˜ // XProof. Let Γ be the Galois group of X˜ →X and Y˜i a connected component of Y˜ . Let Γibe the stabilizer of Y˜ as a subgroup of Aut(Y˜ /Y). All Γi are conjugate to each otherand all Y˜i are isomorphic to each other. Note that since X is the form of BG twistedby ϕ : Γ →Aut(BG), thenX|Y › Y˜i×Γi,ϕi Aut(BG).where ϕi is the restriction of the homomorphism ϕ to the subgroup Γi. Since ϕ isinjective, so is ϕi, which implies that Y˜i is the splitting cover for X|Y for any i. 106


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