{"http:\/\/dx.doi.org\/10.14288\/1.0302066":{"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool":[{"value":"Science, Faculty of","type":"literal","lang":"en"},{"value":"Mathematics, Department of","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider":[{"value":"DSpace","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeCampus":[{"value":"UBCV","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/creator":[{"value":"Ronagh, Pooya","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/issued":[{"value":"2016-05-11T20:00:04Z","type":"literal","lang":"*"},{"value":"2016","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#relatedDegree":[{"value":"Doctor of Philosophy - PhD","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeGrantor":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/description":[{"value":"We view the inertia construction of algebraic stacks as an operator on the Grothendieck groups of various categories of algebraic stacks. We are interested in showing that the inertia operator is (locally finite and) diagonalizable over for instance the field of rational functions of the motivic class of the affine line q = [A\u00b9]. This is proved for the Grothendieck group of Deligne-Mumford stacks and the category of quasi-split Artin stacks. Motivated by the quasi-splitness condition we then develop a theory of linear algebraic stacks and algebroids, and define a space of stack functions over a linear algebraic stack. We prove diagonalization of the semisimple inertia for the space of stack functions. A different family of operators is then defined that are closely related to the semisimple inertia. These operators are diagonalizable on the Grothendieck ring itself (i.e. without inverting polynomials in q) and their corresponding eigenvalue decompositions 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. The commutative and non-commutative products of the Hall algebra respect the graded structure defined above. Moreover, the two multiplications coincide on the associated graded algebra. This result provides a geometric way of defining a Lie subalgebra of virtually indecomposables. Finally, for any algebroid, an \u03b5-element is defined and shown to be contained in the space of virtually indecomposables. This is a new approach to the theory of generalized Donaldson-Thomas invariants.","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO":[{"value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/58120?expand=metadata","type":"literal","lang":"en"}],"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note":[{"value":"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\u00a9 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 \u03b5-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\u2019s 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\u2019s virtual projections [27] with eigenprojections of this opera-tor. Our goal is to reproduce Joyce\u2019s project [24\u201329] 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\u00d7M 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 form\u03c7(E,F)=\u2211i(\u22121)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 : \u03c7(E,F)= 0 for all objects F in Coh(M)}and{F : \u03c7(E,F)= 0 for all objects E in Coh(M)}are the same subgroup K(Coh(M))\u22a5 and therefore the quotientN(M)=K(Coh(M))\/K(Coh(M))\u22a5called the numerical Grothendieck group carries a well-defined non-degenerate bilinearform. There is a monoid \u0393 \u2282N(M) consisting of classes of sheaves. Fixing a class \u03b3 \u2208 \u0393yields an open and closed substack M\u03b3 \u2286M of objects of class \u03b3.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 \u2208Coh(M) of coherent sheaves, the Hilbert polynomialPE is the unique polynomial in Q[t] such thatPE(n)= dimH0(E(n)) for all n\u001d 0 .This polynomial only depends on the class of E in \u0393 . Thus we may write P\u03b3 for \u03b3 \u2208 \u0393 .We define \u03c4(\u03b3) = P\u03b3\/r\u03b3 where r\u03b3 is the leading coefficient of P\u03b3 . This means that \u03c4associates to any class in \u0393 a monic polynomial of degree at most 3. If p(t) and p\u2032(t)are two such polynomials we sayp \u2264 p\u2032, if degp > degp\u2032 and in case of equality p(t)\u2264 p\u2032(t) for all t\u001d 0.1This defines a notion of stability; E is stable if for all nonzero subobjects S \u2286 E, wehave \u03c4([S]) < \u03c4([E\/S]) and it is semistable if for all nonzero subobjects S \u2286 E, wehave \u03c4([S])\u2264 \u03c4([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(\u03c4) for the open substack of M consist-ing of semistable objects for \u03c4 . This itself consists of connected components SS\u03b3(\u03c4)of semistable objects of class \u03b3. 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 \u2208 A0(X), the virtual count of points on X isdefined as the rational number#vir =\u222b[X]vir1\u2208Q .On the other hand, by Behrend\u2019s theorem [6] this virtual count coincides with the Eulercharacteristic of X, weighted by the Behrend function \u03bdX of X:\u222b[X]vir1= \u03c7(X,\u03bdX).Here is a few remarks on the concepts that appear in the above definition.\u2022 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).\u2022 The virtual count: If moreover, X is proper, the push-forward of this class overa point is well-defined and produces a rational number,\u222b[X]vir 1. This numberis an integer if X is a scheme or an algebraic space.Let X \u2192 C be a family Xt of smooth projective varieties parametrized by t \u2208 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].\u2022 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 \u03d5 :W \u2192X of relative dimension n, we have\u03d5\u2217(\u03bdX)= (\u22121)n\u03bdW .3For Calabi-Yau 3-folds, Thomas [49] constructs a symmetric (in particular perfect)obstruction theory on St\u03b3(\u03c4). If SS\u03b3(\u03c4) = St\u03b3(\u03c4) then St\u03b3(\u03c4) is proper as well whichis the conventional case in which Donaldson-Thomas invariants are defined:DT\u03b3(\u03c4)=\u222b[St\u03b3(\u03c4)]vir1= \u03c7(St\u03b3(\u03c4),\u03bdSt\u03b3(\u03c4))=\u2211n\u2208Z\u03c7(\u03bd\u22121St\u03b3(\u03c4)(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 DT\u03b3(\u03c4)\u2208Q that correctly count the semistable objects inSS\u03b3(\u03c4) and are (1) invariant under deformations of M ; and, (2) when SS\u03b3(\u03c4)= St\u03b3(\u03c4)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 \u03c4 ,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 DT\u03b3(\u03c4) in terms of DT\u03b3(\u03c4\u2032).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 \u00a75this would be the category of finite type schemes over a noetherian base scheme. Forthe cases of Artin stacks in \u00a76 and \u00a77 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 \u00a75 this category isthat of the Deligne-Mumford stacks DM and in \u00a76 it will be the category St of alge-braic stacks with affine diagonals. In \u00a77 we work with the category QS of quasi-splitalgebraic stacks and finally in \u00a711 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 \u00a73.4 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 \u2192 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\u22121,{(qk\u22121)\u22121 : k \u2265 1}] and the eigenvalue spectrum of it isthe set {qk : k\u2265 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\u220fi=1(qri \u22121).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\u220fi=1(qri \u22121).5. Theorem 11.10: In the case of stacks functions of algebroids over a linear stackM, the operator Iss : K(M)(q)\u2192 K(M)(q) is diagonalizable and the eigenvaluespectrum of it onsists of the set of all polynomials of the formk\u220fi=1(qri \u22121).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 \u00a74.3 and \u00a75.4 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 \u00a711.3. They respect only a coarser filtration than the usual semisimpleinertia (explained in \u00a711.2) but they are simultaneously diagonalizable (Corollary 11.5).This creates an eigenvalue decompositionK(M)=\u2295k\u22650Kk(M).and a filtration by the order of vanishing of the inertia at q = 1,K\u2264n(M)= kerEn+1 =\u2295k\u2264nKk(M),also called the order filtration. Details are explained in \u00a712.1.Hall algebraA noncommutative product denoted by \u2217, is then defined in \u00a712.1 on K(M) in theusual way of defining Hall algebras. The main result of \u00a712 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\u22641(M) is a Lie subalgebra of K(M) whichis our counterpart for the space of virtually indecomposables of Joyce and Song [29].In \u00a712.2 we introduce the elements \u03b5k associated to a stack function and showin Corollary 12.9 that it lives in K\u2264k(M). In particular, \u03b51 SS\u03b3(\u03c4) 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 \u00e9tale 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 \u2192 X of analgebraic stack X, the sheaf of automorphisms Aut(s)\u2192 S is an affine group scheme.2.1 Stratification of group spacesBy a group space G\u2192 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 \u00a76 and \u00a77.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 \u2192X, the setG0(T)= {u\u2208G(T) :\u2200x \u2208X,ux(Tx)\u2282G0x}.Here G0x is the connected component of unity of the algebraic group Gx =G\u2297X \u03ba(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, \u00a75], but we will reframe it for furtherreference:Lemma 2.1. Let G \u2192 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 {X\u03b1}\u03b1\u2208Aof reduced, locally closed subschemes, such that for each \u03b1 \u2208 A, G|X\u03b1 \u2192 X\u03b1 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 \u00e9tale 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, \u00a77.3 (iii)] and[4, Exp. XIX, \u00a75]), however we have the followingLemma 2.3. Let G\u2192 X be a smooth group scheme and assume that G =G\/G0 is finiteand \u00e9tale. Then G0 is a closed subscheme of G.Proof. G\u2192X is finite, hence proper and consequently universally closed. Thus in thecartesian diagramG0\u00d7X G\u0002\u03d5 \/\/\u000f\u000fG\u00d7X G\u000f\u000fG \/\/ G\u00d7X Gthe morphism \u03d5 : (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\u00d7X G\u0002\/\/\u000f\u000fG\u00d7X G\u000f\u000fG0 \/\/ Gthe embedding of G0 in G is also a closed immersion. The vertical right hand arrow isdescribed by (g,h), gh\u22121. \u0002Corollary 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 = {X\u03b1}\u03b1\u2208A such thatfor all \u03b1\u2208 A and all group schemes G\u03b1 =G|X\u03b1 , (1) G0\u03b1 is closed and (2) G\u03b1\/G0\u03b1 is finiteand \u00e9tale over X\u03b1.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)|X\u03b1 is finite over X\u03b1. By the above remarks on thebase change this means that with the notation G\u03b1 =G|X\u03b1 , each G\u03b1 =G\u03b1\/G0\u03b1 is a finite\u00e9tale group scheme over X\u03b1. The assertion now follows from Lemma 2.3. \u00029For a group scheme G \u2192 X and a closed subscheme Y of it, the functorial cen-tralizer ZG(Y) is defined as in [13, \u00a72.2]. 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 = {X\u03b1}\u03b1\u2208A such that each restricted group schemeG|X\u03b1 has a closed subscheme representing its centre.Before stating a proof, we recall a Galois theoretic fact about finite \u00e9tale covers ofschemes:Remark 2.6. If C \u2192 X is a connected degree d \u00e9tale cover and L\/K is a separableextension of the residue field of the generic point of X, then CXL \u2192XL is the union of ddegree one coverings. More generally let {Ci}i=1,...,k be all the connected componentsof C with corresponding generic points {\u03b7i} and residue fields {\u03ba(\u03b7i)}. Let L be acommon separable closure of the latter. Then XL\u2192X is a finite \u00e9tale 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 \u00e9tale group scheme over X. We will find a finite \u00e9talecover X\u02dc \u2192X such that G|X\u02dc has a scheme theoretic centre and then use affine descent.By the above remark we may take a covering X\u02dc\u2192X such that G\u00d7X X\u02dc\u2192 X\u02dc is a unionof connected components all of which are isomorphically mapped to X\u02dc. Moreover Gis a torsor for a connected group over G, and thus so is G|X\u02dc over G|X\u02dc .G0|X\u02dc \/\/\/\/ G|X\u02dc \/\/\u000f\u000fG|X\u02dc\u000f\u000f\/\/\u0002X\u02dc\u000f\u000fG0 \/\/\/\/ G \/\/ G \/\/ XTherefore the connected components of the source and target correspond bijectively.But every connected component of G|X\u02dc maps isomorphically to X\u02dc, thus each con-nected component of G|X\u02dc is isomorphic to G0. Now by [13, Lem. 2.2.4], the centralizerof each connected component exists over X\u02dc and their intersection is the centre of G|X\u02dc .Finally X\u02dc\u2192X is \u00e9tale and in particular an fpqc covering and the centre of G|X\u02dc is affineover X\u02dc, so affine descent finishes the proof. \u0002102.2 Groups of multiplicative type2.2.1 Quasi-split toriA commutative group scheme T \u2192 X is said to be of multiplicative type if it is locallydiagonalizable over X in the fppf topology (and therefore in the \u00e9tale topology [4, Exp.X, Cor. 4.5]). For the general theory of group schemes of multiplicative type we referthe reader to [4, Ch. IIIV\u2013X], however we recall a few preliminary facts here. Associatedto T there exists [4, Exp. X, Prop. 1.1] a locally constant \u00e9tale abelian sheafT ,M =HomX\u2212gp(T ,Gm), (2.1)and T is the scheme representing the sheaf HomX\u2212gp(M,Gm). This is an anti-equivalenceof the categories of X-group schemes of multiplicative type and locally constant \u00e9talesheaves 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 \u00e9tale 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 \u2286 X such that after pulling back TU along a finite \u00e9talemorphisms U\u02dc \u2192 U , TU\u02dc is isomorphic to HU\u02dc \u00d7Grm,U\u02dc where H is a finite commutativegroup.Proof. This immediately follows from the fact that every \u00e9tale morphism is Zariskilocally quasi-finite and every quasi-finite morphism is Zariski locally finite [46, Lem.03I1]. \u0002So we may associate to T a Galois cover X\u2032 \u2192 X with group \u0393 such that TX\u2032 isisomorphic to HomX\u2032\u2212gp(MX\u2032 ,Gm,X\u2032) where M is a finitely generated abelian group.The action of \u0393 on X\u2032 induces an action of it on MX\u2032 .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\u2032 if TX\u2032 = Gm,X\u2032 and we denote MX\u2032 by \u03c7T . Our anti-equivalence of categories isnow between isotrivial X-tori that split over X\u2032 and \u0393 -lattices (i.e. a finitely generatedtorsion free abelian groups equipped with the structure of some \u0393 -module of finitetype) given byT ,H0(X\u2032,HomX\u2032\u2212gp(X\u2032\u00d7X T ,Gm))and byA,HomX\u2212gp(X\u2032\u00d7A\/\u0393 ,T )in the reserve direction where A is a \u0393 -lattice.11In view of 2.1, the \u0393 -module MX\u2032 is called the character lattice of T and also de-noted by \u03c7T . Note that, if \u03c1 is the Galois action of \u0393 on TX\u2032 , the induced action on \u03c7Tis via pre-composition: \u03c1\u03b3(m)=m\u25e6\u03c1\u03b3 .Definition 2.8. Let T be an isotrivial X-torus splitting on the Galois cover X\u2032\u2192X withGalois group \u0393 . T is called a quasi-split torus if \u03c7T is a permutation \u0393 -lattice (i.e. theaction of \u0393 on \u03c7T is by permutation of the elements of a Z-basis).2.2.2 Maximal tori of group schemesWe recall [4, Expos\u00e9 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:\u03c1r (G)= dimkT .Likewise, the unipotent rank of G is the dimension of the unipotent radical U of Gkand denoted by\u03c1u(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 \u2208X,the corresponding\u03c1r (x)= \u03c1r (Gx), and \u03c1u(x)= \u03c1u(Gx).The function \u03c1r is lower semi-continuous in the Zariski topology [4, Expos\u00e9 XII,Thm. 1.7]. Moreover, the condition of \u03c1r being a locally constant function in theZariski topology, is equivalent to existence of a global maximal X-torus for G in the\u00e9tale topology by [4, Expos\u00e9 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, Expos\u00e9 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 \u03c1r is lower semi-continuous and integer valued there exists a stratifica-tion by locally closed subspaces on which \u03c1r is constant. We can further refine sucha stratification by Lemma 2.7 for the global tori to be trivial after finite \u00e9tale basechange. \u0002122.3 Spreading out argumentsWe say a group scheme H\u03b7 \u2192 X (or a property of it with respect to another groupscheme G\u03b7) spreads out to a neighborhood of the generic point \u03b7\u2208X if there exists adense open subset U \u2282 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 ofG\u03b7).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 \u03b7 bethe generic point of X and H\u03b7 a closed subgroup of G\u03b7. Then there exists a non-emptyopen set U \u2286X such that G|U contains a subgroup schemeHU fitting in the commutativediagramH\u03b7\u000f\u000f\/\/ HU\u000f\u000fG\u03b7 \/\/ 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\u2192 K corresponding tothe inclusion of the generic point \u03b7\u2192 X. Then G = SpecS, where S = R[x1, . . . ,xk]\/Ifor some finitely generated ideal I = \u3008p1, . . . ,p`\u3009 \u2282 R[x1, . . . ,xk] and therefore G\u03b7 =SpecK\u2297R S. With this notation, H\u03b7 is cut out as a subscheme by a finitely generatedideal p \u2286 K\u2297R 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 \u2208R 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\u2192Gandm :G\u00d7XG\u2192G, respectively be the inversion and multiplication morphisms on G.Considering the inversion morphism, existence of group structure on H\u03b7 means thatin level of coordinate rings, we are given a commutative diagramRf [x1, . . . ,xk]\/IRfi# \/\/\u000f\u000fRf [x1, . . . ,xk]\/IRf\u000f\u000fK[x1, . . . ,xk]\/IKi\u02dc# \/\/ K[x1, . . . ,xk]\/IKq \/\/ K[x1, . . . ,xk]\/pIKand the composition of the induced morphism i\u02dc# and the quotient map q has precisely13pIK as its kernel. Hence by a similar argument, there exists some g \u2208 Rf lifting thiscomposition as in the cartesian diagramRfg[x1, . . . ,xk]\/pIRfg \/\/\u000f\u000fRfg[x1, . . . ,xk]\/pIRfg\u000f\u000fK[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 \u00d7HU \/\/ _\u000fHU _\u000fGU \u00d7GU \/\/ GUwhere the horizontal arrows are morphisms (x,y), xy\u22121 is now obvious. Thus HUis the desired subgroup scheme of GU . \u0002Lemma 2.11. Group homomorphisms (repectively isomorphisms) spread out. Let G\u2192Xand \u03b7\u2208X be as in previous lemma. If G\u2032\u2192X is another group scheme and\u03d5\u03b7 :G\u03b7\u2192G\u2032\u03b7is a group scheme homomorphism (resp. isomorphism), then there exists a non-emptyU \u2286X and a homomorphism (resp. isomorphism) \u03d5 :G\u2032U \u2192GU such that \u03d5|\u03b7 =\u03d5\u03b7.Proof. The proof is by similar arguments as in previous lemma. \u00022.4 Tori and unipotent group schemesLemma 2.12. The property of being a quasi-split torus spreads out. Let G \u2192 X and\u03b7 \u2208 X be as in the previous lemmas. If G\u03b7 is a quasi-split torus, then there exists anon-empty U \u2286X such that GU is a quasi-split torus.Proof. Since G\u03b7 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 (\u00e9tale lo-cally constant) sheaf \u03c7(G) = HomU\u2212gp(G,Gm,U ). Restriction from U to \u03b7 induces ahomomorphism of finitely generated Z-modulesHomU\u2212gp(G,Gm,U )\u2192HomK\u2212gp(G\u03b7,Gm,\u03b7)and by Lemma 2.11 we may assume that this is an isomorphism of Z-modules. We alsonote that if U\u02dc \u2192U is a finite \u00e9tale covering that trivializes GU and restricts to the finite14separable extension L\/\u03ba(\u03b7), then \u0393 = Gal(L\/K) is at the same time the fundamentalgroup of this covering and the associated action of \u0393 on \u03c7(G\u03b7) induces same action ofthis group on \u03c7(G). \u0002Now 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\u2192X 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 \u2282 X such that GU has a filtration in sub-groups 1\u2282G1\u2282 . . .\u2282Gr\u22121\u2282Gr =G with all factors G`\/G`\u22121 isomorphic to the constantU -group scheme Ga,U .Proof. Let H\u03b7 be a subgroup of the generic fiber G\u03b7. 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 H\u03b7 is isomorphic toGa,\u03b7, then there exists a non-empty open U \u2286X such that H\u03b7 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 \u03d5 :K\u2297R S \u2192 K[t]. It is easy to check that there exists a localization Rf of R that extends\u03d5 to an isomorphism \u03d5\u02dc : Rf \u2297R S \u2192 Rf [t]. The claim now follows by induction on thequotient scheme GU\/Ga,U .1 \u0002Corollary 2.15. Let X be an integral scheme and G a finitely presented smooth affinecommutative group scheme over X. Let \u03b7 be the generic point of X. Then the decom-position of G\u03b7 as T\u03b7\u00d7U\u03b7 to a maximal torus and the unipotent radical spreads out;i.e. there exists a non-empty Zariski open V \u2286 X such that GV is isomorphic to TV \u00d7UVwhere TV is a maximal torus with T\u03b7 as generic fiber and UV is a unipotent V -groupscheme with U\u03b7 as generic fiber. Moreover, if T\u03b7 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 T\u03b7\u00d7U\u03b7 also spreads out by Lemma 2.11. \u00021A 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 \u00e9tale 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 \u2282X a closed subscheme,equipped with structure of a commutative unital ring according to the fiber productin Sch\/B,[X] \u00b7 [Y]= [X\u00d7Y].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\u21a9 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\u00d7Z, 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\u0002\u000f\u000f\/\/ X\u2206\u000f\u000fX \u2206 \/\/ X\u00d7Xwhere \u2206 is the diagonal morphism. IX is isomorphic to the stack of objects (x,f ),where x is an object of X and f : x \u2192 x is an automorphism of it. Here a morphismh : (x,f )\u2192 (y,g) is an arrow h : x\u2192y in X satisfying g \u25e6h= h\u25e6f .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,\u00b7\u00b7\u00b7 ,fk) of an object x inX and pairwise commuting automorphisms f1,\u00b7\u00b7\u00b7 ,fk. A morphism (x,f1,\u00b7\u00b7\u00b7 ,fk)\u2192(y,g1,\u00b7\u00b7\u00b7 ,gk) is an arrow h : x\u2192y of X satisfying h\u25e6fi = gi \u25e6h for all i= 1,\u00b7\u00b7\u00b7 ,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\u2192X is always an fppf (in particular, \u00e9tale) 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\u2192 S,s) of a line bundle L on S with a section s : S\u2192L. 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\u2192D. The quotient morphism p :A1\u2192Dmaps every morphism f : S\u2192A1, to the pair (S\u00d7A1, id\u00d7f). Restricting p to Gm \u2282A1,we the get a morphism p :Gm \u2192 [Gm\/Gm] where the latter quotient stack is a point.We conclude that m\u25e6p : A1 \u2192D 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 \u2192 X be an \u00e9tale gerbe. Then there exists a sheaf of abeliangroups Z \u2192X and a morphism of sheaves of groups \u03d5 : Z\u00d7X X\u2192 IX such that1. For every s : S \u2192 X, the induced morphism of sheaves of groups s\u2217\u03d5 : Z|S \u2192Aut(s) identifies Z|S with the centre of the sheaf of groups Aut(s); and,2. The pair (Z,\u03d5) is unique, up to isomorphism of sheaves of groups over X.Proof. This is explained in [18, Ch. IV, \u00a71.5], specifically refer to [18, Ch. IV, \u00a71.5.3.2]for existence of the sheaf and to [18, Ch. IV, Cor. 1.5.5] for the properties of it. \u0002Definition 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\u0002\/\/\u000f\u000fX\u000f\u000fZ \/\/ X.Lemma 3.3. The map \u03d5 : IzX\u2192 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 \u2208 X, Aut(s) is a group scheme and its functorial centralizeras defined in [13, \u00a72.2] is representable by a closed (resp. open and closed) subscheme.Proof. For any scheme U , any U -point u\u2208 IX(U) is determined by a pair (u,\u03c3) withu \u2208 X(U) and \u03c3 \u2208 Aut(u). Thus the S-points (for any scheme S) of the fiber productU \u00d7IX IzX is given by objects\u3008f \u2208U(S),s \u2208X(S),\u03c4 \u2208 Z(Aut(s)),\u03b9 : f\u2217u \u009b----\u2192 s\u3009 such that \u03b9\u25e6f\u2217\u03c3 = \u03c4 \u25e6 \u03b9.This stack is then 1-isomorphic to the stack of objectsf \u2208U(S), such that f\u2217\u03c3 \u2208 Z(Aut(f\u2217u)).The group sheaf Aut(u) is represented by an affine U -group scheme G and \u03c3 =\u03c3(U) :U \u2192G is a section of the structure morphism. Thus the stack of the objects above, isrepresented by the fiber product Z(G) j\u00d7G,\u03c3 U where j is the closed immersion of thecentre, Z(G)\u21a9G. \u000218Discrete central inertiaSuppose we are in the case that Z \u2192 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\u0002\/\/\u000f\u000fX\u000f\u000fZ0 \/\/ XandIz\/z,0X\u0002\/\/\u000f\u000fX\u000f\u000fZ\/Z0 \/\/ X.Here Z\/Z0\u2192X is called the discrete central band of X\u2192X.3.4 The semisimple and unipotent inertiaRecall that if G is an affine group scheme of finite type on base scheme S, an elementg \u2208G(S) is defined to be semisimple if for all scheme points s \u2208 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,\u03d5) such that \u03d5 \u2208 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 \u2192 S is an \u00e9tale surjection and \u03d5 \u2208 Aut(x) is an automorphismof an object x over S where f\u2217\u03d5\u2208Aut(f\u2217x) 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\u2192 S, if g is a k-valuedpoint of Gk, and K\/k is an algebraically closed extension, and g\u2032 the K-valued pointof GK obtained by pullback, then g is semisimple if and only if g\u2032 is [4, Expos\u00e9 XII].Let u :U\u2192X be an \u00e9tale 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,Expos\u00e9 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\u2229 IzX. We may also write this is afiber productIss,zX \/\/\u000f\u000fX\u000f\u000fZss \/\/ Xwhich is a relative group scheme, since the set of semisimple elements Zss of thecentral band Z \u2192X form a group scheme of multiplicative type. Equivalently,Iss,zX \/\/\u000f\u000fX\u000f\u000fZ\/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,\u03d5) such that \u03d5 is a unipotent element of Aut(x). It is easy tocheck that IuX\u2282 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)=\u221e\u2295n=1\u221e\u2295i=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\u2264n(gpd) and a descendingfiltration K\u2265i(gpd).Denote by I : K(gpd) \u2192 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\u2264n(gpd) andK\u2265i(gpd). Moreover, on the associated graded, K\u2265i\/K>i(gpd), the endomorphism I ismultiplication by i.Proof. Recall thatI(BG)\u009b\u2294g\u2208C(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]=\u2211g\u2208C(G)[BZG(g)]= #Z(G)[BG]+\u2211g\u2208C(G)\u2217[BZG(g)],where C(G)\u2217 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. \u00024.2 Local finiteness and diagonalizationProposition 4.2. The endomorphism I : K(gpd)\u2192 K(gpd) is diagonalizable, with spec-trum of eigenvalues equal to the positive integers.Proof. Every subspace K\u2264n(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\u2264n(gpd) for all n. \u0002Thus 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]\u2212 [BZ2]\u2212 12 [BZ3]23is an eigenvector for I, with eigenvalue 1. Thuspin[BS3]=\uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3[BS3]\u2212 [BZ2]\u2212 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]\u2212 12 [BZ4]\u2212 [BD2]is an eigenvalue of I with eigenvalue 2. It follows thatpin[BD4]=\uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3[BD4]\u2212 12 [BZ4]\u2212 [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\u00d7r )\u2217\/G],where the brackets are used as the notation for quotient algebroids. In K(gpd) wewriteIr [BG]= [(G\u00d7r )\u2217\/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 \u2265 0, preserve the filtration K\u2265k(gpd). On thequotient K\u2265k(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]+\u2211S\u2208(G\u00d7r )\u2217S 6\u2286Z(G)[BZG(S)].\u0002Corollary 4.4. The operators Ir , for r \u2265 0 are simultaneously diagonalizable. The com-mon eigenspaces form a family \u03a0k(gpd) of subspaces of K(gpd) indexed by positiveintegers k\u2265 0, andK(gpd)=\u2295k\u22650\u03a0k(gpd).Let pik denote the projection onto \u03a0k(gpd). We haveIrpik = r !(kr)pik ,for all r ,k\u2265 0.Corollary 4.5. For r \u2265 0, we havekerIr =\u2295kn, the inertia endomorphism in-duces multiplication by the integer n.Proof. Consider an irreducible gerbe X, with split central number n and {Y\u03b1}\u03b1\u2208A bethe stratification of IX by connected components (hence irreducible gerbes). There areprecisely n of the Y\u03b1 which are contained in IzX and map isomorphically to X (i.e. aredegree one connected \u00e9tale 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\u00d7X IzX\u0002\/\/pi2\u000f\u000f? _joo IzXpi1\u000f\u000fIX \/\/ X(5.1)where the square is cartesian. For any object x of X, elements of IX over x are pairs(x,\u03d5) such that \u03d5 \u2208 Aut(x) and objects of IzX over x are pairs (x,\u03c8) where \u03c8 \u2208Z(Aut(x)). The fibered product IX\u00d7X IzX is hence the stack of triples (x,\u03d5,\u03c8) withx,\u03d5 and \u03c8 as above. On the other hand, Iz(IX) is the stack of the objects (x,\u03d5,\u03c8)such that \u03d5 \u2208Aut(x),\u03c8\u2208 Z(ZAut(x)(\u03d5)). Hence there is an embedding of the fiberedproduct into Iz(IX). Restricting to a substack Y\u2282 IX we get the following diagram.IzYpi3$$Y\u00d7X IzX\u0002\/\/pi2\u000f\u000f? _joo IzXpi1\u000f\u000fY \/\/ 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 \u00e9talecovering maps. Note also that there is a canonical section, \u03b4, to pi3 : I(Y) \u2192 Y viathe diagonal Y\u2192 Y\u00d7XY 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\u2265\u2022.For the action of inertia on the graded piece K\u2265n\/K>n we show that if Y is acomponent of IX which is not a section of pi1, then the associated section \u03b4 is notinduced by pulling back sections of pi1. In fact, if Y is not contained in IzX then \u03b4does not lift to pi2 and we are done. Otherwise, (when Y is completely contained inIzX), \u03b4 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. \u00025.3 Local finiteness and diagonalizationBefore we can deduce that I : K(DM) \u2192 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\u220fkXY to denote the k-fold fiber product of a stack Y by itself overX.29Lemma 5.7. Let Y\u2192 X be a finite \u00e9tale representable morphism of algebraic stacks.Then the following family is a finite set up to isomorphism of stacks.C(Y\u2192X)= {W :W is a connected component ofk\u220fXY for some k\u2265 0}Proof. This is trivial since the Galois closure of Y with respect to X is a finite \u00e9tale X-stack Y\u2192 X. And every element in the above family is isomorphic to an intermediatecover, in between Y and Y. \u0002Corollary 5.8. Let Y1,\u00b7\u00b7\u00b7 ,Ys be finitely many algebraic stacks, finite \u00e9tale over X. Thethe following family is finite up to isomorphism{W :W is a connected component ofk1\u220fXY1\u00d7X \u00b7\u00b7\u00b7\u00d7Xks\u220fXYs for some k1,\u00b7\u00b7\u00b7 ,ks \u2265 0}Proof. There are s projection mapsp` :k1\u220fXY1\u00d7X \u00b7\u00b7\u00b7\u00d7Xks\u220fXYs \u2192k\u220f`XY`, ` = 1,\u00b7\u00b7\u00b7 ,swhich are all finite \u00e9tale and in particular closed and open. The immersioni :k1\u220fXY1\u00d7X \u00b7\u00b7\u00b7\u00d7Xks\u220fXYs \u2192k1\u220fXY1\u00d7\u00b7\u00b7\u00b7\u00d7ks\u220fXYsis similarly closed and open. Hence W is isomorphic to its image i(W) which is aconnected component of pi1(W)\u00d7\u00b7\u00b7\u00b7\u00d7pis(W). However any fiber product W1\u00d7\u00b7\u00b7\u00b7\u00d7Ws where Wi \u2208 C(Yi \u2192 X) has finitely many connected components and by Lemma5.7 there are only finitely many such fiber products. \u0002Corollary 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\u2265 0}Proof. For an irreducible gerbe IX\u2192X is finite \u00e9tale, hence closed and open and there-fore the inertia stratifies to finitely many connected components Y1,\u00b7\u00b7\u00b7 ,Ys which are30finite \u00e9tale over X. In the commutative diagramI(m)X \u001f j \/\/\u001d\u001d\u220fmX IX\u0001\u0001IXthe downward arrows are finite \u00e9tale 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 \u00d7X \u00b7\u00b7\u00b7\u00d7XYim \u2282m\u220fXIXfor a choice of i1,\u00b7\u00b7\u00b7 , im \u2208 {1,\u00b7\u00b7\u00b7 ,s}. The claim now follows from Corollary 5.8. \u0002This 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\u2208A, 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 \u2192 IXi is finitely generated, invariant under inertia endomorphism,and contains [X].Corollary 5.11 (Diagonalization). The endomorphism I : K(DM)\u2192 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 \u2192 X is an X-point of X, andG = Aut(x) is the X-group scheme of automorphisms of x, then si are sections ofG \u2192 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 \u00a73.2. In fact itis easy to see that by an inclusion-exclusion argument that they satisfy the followingidentity,Ir =r\u2211k=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\u2192 X an \u00e9tale morphism of degree n. Thenthere exists a Galois covering X\u02dc\u2192X of X such that IzX|X\u02dc is a disjoint union of n copiesof X\u02dc. So we have[ZIrX|X\u02dc]= r !(nr)[X\u02dc].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 \u2265 0, preserve the filtration K\u2265k(DM) by splitcentral number. On the quotient K\u2265k(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\u2192 X be ofdegree n. Let X\u02dc\u2192X be a splitting cover for IzX\u2192X with Galois group \u0393 . Hence \u0393 actson n.IzX|X\u02dc\u03b9 \/\/\u000f\u000fIzX\u000f\u000fX\u02dc \/\/ XThen IzX|X\u02dc is the disjoint union of n copies of X\u02dc and \u0393 acts on it by permuting thesecopies. Let us rename the i-th copy to Y\u02dc and the image to Y. The integer i\u2208n is fixedunder the action of \u0393 precisely when Y \u009b Y\u02dc\/\u0393 via the horizontal morphism. SinceX\u02dc\/\u0393 \u009b 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 \u0393is of size k. Also,X\u02dc\u00d7 Inj(r ,n) '-\u2192 ZIrX|X\u02dc ,and the action of \u0393 on n induces an action of it on Inj(r ,n) . A morphism \u03d5 : r \u21a9 nis invariant under this action if every element in the image of \u03d5 is so. Therefore the32number of fixed points of Inj(r ,n) is r !(kr). We may hence calculate as follows:ZIr [X]= [X\u02dc\u00d7\u0393 Inj(r ,n)]=\u2211\u03d5\u2208Inj(r ,n)\/\u0393[X\u02dc\/Stab\u0393\u03d5]=\u2211\u03d5\u2208Inj(r ,n)\u0393[X]+\u2211\u03d5\u2208Inj(r ,n)\/\u0393Stab\u0393 \u03d5 6=\u0393[X\u02dc\/Stab\u0393\u03d5]Thus, we conclude,ZIr [X]= r !(kr)[X]+\u2211\u03d5\u2208Inj(r ,n)\/\u0393Stab\u0393 \u03d5 6=\u0393[X\u02dc\/Stab\u0393\u03d5].Finally note that each intermediate cover Y= X\u02dc\/Stab\u0393\u03d5 has a strictly larger split cen-tral number k. In fact, IzY = IzX|Y so every section of IzX\u2192 X pulls back to a sec-tion of IzY\u2192Y but also IzY\u2192Y has sections induced by \u03d5 that do not descend toIzX\u2192X.Finally for every irreducible gerbe Y \u2286 NZIrX, the split central rank is strictlylarger than n, because at least one of the sections si is noncentral. \u0002Corollary 5.13. The operators Ir , for r \u2265 0 are simultaneously diagonalizable. Thecommon eigenspaces form a family \u03a0k(DM) of subspaces of K(DM) indexed by non-negative integers k\u2265 0, andK(DM)=\u2295k\u22650\u03a0k(DM).Let pik denote the projection onto Kk(DM). We haveIrpik = r !(kr)pik ,for all r \u2265 0, k\u2265 0.Corollary 5.14. For r \u2265 1, we havekerIr =\u2295k \u03c1(X) or\u03bd(Y) > \u03bd(X). And if Y is contained in IzX, and maps to a component of Iz\/z,0X, notof degree 1 over X, then \u03c1(X)= \u03c1(Y) but \u03bd(Y) > \u03bd(X).Proposition 6.9. Let Y be a stratum of IX not (completely) contained in IzX. Then\u03c1(Y)\u2265 \u03c1(X) and if \u03c1(Y)= \u03c1(X) then \u03bd(Y) > \u03bd(X).Proof. We use diagram 5.2 here again.IzYpi3$$Y\u00d7X IzX\u0002\/\/pi2\u000f\u000f? _joo IzXpi1\u000f\u000fY \/\/ XFrom this diagram, it is obvious that \u03c1(Y)\u2265 \u03c1(X).Now suppose \u03c1(Y) = \u03c1(X). Here j is a monomorphism of commutative groupspaces over Y thus by the previous lemma we have an induced monomorphism ofdiscrete central group schemes\u03b9 : (IzX)Y\/(IzX)0Y\u21a9 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\u000f\u000f\/\/ Iz\/z,0Xpi1\u000f\u000fY \/\/ 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\u2192Y\u00d7XY (explicitly (x,\u03d5), (x,\u03d5,[\u03d5]), where x is an object of X, \u03d5 \u2208Aut(x) and [\u03d5] is the orbit of \u03d5 by the action of connectedness component of unity).Since Y is ultra-central, this section is not in the image of \u03b9\u02dc. Therefore \u03bd(Y) > \u03bd(X). \u0002When X is a clear gerbe, the connectedness components of the central inertiaIzX\u2192X 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\u2192X 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 \u2286 IzX is an (Iz,0X)-torsor over the associatedconnected component Y\u2032 \u2282 Iz\/z,0X.Proof. It is easy to verify the isomorphism Iz(IzX)\u009b IzX\u00d7X IzX of stacks which alsofit in the commutative diagramIz(IzX)\u0002\/\/\u000f\u000fIzX\u000f\u000fZ\u00d7X Z \/\/ ZThus for any locally closed substack Y is IzX which descends to a locally closed sub-space Y of Z , we haveIz(Y)\u0002\/\/\u000f\u000fY\u000f\u000fZ\u00d7X 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 -\u2192 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 \/\/ _\u000f\u000fY\u2032 _\u000f\u000fIzX \/\/ Iz\/z,0Xtogether with a finite \u00e9tale mapping Y\u2032\u2192X proving the lemma. \u000239Proposition 6.11. Let Y be a connected component of IzX. We always have \u03bd(Y) \u2265\u03bd(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\u2032\u2286 Iz\/z,0X. The homo-morphism of commutative group schemes (IzX)Y \u2192 IzY over Y is an isomorphismgiving the left cartesian square of homomorphisms of commutative group schemesand inducing the right hand one:IzY\u0002\/\/pi2\u000f\u000fIzXpi1\u000f\u000fY \/\/ XIz\/z,0Y \/\/pi2\u000f\u000f\u0002Iz\/z,0Xpi1\u000f\u000fY \/\/ X(6.3)So distinct sections of pi1 pull back to distinct sections of pi2, therefore \u03bd(Y)\u2265 \u03bd(X).Now suppose Y\u2032 is not isomorphic to X. Then, as the structure map Y\u2192X factorsthrough Y\u2032, and yields a section of pi2 that is not induced by pi1. In this case we have\u03bd(Y) > \u03bd(X). If Y\u2032 maps isomorphically to X, the structure map Y\u2192 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, \u03bd(Y)= \u03bd(X). \u0002Remark 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\u2192 Iz\/z,0X is the pull-back of the Z0-principal bundle Z \u2192 Z\/Z0. Passing to a strata Y, we likewise observe that Y\u2192Y\u2032 isthe pull back of a Z0-torsor.6.3 An ascending filtration and local finitenessLet K\u2264(d,c,t) be the sub-module of K(St) spanned by clear gerbes X\u2192X, for which thecentral corank, is at most d, and if this number is exactly d then the maximum numberof geometrically connected components of IX\u2192 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)\u2264 corank(X) and equality happens only if c(Y) < c(X).Proof. We consider a different commutative diagramIY\"\"\u001f \/\/ IX|Y\u000f\u000f\/\/ IX\u000f\u000fY \/\/ X(6.4)40The injective morphism is given on the fiber of any point x \u2208 X by mapping of sub-group ZG(g)\u21a9G where G=Aut(x) and g \u2208G Z(G). Thus we always have dimX IX\u2265dimY IY. This together with Proposition 6.9 show that corank(Y)\u2264 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)\u21a9 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)\u21a9Gmaps connected components of ZG(g) to that of G. Thus the number of connectedcomponents of ZG(g) in strictly less than that of G. \u0002Proposition 6.14. LetY be a connected component of IzX. Then corank(Y)= corank(X)and c(X)= c(Y). Also \u03c4(Y)\u2264 \u03c4(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) \u009b IX\u00d7X IzX. But from Proposi-tion 6.10 we already know that \u03c1(Y) = \u03c1(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.\u0002Corollary 6.15. The filtration K\u2264(\u00b7,\u00b7,\u00b7) 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\u00d7U 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 \u2192 X be a smooth connected commutative group scheme of di-mension r with constant reductive rank t and maximal torus T . Then [Z]= qr\u2212t[T].Proof. Let \u03b7\u2208X be the generic point with residue field K. Over the algebraic closure,we have a decomposition ZK =TK\u00d7UK where UK is the unipotent radical and descentson K to the quotient group scheme Z\u03b7\/T\u03b7. 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\u2192 T\u03b7\u2192 Z\u03b7\u2192 Z\u03b7\/T\u03b7\u2192 141to an open subscheme U of X. This implies that [ZU ] = qr\u2212t[TU ] since the quotientZU\/TU has a composition series with r \u2212 t line bundle factors. The claim now followsby noetherian induction. \u0002Let X be a clear gerbe in K\u2264(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\u00d7XX]by Remark 6.12. Alternatively by condition (C7) we may write[Y]= q\u03c1u(X)[T \u00d7X 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\u2265r be the sub-module of K(St) spanned by clear gerbes X\u2192 X, for which theunipotent central rank, is at least r . This defines an descending filtration K\u2265\u00b7 on K(St)and we show that it is preserved by the unipotent inertia. We also define an ascendingfiltration K\u2264\u00b7 by declaring a clear gerbe X\u2192 X to be in K\u2264s if the unipotent co-rankdefined as the difference dimX IX\u2212\u03c1u(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~~ \u03c1u(X).Now the diagram 6.4 showed that dimY(IY)\u2264 dimX(IX). Therefore the unipotentcorank of Y is strictly less that that of X, proving the second claim. \u0002Corollary 6.18. The endomorphism Iu :K(St)\u2192K(St) is locally finite and triangulariz-able and the eigenvalue spectrum of it is the set of all monomials qu for u\u2265 0.42Proof. By Remark 6.12 we have Iu,z[X]= q\u03c1u(X)[X]. The rest is clear. \u0002Corollary 6.19. The endomorphism Iu is diagonalizable on K(St)[q\u22121,{qk\u22121 :k\u2265 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\u2019s structure theorem. For some pre-liminaries on group schemes of multiplicative type and unipotent group schemes werefer the reader to \u00a72.2.7.1 Motivic classes of quasi-split toriLet \u0393 be a finite group acting on the finite set r with orbit spacer\/\u0393 = {O1,\u00b7\u00b7\u00b7 ,O`} .The polynomial\u220f`i=1(q|Oi|\u22121) depends only on the sizes of the orbits. In fact, to \u0393we may associate a partition \u03bb = (\u03bbi)i\u22651 of the integer r with declaring \u03bbi to be thenumber of elements of r\/\u0393 of size i. The the above polynomial is identical toQ\u03bb =\u220f`i=1(qi\u22121)\u03bbi .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 \u0393 . Let \u039b= {b1,\u00b7\u00b7\u00b7 ,br }be a choice of basis for \u03c7T permuted by the \u0393 -action. Then the motivic class of T is given44by[T]=Q\u03bb(q)[X]+\u2211I\u2022\u2208F(r)\/\u0393Stab\u0393 (I)\u228a\u0393(\u22121)`(I\u2022)q|Imax|[X\/Stab\u0393 (I)] . (7.1)where \u03bb` r is the partition of integer r induced by the action of \u0393 on \u039b.Here, for a subset I \u2282 r , we denote by AI \u2282 Ar the subset of all (x1, . . . ,xr ) suchthat xi = 0, for i 6\u2208 I, and by GIm \u2282AI , the set of all (x1, . . . ,xr )\u2208AI , such that xi 6= 0,for all i\u2208 I. For every subset I \u2282 r , we have a \u0393 -equivariant splittingAI =\u2294J\u2282IGJm ,and hence,[GIm]= [AI]\u2212\u2211J\u228aI[GJm].By induction, we get an equivariant inclusion-exclusion principle[Grm]=\u2211k\u22650(\u22121)k\u2211I\u2022\u2208Fk(r)[AIk].Here Fk(r) is the set of all flags I\u2022 = Ik \u228a . . .\u228a I1 \u228a I0 = r in r . We denote the length ofa flag I\u2022 by k= `(I\u2022), the maximal index by k=max, and the set of all flags, regardlessof length, by F(r).Proof. \u0393 acts on the split torus TX = Spec(OX[\u03c7T ]) and by [35, Prop. 5.21] we canrevive T from this pull-back as T \u009b (X \u00d7X T)\/\u0393 . Then the surjection of Z-module,\u2295ri=1biZ\u2192 \u03c7T , induces a sheaf homomorphism OX[b1,\u00b7\u00b7\u00b7 ,br ]\u2192OX[\u03c7T ] and con-sequently an open immersion Grm,X \u009b TX \u2192 ArX , which is equivariant for the \u0393 -action.Hence we may pass to quotient schemes and getT =X\u00d7\u0393 Grm \u21a9X\u00d7\u0393 Ar .45Thus we have[T]= [X\u00d7\u0393 Grm]=\u2211k\u22650(\u22121)k[X\u00d7\u0393 \u2294I\u2022\u2208Fk(r)AIk]=\u2211k\u22650(\u22121)k\u2211I\u2022\u2208Fk(r)\/\u0393[X\u00d7Stab\u0393 (I\u2022)AIk]=\u2211I\u2022\u2208F(r)\/\u0393(\u22121)`(I\u2022)q|Imax|[X\/Stab\u0393 (I\u2022)]=\u2211I\u2022\u2208F(r)\u0393(\u22121)`(I\u2022)q|Imax|[X]+\u2211I\u2022\u2208F(n)\/\u0393Stab\u0393 (I)\u228a\u0393(\u22121)`(I\u2022)q|Imax|[X\/Stab\u0393 (I)]=Q\u2126(q)[X]+ \u2211I\u2022\u2208F(r)\/\u0393Stab\u0393 (I)\u228a\u0393(\u22121)`(I\u2022)q|Imax|[X\/Stab\u0393 (I)] .Note that all forms of affine spaces occurring in this computation are vector bundlesover their base by Hilbert\u2019s Theorem 90. This is the reason for the appearance of theterms q|Ik| in the calculation. \u00027.2 Quasi-split stacksDefinition 7.2. An algebraic stack X, is called quasi-split if for any point of its coarsemoduli space, x \u2208X, 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 \u00a79, \u00a710,and \u00a711, 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\u2192X with central group scheme Z \u2192X, 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\u2192 X. Let \u03b7= 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 \u2192 X, pullsback over \u03b7 to the algebraic K-group Z\u03b7 where the connected component of unity hasa decomposition as Z0 = T\u03b7\u00d7U\u03b7. Here T\u03b7 is a quasi-split maximal torus of Z and U\u03b7 isa unipotent algebraic K-group. It remains to observe that this decomposition spreadsout to an open neighborhood of \u03b7 in X by Corollary 2.15. \u0002Let 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 \u2192 X be a Z0-torsorand Y be the pullback Iz,0X-torsor over X. Then[Y]= qr\u2212tQ\u03bb`t(q)[X]+\u2211I\u2022\u2208F(r)\/\u0393Stab\u0393 (I)\u228a\u0393(\u22121)`(I\u2022)qr\u2212t+|Imax|[X\/Stab\u0393 (I)] , (7.2)for some finite \u00e9tale covering X\u2192X and an action of \u0393 =pi1(X\/X) on the set {1,\u00b7\u00b7\u00b7 , t}.Here \u03bb ` t is the integer partition type of the orbit space t\/\u0393 in the sense of \u00a77.1 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\u2192X. \u00027.3 An ascending filtration and local finitenessRecall the ascending filtration of \u00a76.3 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\u2192X, 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\u2286 IX is contained in K<(d,c,t) unless Y isan Iz,0X torsor. We denote the associated graded piece by K\u2264(d,c,t)\/K<(d,c,t).Proposition 7.7. The endomorphism I : K(QS)\u2192 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 \u0393 be defined as in Proposition 7.6. Consider the finite-dimensionalZ[q]-submodule, L generated by the finitely many intermediate covers X\/Stab\u0393 (I).Then the results of \u00a76.3 and Proposition 7.6 show thatI(L)\u2286 L modK<(d,c,t).The claim is now clear by induction. \u00027.4 A descending filtration and diagonalizationLet X\u2192 X be a quasi-split clear gerbe with a central X-group scheme Z of reductiverank t. The Galois group \u0393 = 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 \u03bb` t as explained in \u00a77.1.Definition 7.8. For a quasi-split clear gerbe X, the partition \u03bb 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\u2265(\u00b7,\u00b7). We now introduce a refinement, K\u2265(\u00b7,\u00b7,\u00b7) 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 \u2265 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 \u03bb` t and \u00b5 ` s and t < s then \u03bb < \u00b5; and2. if \u03bb,\u00b5 ` t and b(\u03bb) < b(\u00b5) then \u03bb < \u00b5.Now, given integers r ,n\u22650 and an integer partition \u03bb` t, the submodule K\u2265(r ,n,\u03bb)is generated by those quasi-split clear gerbes in K\u2265(r ,n) that have twist type is at least\u03bb.Lemma 7.9. The inertia endomorphism of K(QS) respects the filtration K\u2265(\u00b7,\u00b7,\u00b7).Proof. In view of results of section 6.2 it suffices to consider a tuple (r ,n,\u03bb), and aquasi-split clear gerbe X\u2192X with central rank r , discrete split central number n, andtwist type \u03bb. 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\/Stab\u0393 (I) for aminimal splitting cover X\u2192X of the maximal torus T \u2192X 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\/Stab\u0393 (I). Since \u0393 \u2032 = StabI is a proper subgroup of the Galois group \u0393 =pi1(X\/X),the orbit space r\/\u0393 \u2032 has strictly more elements than r\/\u0393 . Therefore the twist type ofX\/Stab\u0393 (I) is strictly bigger than that of X in the well-ordering defined above.\u0002We can now finish the proof of our main result.Theorem 7.10. The operator I is diagonalizable on K(QS)(q)=Q(q)\u2297Q[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\u220fi=1(qri \u22121).Proof. In Lemma 7.9 we showed that I is triangularizable and by Proposition 7.6 theaction of it on each graded piece K\u2265(r ,n,\u03b3)\/K>(r ,n,\u03b3) is multiplication by the polyno-mial nqr\u2212tQ\u03b3`t(q). This polynomial uniquely determines the triply (r ,n,\u03bb) resultingdistinct eigenvalues associated to each graded piece. \u0002Remark 7.11. In fact I is diagonalizable as an endomorphism of the Z[q]-moduleK(QS)[q\u22121,{(Q\u03bb\u2212Q\u00b5)\u22121 :\u2200\u03bb` t,\u00b5 ` 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 \u03bd(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. Then\u03c1(Y)\u2265 \u03c1(X) and if \u03c1(Y)= \u03c1(X) then \u03bd(Y) > \u03bd(X).Proof. Let Y\u2286 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 \u2192 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\u00d7X Iss,zX\u0002\/\/pi2\u000f\u000f? _joo Iss,zXpi1\u000f\u000fY \/\/ X(7.3)From previous diagram it is obvious that \u03c1(Y)\u2265 \u03c1(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\u000f\u000f\/\/ Iss,z\/ss,z,0Xpi1\u000f\u000fY \/\/ X(7.4)in level of stacks. Same analysis as in case of Proposition 6.9 shows that pi3 has strictlymore sections that pi1. \u0002When X is a quasi-split clear gerbe, the connectedness components of the semisim-ple central inertia Iss,zX\u2192 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\u2192 X. Then \u03c1(Y) =\u03c1(X). We always have \u03bd(Y)\u2265 \u03bd(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)\u0002\/\/\u000f\u000fIss,zX\u000f\u000f\/\/ X\u000f\u000fZss \u00d7X Zss \/\/ Zss \/\/ Xfor the first claim where by restricting to Y we getIss,z(Y)\u0002\/\/\u000f\u000fY\u000f\u000fZss \u00d7X Y \/\/ Y .50Note that Iz,ssX -\u2192 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\u2032 \u2286 Iz\/z,0X. Similar to the case in Proposition 6.11 we now can form thefollowing cartesian diagramIss,z\/ss,z,0Y \/\/pi2\u000f\u000f\u0002Iss,z\/ss,z,0Xpi1\u000f\u000fY \/\/ Xand the rest of the proof is now similar to Proposition 6.11. \u0002The 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\u2192 X is the base change of aquasi-split torusIss,z,0X \/\/\u000f\u000fX\u000f\u000fT \/\/ 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]= \u03bd(X)[Iss,z,0X]+ terms of lower co-untwistedness= \u03bd(X)Q\u03bb\u00eer (q)[X]+ terms with finer twist types \u03b3 > \u03bb.We may now consider a simpler filtration, K\u2265(\u00b7,\u00b7,\u00b7) than before: given integersr ,n\u22650, and a partition \u03bb` r the submodule K\u2265(r ,n,\u03bb) 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 \u03bb. This filtration is preserved by the semisimple51inertia endomorphism and leads toTheorem 7.14. The endomorphism Iss :K(QS)\u2192K(QS) is locally finite and triangular-izable and the eigenvalue spectrum of it is the set of all polynomials of the formnk\u220fi=1(qri \u22121).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 \u2192 GL2 via x,(x 00 x). This is equivariantwith respect to the natural GL2-action, so we get an induced morphism of stacksGm\u00d7BGL2\u2192 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 \u2206 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 \u00cfZ2 is in fact a semi-directproduct, by taking(0 11 0)as the nontrivial element of Z2 \u2282 N . The induced action ofthe Weyl group Z2 on T is by swapping the two entries. The natural inclusion mapT \u2206\u2192 GL2 is equivariant for the inclusion of groups N \u2282 GL2, so we get an inducedmorphism of stacks(T \u2206)\/N -\u2192 GL2 \/GL253which is an isomorphism onto the second stratum. We will abbreviate this asX= (T \u2206)\/N.Third stratum: Let H be the (commutative) subgroup of all matrices of the form(\u03bb \u00b50 \u03bb). 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\u00d7BH.We conclude that in the level of motivic classes the inertia of the class [BGL2] isgiven byI[BGL2]= (q\u22121)[BGL2]+ [X]+ (q\u22121)[BH].Since H is commutative, we also have I[BH]= q(q\u22121)[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 \u2206\/Z2. Note also that IX = IzX as the stabilizer of any pointin T \u2206 is commutative. We will write X\u02dc = T \u2206 to emphasize the fact that T \u2206 is adegree 2 cover of X.Associated to the Z2-action on the group T , there exists a commutative X-groupschemeT \u2032 = X\u02dc\u00d7Z2 T ,with fibre T . By Lemma 8.1, X is the neutral gerbe, X= BXT \u2032. And IzX= IX fits in thecartesian diagramIX \/\/\u000f\u000fX\u000f\u000fT \u2032 \/\/ X.The representation of Z2 on A2 given by swapping entries, yields a canonicalclosed embedding T \u2032 \u2282 V , into a rank 2 vector bundle over X. As in Proposition 7.1,this leads toI[X]= (q2\u22121)[X]\u2212 (q\u22121)2(q\u22122)[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\u22121,q(q\u22121),q2\u22121,(q\u22121)2}. Inertia endomorphism is lower triangularizable54on L and we haveI =\uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8edq\u22121 0 0 0q\u22121 q(q\u22121) 0 01 0 q2\u22121 00 0 \u2212(q\u22121)2(q\u22122) (q\u22121)2\uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8with a set of eigenvectorsEigenvalues Eigenvectorsq\u22121 \u2212q(q\u22121)[BGL2]+q[BH]+ [X]+ (q\u22121)[BG2m]q(q\u22121) [BH]q2\u22121 [X]\u2212 (q\u22121)(q\u22122)2 [BG2m](q\u22121)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\u22121,(q\u22121)\u22121] and the eigenprojections of [BGL2]areEigenvalues Eigenvectors\u03a0q\u22121 [BGL2]\u2212 qq\u22121 [BH]\u2212 1q(q\u22121) [X]\u2212 1q [BG2m]\u03a0q(q\u22121) qq\u22121 [BH]\u03a0q2\u22121 1q(q\u22121) [X]\u2212 (q\u22122)2q [BG2m]\u03a0(q\u22121)2 12 [BG2m]Table 8.2: Eigenprojections of [BGL2]Lemma 8.1. Let \u0393 be a group acting on the group T by automorphisms, \u0393 \u2192Aut(T), andlet G=N\u00cfH be the associated semi-direct product of groups. Let X be a variety, X\u02dc\u2192Xa principal \u0393 -bundle, and T \u2032\u2192X is the associated form of T over X. The BXT \u2032 = X\u02dc\/G.Proof. Consider the diagramX\u02dc\u00d7T \/\/\u000f\u000fX\u02dc\u00d7T\u000f\u000fX\u02dc \/\/ X\u02dc55where \u0393 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 \u0393 \u2192G equivariant. Thuswe get an induced cartesian diagram of stacks(X\u02dc\u00d7T)\/\u0393 \/\/\u000f\u000f(X\u02dc\u00d7T)\/G\u000f\u000fX\u02dc\/H \/\/ X\u02dc\/Gwhich we may rewrite asT \u2032 \/\/\u000f\u000fX\u000f\u000fX \/\/ X\u02dc\/G.Then the latter diagram induces a morphism X \u2192 X\u02dc\/G, which is then obviously anisomorphism. \u0002Example 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 = G2m\u00cfZ2 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\u00d7{\u03c3}\/Nwhere \u03c3 is the nontrivial element of Z2.First stratum: This stratum is not already a gerbe (as the stabilizer of points ondiagonal \u2206 \u2282 G2m is not isomorphic to the stabilizer of other points). However thefollowing is a stratification of it into clear gerbes:G2m\/N =\u2206\/NunionsqX,where X = G2m \u2206\/N is the same quotient stack that appeared in previous example.The action of N on \u2206 is trivial so we have[G2m\/N]= (q\u22121)[BN]+ [X].Second stratum: This stratum is already a clear gerbe and we will denote it as Y.Any point \u3008(\u00b5,\u03b3),\u03c3\u3009 of G2m\u00d7{\u03c3} is conjugate to \u3008(\u00b5\u03b3,1),\u03c3\u3009 which is canonical forthe orbit. Thus the subscheme Y representing the points,{\u3008(x,1),\u03c3\u3009} \u2282G2m\u00d7{\u03c3},is a coarse moduli space for this gerbe. This is isomorphic to A1 {0}. The stabilizerof this subscheme is a subgroup scheme N\u2032 \u2286N , the fiber of which over a geometric56point x of Y is the \u03ba(x)-algebraic groupN\u2032x = {\u3008(t,t),1\u3009 : t \u2208 \u03ba(s)\u00d7}\u222a{(xt,t),\u03c3\u3009 : t \u2208 \u03ba(s)\u00d7}.We notice that the mapping Y\/N\u2032\u2192Y is an isomorphism of stacks and also that N\u2032 isa commutative Y -group scheme, acting trivially on Y . Therefore[Y]= [Y][BN\u2032]= (q\u22121)[BN\u2032]I[BN\u2032]= [N\u2032\/N\u2032]= [N\u2032][BN\u2032]Finally N\u2032\/(N\u2032)0\u2192 Y is a degree two covering of X. The image of (N\u2032)0 in N\u2032\/(N\u2032)0 isisomorphic to Y and therefore so is the image of the other connected component. SoN\u2032 is Zariski locally the union of two Gm-torsors over Y . Pulling back along Y\u2192 Y wehaveI[BN\u2032]= 2(q\u22121)[BN\u2032].We conclude that the K(Var)-submodule of K(St) generated by[BN],[BN\u2032],[X], and [BG2m]is invariant under inertia endomorphism. The first two generators have central rankone, and [BN] has split central number one whereas [BN\u2032] has split central numbertwo. The spectrum of I restricted to this submodule is the set{(q\u22121),2(q\u22121),q2\u22121,(q\u22121)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 Jk\u03bb be the subscheme of all general linear matrices withk-distinct eigenvalues and \u03bb \u00ee 3 is a partition of 3 indicating format of the Jordanblocks and Rk\u03bb \u21d2 Jk\u03bb is the groupoid representation of restriction of [GL3 \/GL3] to Jk\u03bb .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 Rk\u03bb on Jk\u03bb 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)\u00d7BR1(3)unionsqJ1(2,1)\u00d7BR1(2,1)unionsqJ1(1,1,1)\u00d7BR1(1,1,1)unionsqJ2(2,1)\u00d7BR2(2,1)unionsqJ2(1,1,1)\u00d7BR2(1,1,1)unionsqJ3(1,1,1)\u00d7BR3(1,1,1)We recall the notation of Example 5 for the subgroup of upper-triangular 2\u00d72 matriceswith a single eigenvalue of multiplicity two:H =\uf8f1\uf8f2\uf8f3\uf8eb\uf8eda b0 a\uf8f6\uf8f8 : a,b \u2208Gm\uf8fc\uf8fd\uf8fe .This represents a commutative group scheme. Now, easy computations show that allRk\u03bb\u2019s are subgroup schemes of GL3 and in factGroupoid Group scheme structure Commutative?R1(3)\uf8f1\uf8f4\uf8f2\uf8f4\uf8f3\uf8eb\uf8ec\uf8eda b c0 a b0 0 a\uf8f6\uf8f7\uf8f8 : a\u2208Gm,b,c \u2208A1\uf8fc\uf8f4\uf8fd\uf8f4\uf8fe YesR1(2,1)\uf8f1\uf8f4\uf8f2\uf8f4\uf8f3\uf8eb\uf8ec\uf8eda b c0 a 00 d e\uf8f6\uf8f7\uf8f8 : a,e\u2208Gm,b,c,d\u2208A1\uf8fc\uf8f4\uf8fd\uf8f4\uf8fe NoR1(1,1,1) GL3 NoR2(2,1) H\u00d7Gm YesR2(1,1,1) GL2\u00d7Gm NoR3(1,1,1) G3m\u00cfS3 YesTable 8.3: Stratification of GL3[GL3 \/GL3]= (q\u22121)[BR1(3)]+ (q\u22121)[BR1(2,1)]+ (q\u22121)[BGL3]+ (q\u22121)(q\u22122)[BH][BGm]+ (q\u22121)(q\u22122)[BGL2][BGm]+ [G3m\/G3m\u00cfS3]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\u22121)[BR1(3)].58Strata Canonical form for an orbit Centralizer of the canonical form\uf8eb\uf8ec\uf8eda b c0 a 00 d e\uf8f6\uf8f7\uf8f8 : a\u2260 e\uf8eb\uf8ec\uf8eda b+cd\/(a\u2212e) 00 a 00 0 e\uf8f6\uf8f7\uf8f8 G1 =\uf8f1\uf8f4\uf8f2\uf8f4\uf8f3\uf8eb\uf8ec\uf8edx y 00 x 00 0 z\uf8f6\uf8f7\uf8f8 : x,z \u2260 0\uf8fc\uf8f4\uf8fd\uf8f4\uf8fe\uf8eb\uf8ec\uf8eda b c0 a 00 d a\uf8f6\uf8f7\uf8f8 : c,d\u2260 0\uf8eb\uf8ec\uf8eda 0 10 a 00 cd a\uf8f6\uf8f7\uf8f8 G2 =\uf8f1\uf8f4\uf8f2\uf8f4\uf8f3\uf8eb\uf8ec\uf8edx y z0 x 00 w x\uf8f6\uf8f7\uf8f8 : x \u2260 0\uf8fc\uf8f4\uf8fd\uf8f4\uf8fe\uf8eb\uf8ec\uf8eda b 00 a 00 d a\uf8f6\uf8f7\uf8f8 : d\u2260 0\uf8eb\uf8ec\uf8eda b+d 00 a 00 d a\uf8f6\uf8f7\uf8f8 G3 =\uf8f1\uf8f4\uf8f2\uf8f4\uf8f3\uf8eb\uf8ec\uf8edx y 00 x 00 z x\uf8f6\uf8f7\uf8f8 : x \u2260 0\uf8fc\uf8f4\uf8fd\uf8f4\uf8fe\uf8eb\uf8ec\uf8eda b c0 a 00 0 a\uf8f6\uf8f7\uf8f8 : c \u2260 0\uf8eb\uf8ec\uf8eda 0 10 a 00 0 a\uf8f6\uf8f7\uf8f8 G4 =\uf8f1\uf8f4\uf8f2\uf8f4\uf8f3\uf8eb\uf8ec\uf8edx y z0 x 00 0 x\uf8f6\uf8f7\uf8f8 : x \u2260 0\uf8fc\uf8f4\uf8fd\uf8f4\uf8fe\uf8eb\uf8ec\uf8eda b 00 a 00 0 a\uf8f6\uf8f7\uf8f8 itself GTable 8.4: Stratification of R1(2,1)The case of Y = [G3m\/G3m\u00cfS3] is similar to that of [G2m\/G2m\u00cfZ2]. 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\u22121)[BG]+q3(q\u22121)(q\u22122)[BG1]+q(q\u22121)3[BG2]+q(q\u22121)2[BG3]+q(q\u22121)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\u221212 2 (2,0) [BGL2][BGm] (q\u22121)21 (1) [BG] q(q\u22121)3 3 (0,0,1) [Y] q3\u22121(1,1,0) [X][BGm] (q2\u22121)(q\u22121)(3,0,0) [BG3m] (q\u22121)32 (2,0) [BH][BGm],[BG1] q(q\u22121)21 (1) [BR1(3)],[BG3],[BG4] q2(q\u22121)4 1 (1) [BG2] q3(q\u22121)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 \u00e9tale 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 \u21d2 X0, where X0 and X1 are algebraic spaces, locally of finite type, the source andtarget morphism s,t : X1 \u2192 X0 are smooth, and the diagonal X1 \u2192 X0\u00d7X0 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\u2192S. The category X inheritsa topology from S, called the \u00e9tale topology, and X endowed with this topology is thebig \u00e9tale site of X. Sheaves over X are by definition sheaves on this big \u00e9tale 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) \u00e9tale site of the scheme U , denoted FU . Moreover, forevery morphispm \u03b1 : y \u2192 x lying over f : V \u2192 U , we obtain a morphism of sheaves\u03b1\u2217 :FU \u2192f\u2217FV . (The \u03b1\u2217 satisfy an obvious cocycle condition.) The data of the sheaves61FU , together with the compatibility morphisms \u03b1\u2217, 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 \u03b1\u2217 : f\u2217FU \u2192FV are isomorphisms ofsheaves of OV -modules.For example, a groupoid presentation X1 \u21d2 X0 of X, and a coherent sheaf F0 onX0, together with an isomorphism s\u2217F0\u2192 t\u2217F0, satisfying the usual cocycle conditionon X2 =X1\u00d7X0 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 \u2192X, with repre-sentable structure morphism Y \u2192X, such thatF is isomorphic to the sheaf of sectionsof Y \u2192X.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\u2228\u2228U . So F isrepresentable if and only if F is isomorphic to F\u2228\u2228 as an OX -module. If F is repre-sentable, it is represented by the finite type affine X-scheme Y = SpecX(SymOXF\u2228).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,Expos\u00e9 VI].Suppose X\u2192S is a category over S. We write X(S) for the fiber of X over theobject S of S. If f : S\u2032 \u2192 S is a morphism in S, and x\u2032 \u2208 X(S\u2032) and x \u2208 X(S) are X-objects lying over S\u2032 and S, respectivly, we write Homf (x\u2032,x) for the set of morphismsfrom x\u2032 to x in X, lying over f . For S\u2032 = S and f = idS , we write HomS(x\u2032,x).Recall that a morphism \u03b1 :x\u2032\u2192x lying over f : S\u2032\u2192 S is cartesian, if for every ob-ject x\u2032\u2032 of X(S), composition with\u03b1 induces a bijection HomS(x\u2032\u2032,x\u2032)'-\u2192Homf (x\u2032\u2032,x).Recall further that X\u2192S is a fibered category, if every composition of cartesian mor-phisms is cartesian, and if for every f : S\u2032 \u2192 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\u2032 \u2192 S in S and all x\u2032 \u2208 X(S\u2032), x \u2208 X(S), the set Homf (x\u2032,x) is endowed with the62structure of an O(S\u2032)-module, in such a way that for every pair of morphisms g :S\u2032\u2032 \u2192 S\u2032, f : S\u2032 \u2192 S, and every triple of objects x\u2032\u2032 \u2208 X(S\u2032\u2032), x\u2032 \u2208 X(S\u2032), x \u2208 X(S), thecompositionHomf (x\u2032,x)\u00d7Homg(x\u2032\u2032,x\u2032) -\u2192Homf\u25e6g(x\u2032\u2032,x)is O(S\u2032)-bilinear. Here the O(S\u2032\u2032)-modules, Homg(x\u2032\u2032,x\u2032) and Homf\u25e6g(x\u2032\u2032,x) inheritthe structure of O(S\u2032)-modules via pullback along g.An O-linear functor F : X \u2192 Y between O-linear categories is a functor of cate-gories over S, such that for every f : S\u2032 \u2192 S, and all x\u2032 \u2208 X(S\u2032), x \u2208 X(S) the mapHomf (x\u2032,x)\u2192Homf(F(x\u2032),F(x))is O(S\u2032)-linear.Let X be an O-linear fibered category over S. Pullback in X is O-linear, i.e., iff :S\u2032\u2192S is a morphism inS, and x,y \u2208X(S) are objects with pullbacks x\u2032,y\u2032\u2208X(S\u2032),the pullback map f\u2217 : HomS(x,y)\u2192HomS\u2032(x\u2032,y\u2032) is O(S)-linear. So if we fix objectsx,y \u2208X(S), the presheaf HomS(x,y) over the usual (small) \u00e9tale site of S, defined byHomS(x,y)(T)=HomT (x|T ,y|T ), for every \u00e9tale T \u2192 S, is a presheaf of OS -modules.Moreover, for any morphism f : S\u2032 \u2192 S in S, we have a natural homomorphism ofsheaves of OS\u2032 -modules f\u2217HomS(x,y)\u2192HomS\u2032(x\u2032,y\u2032).Definition 9.2. A linear algebraic stack is an O-linear fibered category X over S, suchthat(i) for every object S \u2208S, and every pair x,y \u2208 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\u2032 \u2192 S the pullback homomorphism f\u2217HomS(x,y)\u2192HomS\u2032(x\u2032,y\u2032) is an isomorphism of coherent OS\u2032 -modules; and,(iii) the underlying category fibered in groupoids Xcfg\u2192S 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\u00d7X, which represents the sheafover X \u00d7X, whose set of sections over the pair x,y \u2208 X(S) is the O(S)-moduleHomS(x,y). The sheaf H is the universal sheaf of homomorphisms. The subsheafI\u2282H representing isomorphisms is naturally identified with X, and the projection toX\u00d7X 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 \u2208 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\u00d7X -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 \u00d7S, which are flat over S. Fora morphism of R-schemes f : S\u2032 \u2192 S, and F\u2032 \u2208 CohX(S\u2032), and F \u2208 CohX(S), we setHomf (F\u2032,F)=HomOX\u00d7S\u2032 (F\u2032,f\u2217F). A morphismF\u2032\u2192F in CohX over f in S is carte-sian, if it induces an isomorphism F\u2032 \u009b f\u2217F.The linear stack CohX is algebraic. Let pi : X \u00d7 S \u2192 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\u2217HomS(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 [34,4.6.2.1].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\u2294n\u22650BGLn. Thesheaf H over \u2294n\u22650BGLn\u00d7\u2294n\u22650BGLn =\u2294n,m\u22650B(GLn\u00d7GLm)is given by the natural representation M(m\u00d7n) of GLn\u00d7GLm over the componentB(GLn\u00d7GLm).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\u2032 \u2192 S of R-schemes we have Homf (F\u2032,F) is the O(S\u2032)-module of homomorphisms F\u2032\u2192 f\u2217F of diagrams of locally free OS\u2032 -modules.Example 11. As a toy example, let A be an R-algebra scheme of finite type, with groupscheme of units A\u00d7, also of finite type. Then we define the linear stack of A\u00d7-torsors64to have as objects over the R-scheme S the right A\u00d7-torsors over S, and for f : S\u2032 \u2192S and A\u00d7-torsors P \u2032 over S\u2032 and P over S, we set Homf (P \u2032,P) = HomS\u2032(P \u2032,f\u2217P) =P \u2032\u00d7A\u00d7 A\u00d7A\u00d7 f\u2217P . In this example, the underlying algebraic stack is BA\u00d7 and we haveH =A\u00d7\\A\/A\u00d7.The case A = 0 is not excluded. The associated linear stack is id : S \u2192 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 \u2282 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\u2192 Z and G : Y\u2192 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,\u03b1,y), where x is an X-object over T , y is a Y-object over T , and \u03b1is an isomorphism \u03b1 : F(x)\u2192G(y), over T . A morphism from (x\u2032,\u03b1\u2032,y\u2032) to (x,\u03b1,y)over T \u2032\u2192 T is a pair of morphisms f : x\u2032\u2192 x over T \u2032\u2192 T and g :y\u2032\u2192y over T \u2032\u2192 T ,such that \u03b1\u25e6F(f)=G(g)\u25e6\u03b1\u2032.In other words, we can write the set of morphisms from (x\u2032,\u03b1\u2032,y\u2032) to (x,\u03b1,y)over \u03d5 : T \u2032\u2192 T as the fibered productHom\u03d5(x\u2032,x)\u00d7Hom\u03d5(F(x\u2032),G(y))Hom\u03d5(y\u2032,y),and as each of the sets in this fibered product is an O(T \u2032)-module, and the maps arelinear, this fibered product is also an O(T \u2032)-module. We leave it to the reader to verifythat composition is bilinear.Let us verify that W is a fibered category. Suppose that (x,\u03b1,y) is a triple over T ,and \u03d5 : T \u2032 \u2192 T a morphism in S. We construct a triple (x\u2032,\u03b1\u2032,y\u2032) over T \u2032 by takingas x\u2032 a pullback of x via \u03d5, and for y\u2032 a pullback of y via \u03d5. Then, as G is cartesian,G(y\u2032) is a pullback of G(y) via \u03d5. Hence there exists a unique morphism \u03b1\u2032 : F(x\u2032)\u2192G(y\u2032) covering T \u2032, such that \u03b1 \u25e6 F(x\u2032 \u2192 x) = G(y\u2032 \u2192 y) \u25e6\u03b1\u2032. Then \u03b1\u2032 is cartesian,because cartesian morphisms satisfy the necessary two out of three property. Then\u03b1\u2032 is invertible, because cartesion morphisms covering an identity are invertible. Thetriple (x\u2032,\u03b1\u2032,y\u2032) comes with a given morphism to (x,\u03b1,y) which covers\u03d5. It is easilyverified that this morphism is cartesian.65Therefore, W is an O-linear fibered category. By construction, the two projectionsW\u2192X and W\u2192Y 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\u2032,\u03b1\u2032,y\u2032) and (x,\u03b1,y) over S, the presheafHomS((x\u2032,\u03b1\u2032,y\u2032),(x,\u03b1,y))is equal to the fibered productHomS(x\u2032,x)\u00d7HomS(Fx\u2032,Gy)HomS(y\u2032,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\u00d7Z 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\u2192 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\u2192 X is an algebra over the algebraic stack X, we have a tautological mor-phism of sheaves of groups over XIX -\u2192AutX(A). (10.1)Here IX is the inertia stack of X, i.e., the stack of pairs (x,\u03d5), where x is an object of X,and\u03d5 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) \u00e9tale site of S. A morphism from the sheaf of OS\u2032 -algebras A\u2032 over S\u2032,covering the morphism of schemes f : S\u2032 \u2192 S, to the sheaf of OS -algebras A over S,is, by definition, an isomorphism of sheaves of OS\u2032 -algebras A\u2032 \u2192 f\u2217A. The sheaf ofalgebras A\u2192 X gives rise to a morphism of S-stacks a : X \u2192 Alg. We get an inducedmorphism on inertia stacks IX \u2192 IAlg, and notice that a\u2217IAlg =AutX(A).With this definition, an automorphism \u03d5 of the object x of the stack X is mappedto the inverse of the restriction morphism \u03d5\u2217 :A(x)\u2192A(x).67Lemma 10.2. Suppose X is a gerbe over the algebraic space S, and A\u2192X is an algebra.Then there exists a sheaf of OY -algebras B, and an isomorphism A\u009b B|X if and only ifthe inertia representation IX \u2192AutX(A) is trivial.If this is the case, then A is representable or of finite type if and only if B is. \u0002We can pull back the sheaf of algebras A over X, via the structure morphismIX \u2192X, 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 \u2208 X an action of Aut(x) on Ax . Therefore the objects of AIX overan object x of X, are triples (x,\u03d5,a) where \u03d5 is an automorphism of x, and a \u2208 Axis an object of A lying over x. The tautological automorphism of AIX maps (x,\u03d5,a)to (x,\u03d5,\u03d5(a)). The algebra of invariants for this automorphism consists of objects(\u03d5,a) such that \u03d5(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 \/\/\u000f\u000fA\u000f\u000fIX \/\/ Xwhich identifies IA with a substack of A|IX . In fact, the factorization IA \u2192 A|IX is amonomorphism if and only if A\u2192X is representable [46, Tag 04YY]. The algebra A|IXis the stack of triples (x,\u03d5,a), where x is an object of X, \u03d5 is an automorphism ofx, and a \u2208 A(x) is an object of A lying over x. Such a triple is in IA, if and only if\u03d5\u2208Aut(x) is in the subgroup Aut(a)\u2282Aut(x). This is equivalent to\u03d5 fixing a underthe action of Aut(x) on A(x). This is the claim. \u0002In fact, the fibre of IA over the objects x of X is equal toIA(x)= {(\u03d5,a)\u2208Aut(x)\u00d7A(x) |\u03d5\u2217(a)= a} .The fibre of IA(x) over \u03d5 \u2208 Aut(x) is the subalgebra A(x)\u03d5 \u2282 A(x), and the fibre ofIA(x) over a\u2208A(x) is the subgroup StabAut(x)(a)\u2282Aut(x).68Algebra bundlesDefinition 10.4. We call a finite type algebra A\u2192X an algebra bundle, if the underly-ing OX -module is locally free (necessarily of finite rank).Lemma 10.5. Let A\u2192 X be a finite type algebra over the algebraic stack X. There is anon-empty open substack U \u2282X, 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 \u2192 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. \u0002Remark 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\u2228,Hom(E,O))=Hom(E\u2228\u2297E,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 \u2192 EndO(A), given by a , [a,\u00b7]. As such, it is a representablecoherent sheaf itself. \u0002For a representable algebra A over X, we denote the closed substack of idempo-tents in A by E(A).Lemma 10.8. Suppose A\u2192X is a commutative finite type algebra. The stack E(A)\u2192Xof 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 \u2282 T \u2032 be a square zero extension of affineschemes, with ideal I, and e,f two n\u00d7n matrices with entries in O(T \u2032), which agreeon T . Hence the entries of e\u2212f are in I, which implies that (e\u2212f)2 = 0. Therefore, wehave0= e(e\u2212f)2 = e(e\u22122ef +f)= e\u22122ef +ef = e\u2212ef .So we have e = ef , and by symmetry also f = fe, and as e and f commute, we havee= f . \u0002Corollary 10.9. If A\u2192 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 )\u2192U if finite \u00e9tale.Proof. First we use generic flatness, and the fact that a flat and unramified morphismsis necessarily \u00e9tale, to prove that E(A)\u2192 X is generically \u00e9tale. Then we use Zariski\u2019smain theorem to prove that E(A)\u2192X is generically finite. \u0002By 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 \u00e9tale over X.Also note that a finite \u00e9tale 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 \u00e9talestack of idempotents E(A)\u2192X. An idempotent local section of A\u2192X is primitive, ifit is a primitive idempotent locally in the smooth topology.Lemma 10.11. Let A\u2192 X be a commutative finite type algebra, with finite \u00e9tale stackof idempotents E(A)\u2192 X. There is an open and closed substack PE(A) \u2282 E(A), suchthat an idempotent factors through PE(A) if and only if it is primitive.Proof. We may assume that I(A)\u2192 X is constant. Then the multiplication operationand the partially defined addition operation on E(A) are also constant. The claimfollows. \u000270Definition 10.12. Let A\u2192 X be a finite type algebra, with centre Z \u2192 X. Let ZE(A)be the stack of idempotents in Z , in other words the stack of central idempotents inA. Assume that ZE(A)\u2192 X is finite \u00e9tale. The substack of primitive idempotents inZE(A) is denoted by PZE(A), and called the stack of primitive central idempotentsof A. It is finite \u00e9tale over X. The degree of PZE(A)\u2192 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\u2192X be a commutative finitetype algebra, with finite \u00e9tale stack of idempotents E(A)\u2192 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 \u0393(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\u2192X be a commutative finite type algebra whose stack of idempotents E(A) isfinite \u00e9tale. Consider the finite \u00e9tale cover of primitive idempotents pi : PE(A)\u2192X. Wehave a tautological global section e of A|PE(A), and a, ae defines a homomorphismof OPE(A)-modules OPE(A)\u2192A|PE(A). Pushing forward with pi and composing with thetrace map pi\u2217(A|PE(A))\u2192A defines the morphism of algebras over Xpi\u2217OPE(A) -\u2192A. (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\u2217OPZE(A) -\u2192A, (10.3)whose image is the semisimple centre Z(A)ss .71Permanence of rank and split rankProposition 10.14. LetA\u21a9A\u2032 be a monomorphism of commutative finite type algebraswith finite \u00e9tale stacks of idemtpotents over the smooth and connected stack X. Denotethe ranks of A and A\u2032 by n and n\u2032, and the split ranks by k and k\u2032, repsectively. Thenn\u2264n\u2032 and k\u2264 k\u2032. Morover,(i) if A\u2032 admits a semisimple global section, which is not in A, then n0(M) in K(M)= K\u22650(M), i.e.,K(M)=K(Var)\u2295K>0(M). In particular, we haveK\u22650(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\u2192M] to [Er (X)\u2192X \u2192M], 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) \u2282 K(M), the image (1-eigenspaces) is K(Var) \u2282K(M).For r = 1, the operator E1 vanishes on stack functions [X \u2192M], 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 \u2265 0, preserve the filtration K\u2265k(M) by splitcentral rank. On the quotient K\u2265k(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 \u2192M defining the stackfunction [X\u2192M] in K(M). Let n be the central rank of X, and k the split central rankof X. The filtered piece K\u2265k(M) is generated by such [X \u2192M].Denote by X \u2192 X the coarse space of X, which is a non-singular variety, by as-sumption.Let X\u02dc \u2192 X be a connected Galois cover with Galois group \u0393 , which trivializesPZE(A) \u2192 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\u02dc \u2192X is inertiapreserving and hence X\u02dc inherits, via pullback, the structure of an algebroid, and hence[X\u02dc \u2192X \u2192M] is a stack function.Recall that the degree of the cover PZE(A)\u2192 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\u02dc, we obtain an actionof \u0393 on the set n= {1, . . . ,n} and an isomorphism of finite \u00e9tale covers of XX\u02dc\u00d7\u0393 n '-\u2192 PZE(A)[x,\u03bd] 7 -\u2192 e[x,\u03bd] .Both source and target of this isomorphism support natural algebroids and the iso-morphism preserves them.The number of orbits of \u0393 on n is k.Then we also have an isomorphismX\u02dc\u00d7\u0393 Epi(n,r) '-\u2192 ZEr (A)[x,\u03d5] 7 -\u2192( \u2211\u03d5(\u03bd)=\u03c1e[x,\u03bd])\u03c1=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 \u2192M]= [X\u02dc\u00d7\u0393 Epi(n,r)\u2192M]= [X\u02dc\u00d7\u0393 unionsq\u03d5\u2208Epi(n,r)\/\u0393 \u0393\/Stab\u0393\u03d5\u2192M]=\u2211\u03d5\u2208Epi(n,r)\/\u0393[X\u02dc\/Stab\u0393\u03d5\u2192M]=\u2211\u03d5\u2208Epi(n,r)\u0393[X \u2192M]+\u2211\u03d5\u2208Epi(n,r)\/\u0393Stab\u0393 \u03d5 6=\u0393[X\u02dc\/Stab\u0393\u03d5\u2192M].Now, we have Epi(n,r)\u0393 = Epi(n\/\u0393 ,r ), and hence#Epi(n,r)\u0393 = r !S(|\u03bb|,r )= r !S(k,r).Thus, we conclude,ZEr [X \u2192M]= r !S(k,r)[X \u2192M]+\u2211\u03d5\u2208Epi(n,r)\/\u0393Stab\u0393 \u03d5 6=\u0393[X\u02dc\/Stab\u0393\u03d5\u2192M].For any proper subgroup \u0393 \u2032 \u2282 \u0393 , the quotient X\u2032 = X\u02dc\/\u0393 \u2032 is an intermediate cover X\u02dc \u2192X\u2032\u2192X, such that X\u2032 6=X. The pullback of PZE(A) to X\u2032 has more than k components,because the number of orbits of \u0393 \u2032 on n is larger than k. Thus we have proved thetheorem for ZEr , instead of Er .Now observe that ZEr (A) \u2282 Er (A) is a closed substack, because ZEr (A)\u2192 X isproper and Er (A)\u2192X is separated. So we can writeEr [X \u2192M]= ZEr [X \u2192M]+ [NZEr (A)\u2192X \u2192M],where NZEr (A) is the complement of ZEr (A) in Er (A). To prove that [NZEr (A)\u2192M]\u2208K>k(M), let Y \u21a9NZEr (A) be a locally closed embedding, such that the algebroid(Er (A),Afix)|Y is clear.Consider the embedding of algebras Afix|Y \u21a9 A|Y . It induces an embedding ofcommutative algebras Z(A|Y )\u21a9 Z(Afix|Y ), because Z(A|Y )\u2282Afix|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 \u2192M] \u2208 K>k(M) and finishes the80proof. \u0002Corollary 11.5. The operators Er , for r \u2265 0 are simultaneously diagonalizable. Thecommon eigenspaces form a family Kk(M) of subspaces of K(M) indexed by non-negative integers k\u2265 0, andK(M)=\u2295k\u22650Kk(M). (11.1)Let pik denote the projection onto Kk(M). We haveErpik = r !S(k,r)pik ,for all r \u2265 0, k\u2265 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 \u25e6Er \u2032 as-sociates to an algebroid (X,A) the stack of pairs (e,e\u2032), where both e and e\u2032 are com-plete families of non-zero orthogonal idempotents in A, the length of e being r andthe length of e\u2032 being r \u2032, and the members of e commuting with the members of e\u2032.Finally, let us prove that, for every k and every r , the Q-vector space K\u2265k(M) is aunion of finite-dimensional subspaces invariant by Er .For this, define K(M)\u2264N to be generated asQ-vector space by stack functions [X\u2192M], where X is a clear algebroid, such that the rank of the vector bundle underlyingthe algebra AX \u2192 X is bounded above by N . This is an ascending filtration of K(M),which is preserved by Er . SetK\u2265k(M)\u2229K(M)\u2264N =K\u2265k(M)\u2264N .Suppose x = [X \u2192M] 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 \u2264 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\u2265k(M)\u2264N = 0.On the other hand, Theorem 11.4 implies by induction thatEir (x)\u2208Qx+QEr (x)+ . . .+QEi\u22121r (x)+K\u2265k+i(M).Applying this for i=N\u2212k+1, we see thatEr(EN\u2212kr (x))\u2208Qx+QEr (x)+ . . .+QEN\u2212kr (x),and hence that Qx+QEr (x)+ . . .+QEN\u2212kr (x) is invariant under Er .81This proves that any x \u2208 K\u2265k(M) is contained in a finite-dimensional subspaceinvariant under Er . Standard techniques from finite-dimensional linear algebra nowimply the result. \u0002Remark 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 \u2265 1, we havekerEr =\u2295k 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\uf8eb\uf8ed0 01 2\uf8f6\uf8f8 . (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]\u2212 12 [BT] with eigenvalue 0,(ii) v2 = 12 [BT] with eigenvalue 2.Therefore, we have v1 \u2208 K1(Vect) and v2 \u2208 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) -\u2192K(M)[X \u2192M] 7 -\u2192 [IssX \u2192X \u2192M].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] \u2282 K(Var), and extendscalars to the quotient field Q(q). Thus, we will consider Iss as a Q(q)-linear operatorIss :K(M)(q) -\u2192K(M)(q),where K(M)(q)=K(M)\u2297Q[q]Q(q).83Recall the definition of Q\u03bb for a partition \u03bb`n as in \u00a77.1. This is a polynomial inq, of degree n, which vanishes to order |\u03bb| at q= 1. Recall the well-ordering on the setof all partitions of all integers introduced in \u00a77.4.Theorem 11.10. The operator Iss : K(M)(q)\u2192 K(M)(q) is diagonalizable. Its eigen-value spectrum consists of the Q\u03bb \u2208Q(q), for all paritions \u03bb. Denote the eigenspacecorresponding to the eigenvalue Q\u03bb by K\u03bb(M)(q). We haveKk(M)(q)=\u2295|\u03bb|=kK\u03bb(M)(q),where Kk(M)(q)=Kk(M)\u2297Q[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 \u2192M for which X is aclear algebroid.Then define a decreasing filtration K\u2265\u03bb(M)(q) indexed by partitions, by declaringK\u2265\u03bb(M)(q) to be generated by clear stack functions whose central type is \u2265 \u03bb. We willprove(i) the operator Iss preserves the filtration by partitions,(ii) on the quotient K\u2265\u03bb(M)(q)\/K>\u03bb(M)(q), the operator Iss acts as multiplicationby Q\u03bb.(iii) the operator Iss is locally finite.This will prove the claims concerning diagonalizability of Iss .Consider a clear stack function X \u2192M, of central rank n, central type \u03bb` n, andsplit central rank k = |\u03bb|. As in the proof of Theorem 11.4, let X\u02dc \u2192 X be a connectedGalois cover with Galois group \u0393 , acting on the set n, such thatX\u02dc\u00d7\u0393 n '-\u2192 PZE(AX).We get an induced isomorphismX\u02dc\u00d7\u0393 An '-\u2192 Z(AX)ssonto the semisimple centre of AX , by reformulating (10.3). It follows that we have anisomorphismX\u02dc\u00d7\u0393 Gnm '-\u2192 ZIss(X)onto the semisimple central inertia of X. A computation similar to that of \u00a77.1 wouldnow show that84ZIss[X \u2192M]=Q(\u03bb)[X \u2192M]+\u2211I\u2022\u2208F(n)\/\u0393Stab\u0393 (I)\u228a\u0393(\u22121)`(I\u2022)q|Imax|[X\u02dc\/Stab\u0393 (I)\u2192M].Recall that F(n) is the set of all flags I\u2022 = Ik \u228a . . . \u228a I1 \u228a I0 = n in n. The length of aflag I\u2022 is denoted by k= `(I\u2022), and the maximal index is k=max.Note that the cover X\u02dc\u2192X, as well as all intermediate covers X\u02dc\u2192 X\u02dc\/\u0393 \u2032\u2192X, for anysubgroup \u0393 \u2032 \u2282 \u0393 , 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\u02dc\/\u0393 \u2032 \u2192M], for \u0393 \u2032 \u228a \u0393 arecontained in K>k(M)(q), hence in K>\u03bb(M)(q) by the first property of our partitionordering.Let us now consider Iss , instead of ZIss . We have a closed immersion of algebroidsZIssX \u21a9 IssX. Thus we can writeIss[X \u2192M]= ZIss[X \u2192M]+ [NZIss(X)\u2192X \u2192M].Now consider a locally closed embedding Y \u21a9 NZIss(X), such that Y is a clear alge-broid. Over Y , we have the inclusion of algebras Afix|Y \u21a9A|Y , inducing an embeddingof the centres in the opposite direction Z(A)|Y \u21a9 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 \u2192M]\u2208K>\u03bb(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 \u25e6 Iss and Iss \u25e6Er 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 K\u03bb(M)(q) of the Iss . AsK\u2265|\u03bb| \u2283K\u2265\u03bb \u2283K>\u03bb \u2283K>|\u03bb|it follows from Theorem 11.4, that Er acts on K\u03bb(M)(q) by scalar multiplication by85r !S(|\u03bb|,r ), and hence that K\u03bb(M)(q)\u2282K|\u03bb|(M)(q). \u0002Example 20. Let us continue with Example 19. The stack function [BGL2] is clear, itscentral rank is 1. We haveIss[BGL2]= [\u2206\/GL2]+ [T\u2217\/N]= (q\u22121)[GL2]+ [T\u2217\/N],where \u2206 is the central torus of GL2, and T\u2217 = T \u2206. Also, N is the normalizer of Tin GL2. Moreover, [\u2206\/GL2]=Gm\u00d7 [BGL2] is a clear stack function of central rank 1,and [T\u2217\/N] is a clear stack function of central rank 2, and split central rank 1.Applying Iss to [T\u2217\/N], we getIss[T\u2217\/N]= (q2\u22121)[T\u2217\/N]\u2212 (q\u22121)[T\u2217\/T]= (q2\u22121)[T\u2217\/N]\u2212 (q\u22121)[T\u2217][BT]= (q2\u22121)[T\u2217\/N]\u2212 (q\u22121)2(q\u22122)[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\u22121)2[BT].We see that Q(q)[BGL2]+Q(q)[T\u2217\/N]+Q(q)[BT] is invariant under Iss , and thematrix of Iss on this subspace is\uf8eb\uf8ec\uf8ec\uf8edq\u22121 0 01 q2\u22121 00 \u2212(q\u22121)2(q\u22122) (q\u22121)2\uf8f6\uf8f7\uf8f7\uf8f8This 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]\u2212 1q(q\u22121) [T\u2217\/N]\u2212 1q [BT],(ii) v(2) = 1q(q\u22121) [T\u2217\/N]\u2212 q\u221222q [BT],(iii) v(1,1) = 12 [BT],where v(1) \u2208 K(1)(Vect)(q), v(2) \u2208 K(2)(Vect)(q) and v(1,1) \u2208 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 \u2208K(M), we haveIss(x \u00b7y)= Iss(x) \u00b7 Iss(y).Proof. This follows immediately from the fact that, for any two algebroids X, Y , wehave Iss(X\u00d7Y)= Iss(X)\u00d7 Iss(Y), as algebroids over X\u00d7Y . \u0002Denote the disjoint union of two partitions \u03bb and \u00b5 by \u03bb+\u00b5.Corollary 11.12. We have K\u03bb(M)(q) \u00b7K\u00b5 \u2282 K\u03bb+\u00b5(M)(q), and hence also Kk(M)(q) \u00b7K`(M)(q)\u2282Kk+`(M)(q).Remark 11.13. So the Q(q)-vector spaceK(M)(q)=\u2295k\u22650Kk(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 \u2208Kk(M) and y \u2208K`(M), then x.y \u2208Kk+`(M).87Chapter 12Hall Algebra of algebroidsLet M be a linear stack, and A\u2192M 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) \u2192M 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 \u2286M1 \u2286 \u00b7\u00b7\u00b7 \u2286Mn =M of objects of M over the same scheme S,where each factor Mi\/Mi\u22121 is also an object of M over S. Morphisms over f : T \u2192 Sare diagramsf\u2217(M1) \/\/\u000f\u000ff\u2217(M2)\u000f\u000ff\u2217(Mn\u22121) \/\/\u000f\u000ff\u2217(Mn)\u000f\u000fN1 \/\/ N2 Nn\u22121 \/\/ 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 categories\u03b1i :M(n)\u2192M, sending the above flag to its i-th factor Mi\/Mi\u22121. And there is another88morphism b :M(n)\u2192M sending the above flag to An =A.M(n)b \/\/a1\u00d7\u00b7\u00b7\u00b7\u00d7an\u000f\u000fMM\u00d7\u00b7\u00b7\u00b7\u00d7M.In particular M(2) is the linear category of short exact sequences M\u2032 \u2192M \u2192M\u2032\u2032 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\u00d7\u00b7\u00b7\u00b7\u00d7an 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]. \u0002Recall the space K(M) of stack functions as defined in \u00a711. 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] \u00b7 [X \u2192M]= [Z\u00d7X \u2192X \u2192M]. This action turns K(M) into a K(Var)-module.2. Multiplication. We multiply two stack functions [X \u2192M] and [Y \u2192M] by theformula[X \u2192M] \u00b7 [Y \u2192M]= [X\u00d7Y \u2192M\u00d7M \u2295-\u2192M].3. Hall product. The Hall product of the stack functions [X \u2192M] and [Y \u2192M] isdefined by first constructing the fibered productX\u2217Y\u000f\u000f\/\/ M(2)a1\u00d7a2\u000f\u000fX\u00d7Y \/\/ M\u00d7Mand then setting[X \u2192M]\u2217 [Y \u2192M]= [X\u2217Y -\u2192M(2) b-\u2192M].The additive zero in K(M) is given by the empty algebroid 0 = [\u009c\u2192M]. 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-\u2192M], i.e.the 0-object of M.8912.1.1 Filtered structure of the Hall algebraDefinition 12.2. For n\u2265 0, we defineK\u2264n(M)= kerEn+1 =\u2295k\u2264nKk(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\u2264n(M)(q) is the direct sum of all eigenspaces of Iss whose coresspond-ing eigenvalue Q\u2208Q[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 \u03be \u2208 K\u2264m(M) and \u03b7\u2208K\u2264n(M), then \u03be\u2217\u03b7\u2208K\u2264m+n(M) and \u03be.\u03b7\u2208K\u2264m+n(M). On the associated graded theHall product coincides with the commutative product.12.1.2 Proof of Theorem 12.3Analysis of Ep(En\u2217Em)Suppose \u03be = (X \u2192M) and \u03c7 = (Y \u2192M) are stack functions. The stack function \u03be\u2217\u03c7is defined by the cartesian diagram:X\u2217Y\u000f\u000f\/\/ M(2)\u000f\u000f\/\/ MX\u00d7Y \/\/ M\u00d7MExplicitely, X\u2217Y is the stack of triples (x,M,y),x\u000f\u000fy\u000f\u000fM\u2032 \/\/ M \/\/ M\u2032\u2032(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\u2032 \u2192 M \u2192 M\u2032\u2032 of objects in M, and x \u2192 M\u2032 and y \u2192 M\u2032\u2032 areisomorphisms from the images of x and y in M to M\u2032 and M\u2032\u2032, respectively. (We omitthese isomorphisms from the triple to simplify the notation.)90The stack function En(\u03be)\u2217Em(\u03c7) is defined by the enlarged diagram:En(X)\u2217Em(Y) \/\/\u000f\u000fX\u2217Y\u000f\u000f\/\/ M(2)\u000f\u000f\/\/ MEn(X)\u00d7Em(Y) \/\/ X\u00d7Y \/\/ M\u00d7MExplicitly, En(X)\u2217Em(Y) is the stack of 5-tuples(x,(e\u03bd),M,y,(f\u00b5)), where (x,M,y)represents a diagram (12.2), and (e\u03bd) = (e1, . . . ,en) is a complete set of non-zero or-thogonal idempotents in A(x), and (f\u00b5) = (f1, . . . ,fm) is a complete set of non-zeroorthogonal idempotents in A(y).Finally, the stack Ep(En(X)\u2217Em(Y))is the stack of objects of En(X)\u2217Em(Y),endowed with a complete set of p non-zero labelled idempotents. Explicitly, it consitsof 6-tuples (x,(e\u03bd,\u03c1),M,(g\u03c1),y,(f\u00b5,\u03c1)), (12.3)where (x,M,y) is as in (12.2), and (g\u03c1)\u03c1\u2208p is a complete set of non-zero orthogonalidempotents for the short exact sequence M\u2032 \u2192M \u2192M\u2032\u2032. Moreover, (e\u03c1,\u03bd)\u03c1\u2208p,\u03bd\u2208n isa pn-tuple of orthogonal idempotents in A(x), and (f\u03c1,\u00b5)\u03c1\u2208p,\u00b5\u2208m is a pm-tuple oforthogonal idempotents in A(y), such that for every \u03c1 = 1, . . . ,p we have \u2211n\u03bd=1 e\u03c1,\u03bd =g\u03c1|E\u2032 and\u2211m\u00b5=1f\u03c1,\u00b5 =g\u03c1|E\u2032\u2032 . Finally, we require for all \u03bd =1, . . . ,n that e\u03bd =\u2211p\u03c1=1 e\u03c1,\u03bd 6=0 and for all \u00b5 = 1, . . . ,m that f\u00b5 =\u2211p\u03c1=1f\u03c1,\u00b5 6= 0.Decomposing Ep(En\u2217Em)We use the notation \u03c3 \u00ee u for partitions of sets as opposed to \u03c3 \u00ee` r for labelledset partitions (where the order of blocks matter). For more details in labelled andunlabelled partitions we refer the reader to \u00a7A.3. If b(\u03c3) = p then for an element\u03c9 \u2208 u we say \u03c3(\u03c9) = \u03c1 or \u03c9, \u03c1 if \u03c1 \u2208 p is the label of the partition \u03c9 belongs to.Also if \u03c3 and \u03b3 are both labelled set partitions of u, we write \u03c9, (\u03c1,\u00b5) if \u03c9 is in theset with label \u03c1 in \u03c3 and in the set with label \u00b5 in \u03b3.Given non-negative integers p, u, v , and labelled set partitions \u03b3 \u00eeuunionsqv such thatb(\u03b3)= p, we define a new stack function (X\u2217Y)\u03b3 \u2192M, denoted (\u03be\u2217\u03c7)\u03b3 , as follows.Let (X\u2217Y)\u03b3 be the algebraic stack of 6-tuples(x,(e\u03c9),M,(g\u03c1),y,(f\u03b7)), (12.4)where (x,M,y) is as in (12.2), and (e\u03c9)\u03c9\u2208u, (f\u03b7)\u03b7\u2208v and (g\u03c1)\u03c1\u2208p are complete setsof non-zero orthogonal idempotents for x, y and the short exact sequence E, respec-91tively. Moreover, we require that for all \u03c1 = 1, . . . ,p we haveg\u03c1|M\u2032 =\u2211\u03b3(\u03c9)=\u03c1e\u03c9 and g\u03c1|M\u2032\u2032 =\u2211\u03b3(\u03b7)=\u03c1f\u03b7 . (12.5)There is a natural algebroid structure on (X\u2217Y)\u03b3 . 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\u2217Y)\u03b3 into a stack function.There is a morphismEu(X)\u00d7Ev(Y) -\u2192 (X\u2217Y)\u03b3 (12.6)which maps a quadruple(x,(e\u03c9),y,(f\u03b7))to the 6-tuple (12.4) where M =M\u2032\u2295M\u2032\u2032,with M\u2032 denoting the image of x in M, and M\u2032\u2032 the image of y in M. The family ofidempotents (g\u03c1) on M is defined by formulas (12.5).Lemma 12.4. If for every \u03c1 = 1, . . . ,p exactly one of the two preimages \u03b3\u22121(\u03c1)\u2229u and\u03b3\u22121(\u03c1)\u2229v is empty, the morphism (12.6) is an isomorphism.Proof. Given an object (12.4) of (X\u2217Y)\u03b3 , 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 \u03d5 and \u03c8. Therefore the sequence M is split, canonically, too. \u0002Now suppose given \u03d5 \u00ee` u with n labelled blocks and \u03c8 \u00ee` v with m labelledblocks such that \u03d5\u2229\u03b3|u = 0 and \u03c8\u2229\u03b3|v = 0. Then we can define a morphism ofstacks(X\u2217Y)\u03b3 -\u2192 Ep(En(X)\u2217Em(Y)), (12.7)by mapping the 6-tuple (12.4) to the 6-tuple (12.3) by defininge\u03c1,\u03bd =\u2211\u03c9,(\u03c1,\u03bd)e\u03c9 and f\u03c1,\u00b5 =\u2211\u03b7,(\u03c1,\u00b5)f\u03b7 .By our assumptions, these sums are either empty or consist of a single summand, sothe e\u03c1,\u03bd and the f\u03c1,\u00b5 are obtained from the e\u03c9 and the f\u03b7 essentially by relabelling.Lemma 12.5. The morphism (12.7) gives rise to a morphism of stack functions (\u03be\u2217\u03c7)\u03b3 \u2192 Ep(En(\u03be)\u2217Em(\u03c7)), which is both an open and a closed immersion. If we changeany of u, v or \u03b3 or \u03d5 or \u03c8, we get a morphism with disjoint image. The images of allmorphisms (12.7) cover Ep(En(X)\u2217Em(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. \u000292Corollary 12.6. In K(M) we have the equationEp(En(\u03be)\u2217Em(\u03c7))= \u2211u,v,\u03b3\u2211\u03d5,\u03c8(\u03be\u2217\u03b7)\u03b3 =\u2211u,v,\u03b3n!m![n\u03b3|u][m\u03b3|v](\u03be\u2217\u03b7)\u03b3 ,where u,v run over all positive integers and \u03b3 runs over all labelled partitions of uunionsqvwith p blocks. The bracket notation is adapted from \u00a7A.2.The Hall algebra is filteredWe can now calculate as follows:Ep(pik(\u03be)\u2217pi`(\u03c7))= \u2211n,ms(n,k)n!s(m,`)m!Ep(En(\u03be)\u2217Em(\u03c7))=\u2211n,ms(n,k)s(m,`)\u2211u,v,\u03b3[n\u03b3|u][m\u03b3|u](\u03be\u2217\u03c7)\u03b3=\u2211u,v,\u03b3(\u2211ns(n,k)[n\u03b3|u])(\u2211ns(m,`)[m\u03b3|u])(\u03be\u2217\u03c7)\u03b3 .For both brackets to be non-zero, the number of blocks of \u03b3|u must be at most k, andthe number of blocks of \u03b3|v must be at most `, by Lemma A.3. We conclude that forall p>k+` we have Ep(pik(\u03be)\u2217pi`(\u03c7))= 0, which proves the first part of the theorem.If p = k+` then the only possible case is if the number of blocks of \u03b3|u is exactlyk, and the number of blocks of \u03b3|v is exactly `, by Lemma A.3. In this case we haveEp(pik(\u03be)\u2217pi`(\u03c7))= \u2211u,v,\u03b3\u00b5(0u,\u03b3|u)\u00b5(0v ,\u03b3|v)(\u03be\u2217\u03c7)\u03b3 ,where \u03b3 runs over those partitions of uunionsqv such that for every \u03c1 = 1, . . . ,p exactly oneof the two preimages \u03b3\u22121(\u03c1)\u2229u and \u03b3\u22121(\u03c1)\u2229v is non-empty. By Lemma 12.4, wehave thereforeEp(pik(\u03be)\u2217pi`(\u03c7))= \u2211u,v,\u03b3\u00b5(0u,\u03b3|u)\u00b5(0v ,\u03b3|v)Eu\u03be.Ev\u03c7=\u2211u,v(k+`)!u!v !Eu\u03be.Ev\u03c7\uf8eb\uf8ec\uf8ec\uf8ec\uf8ed \u2211\u03b31\u00eeub(\u03b31)=k\u00b5(0,\u03b31)\uf8f6\uf8f7\uf8f7\uf8f7\uf8f8\uf8eb\uf8ec\uf8ec\uf8ec\uf8ed \u2211\u03b32\u00eevb(\u03b32)=`\u00b5(0,\u03b32)\uf8f6\uf8f7\uf8f7\uf8f7\uf8f8= (k+`)!(\u2211us(u,k)u!Eu\u03be)(\u2211vs(v,`)v !Ev\u03c7)= (k+`)!pik(\u03be)pi`(\u03c7).93The associated graded algebraBy what we just proved, we havepik+`(pik(\u03be)\u2217pi`(\u03c7))=\u2211ps(p,k+`)p!Ep(pik(\u03be)\u2217pi`(\u03c7))only nonzero if p = k+`= s(k+`,k+`)(k+`)! Ek+`(pik(\u03be)\u2217pi`(\u03c7))=pik(\u03be).pi`(\u03c7).The proof for the commutative product is similar and givespik+`(pik(\u03be).pi`(\u03c7))=pik(\u03be).pi`(\u03c7).This finishes the proof of the theorem.12.2 Epsilon functionsWe consider a stack function \u03be = (X \u2192M), and an idempotent operator Ek. Let usdenote by FnX the fibered productFnX \/\/\u000f\u000fX\u000f\u000fM(n)b \/\/ MThen we will consider Ek(FnX). This is the stack of triples(x,(e\u03ba),F)where x is an object of X, and F = (F1\u2192 . . .\u2192 Fn) is a flag in M, such Fn =M , the imageof x in M. Moreover, (e\u03ba)\u03ba=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 f\u03ba,\u03bd = e\u03ba|F\u03bd\/F\u03bd\u22121 . They have the properties that\u2211\u03ba f\u03ba,\u03bd = 1, forall \u03bd , and for every \u03ba, at least one of the f\u03ba,\u03bd does not vanish.We will decompose Ek(FnX) according to which of the f\u03ba,\u03bd vanish. For this, con-sider a non-negative integer p and a labelled partition \u03b1\u00ee` p with k blocks. Then wedefine F\u03b1X to be the stack of triples(x,(e\u03ba),(F\u03ba)).Here x is an object of X, and (e\u03ba) 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\u2295 . . .\u2295Mk. For every \u03ba, we have a filtration F\u03ba of E\u03baindexed by \u03b1\u22121(\u03ba).Now let us suppose given another labelled partition \u03b2 \u00ee` p with n blocks suchthat \u03b1\u2229\u03b2= 0. Using \u03b2, we define a morphismF\u03b1X -\u2192 Ek(FnX), (12.8)by defining the flag F in terms of the k-tuple of flags (F\u03ba) byF\u03bd =\u2295\u03ba\u2211\u03b1(\u03c1)=\u03ba\u03b2(\u03c1)\u2264nF\u03c1 .Note that the sum for fixed \u03ba is not really a sum, it is just the largest of the spaces F\u03c1 ,such that \u03b1(\u03c1)= \u03ba and \u03b2(\u03c1)\u2264n.Lemma 12.7. The morphism (12.8) is an isomorphism onto the locus in Ek(FnX), givenby f\u03ba,\u03bd 6= 0 if and only if \u03b1\u22121(\u03ba)\u2229\u03b2\u22121(\u03bd) is nonempty (i.e. is a singleton).Corollary 12.8. We haveEk(Fn\u03be)=\u2211p\u2211\u03b1:b(\u03b1)=k[n\u03b1]F\u03b1\u03be .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, \u00a76.3]).Corollary 12.9. We have\u03b5k(\u03be)=\u2211ns(n,k)n!Fn(\u03be)\u2208K\u2264k(M).Proof. Let us do a calculation:Ek+1\u2211ns(n,k)n!Fn(\u03be)=\u2211ns(n,k)n!Ek+1Fn(\u03be)=\u2211ns(n,k)\u2211p\u2211\u03b1:b(\u03b1)=k+1[n\u03b1]F\u03b1\u03be=\u2211p\u2211\u03b1:b(\u03b1)=k+1(\u2211ns(n,k)[n\u03b1])F\u03b1\u03be= 0 .95The expression in the bracket vanishes, by Corollary A.3.\u0002In particular, in the case of k= 1, we have\u03b51(\u03be)=\u2211n>0(\u22121)nnFn(\u03be).For M = CohX the moduli stack of coherent sheaves on a projective C-scheme X (cf.Example 8) and \u03c4 a stability condition as in \u00a71.1, let \u03be = [SS\u03b3(\u03c4)\u21a9 CohX] be the in-clusion of the substack of semistable sheaves of class \u03b3. Applying the above result tothis case produces\u03b51(\u03be)=\u2211n(\u22121)nn!\u221106=\u03b31,...,\u03b3n\u2208\u0393\u03c4(\u03b3i)=\u03c4,\u2211\u03b3i=\u03b3SS\u03b31(\u03c4)\u2217 . . .\u2217SS\u03b3n(\u03c4)\u2208K\u22641(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)=\u2295n\u22650tnK\u2264n(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),\u2217). Thequotient map K \u2192K\/tK is identified with the map \u2211nxntn,\u2211npin(xn).The graded algebra associated to the filtered algebra(K(M),\u2217), is canonically iso-morphic to the commutative graded algebra(K(M), \u00b7), by Theorem 12.3. The specialfibre inherits therefore a Poisson bracket, which encodes the Hall product to secondorder. This Poisson bracket has degree \u22121, and is given by the formula{x,y} =pik+`\u22121(x\u2217y\u2212y\u2217x), for x \u2208Kk(M), y \u2208K`(M). (12.9)Corollary 12.10. The graded K(Var)-algebra(K(M), \u00b7) is endowed with a Poisson bracketof degree \u22121, 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 \u2208 K1(M), we have that x\u2217y \u2212y \u2217x \u2208 K1(M), soin this case, the Poisson bracket is equal to the Lie bracket.Corollary 12.12. For every stack function \u03be = (X\u2192M), the epsilon element \u03b51\u03be 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. 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Foundations of algebraic geometry. math216.wordpress.com, June2013. Last time retrieved on January 2016. \u2192 pages 9[51] F. Zhang. Matrix theory. Universitext. Springer, New York, second edition, 2011.ISBN 978-1-4614-1098-0. doi: 10.1007\/978-1-4614-1099-7. URLhttp:\/\/dx.doi.org\/10.1007\/978-1-4614-1099-7. Basic results and techniques. \u2192pages101Appendix APartitionsA.1 Mobius numbersWe follow the notation of [9]. Let (S,\u2264) be a poset. The two elements 0,1 \u2208 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 \u00b5(x,z) = 0 when x \u009c z and whenx \u2264 z by \u2211y :x\u2264y\u2264z\u00b5(x,y)= \u03b4(x,z). (A.1)Here \u03b4 is the Kronecker delta function, defined as \u03b4(x,z) = 1 if x = z and is zerootherwise.The meet of two elements x,y \u2208 S is defined by x\u2227y being the unique greatestlower bound for x and y . A poset with well-defined meet operator \u2227 is called a meetsemilattices [15].For every two elements x,y \u2208 S, the notation [x,y] stands for the subset of allnodes z such that z \u2265 x and z \u2264 y . If S is a met semi-lattice then so it [x,y]. S iscalled locally finite if for every two elements x,y \u2208 S, the segment [x,y] has finitelymany nodes.Proposition A.1. Given three fixed elements x,z,w of a locally finite meet semilatticeS \u2211y :y\u2227w=x\u00b5(y,z)=\uf8f1\uf8f2\uf8f3 \u00b5(x,z) w \u2265 z0 w 6\u2265 z.In the following proof we use the \u00b5w(x,z) for this type of summations.Proof. The case of w \u2265 z is easy because if y \u2227w = x, and y \u2264 z \u2264w 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 \u2208 [x,z] is thedisjoint union of the following sets{y :y\u2227w = x} \u2200x \u2208 [x,z]which means\u00b5w(x,z)=\u2211x:x\u2264x\u2264z\u00b5w(x,z)=\u2211y :x\u2264y\u2264z\u00b5(x,y)= 0where the first identity follows from the induction hypothesis and the last identityfollows from (A.1) in the definition of \u00b5. \u0002A.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 \u2126u for the set of all partitions \u03bb\u00eeu. Given two partitions \u03bb,\u00b5 \u00eeu wewrite \u03bb \u2264 \u00b5 if every element of \u03bb is a subset of an element of \u00b5. In this case we say\u03bb is finer than \u00b5 and \u00b5 is coarser than \u03bb. This way, (\u2126u,\u2264) 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 \u03b1\u00eeu the partition \u03b1\u2229\u03bb is the coarsest partition that isfiner than both \u03b1 and \u03bb. This turns \u2126u into a meet semilattice.We use the notation b(\u03bb) for the number of blocks of a partition. In [47, Example3.10.4] the k-th Whitney number of the first kind associated to \u2126u is defined aswk =\u2211\u03c4\u00eeub(\u03c4)=u\u2212k\u00b5(0u,\u03c4)and is it shown to be equal to a Stirling number of the first kind by wk = s(u,u\u2212k).We like to rewrite this result ass(b(\u03b1),k)=\u2211\u03c4\u00eeub(\u03c4)=k\u00b5(\u03b1,\u03c4) (A.2)by substituting \u2126u with the segment [\u03b1,1u]. Note that if \u03c4 6\u2208 [\u03b1,1u] then \u00b5(\u03b1,\u03c4)= 0hence the above identity holds without the need to restrict \u03c4 to be in [\u03b1,1u].Lemma A.2. We have\u2211\u03b1:\u03b1\u2229\u03bb=0s(b(\u03b1),k)=\uf8f1\uf8f4\uf8f2\uf8f4\uf8f30 if k < b(\u03bb)\u00b5(0,\u03bb) if k= b(\u03bb).103Here \u03b1 runs over all partitions of u.Proof. We use the identity (A.2) and have\u2211\u03b1:\u03b1\u2229\u03bb=0s(b(\u03b1),k)=\u2211\u03b1:\u03b1\u2229\u03bb=0\u2211\u03c4 :b(\u03c4)=k\u00b5(\u03b1,\u03c4)=\u2211\u03c4 :b(\u03c4)=k\u2211\u03b1:\u03b1\u2229\u03bb=0\u00b5(\u03b1,\u03c4)=\u2211\u03c4 :b(\u03c4)=k:\u03c4\u2264\u03bb\u00b5(0,\u03c4)where the last identity follows from Proposition A.1. The set of all such \u03c4 can only benonempty if k\u2265 b(\u03bb) proving the lemma. \u0002Let u be a positive integer and \u03bb\u00ee u a set partition. The notation[n\u03bb]means thenumber of partitions \u03b1\u00eeu with n blocks such that \u03b1\u2229\u03bb= 0.Corollary A.3. We have\u2211ns(n,k)[n\u03bb]=\uf8f1\uf8f4\uf8f2\uf8f4\uf8f30 if k < b(\u03bb)\u00b5(0,\u03bb) if k= b(\u03bb).Proof. This follows from previous lemma and the following computation\u2211ns(n,k)[n\u03bb]=\u2211n\u2211\u03b1,b(\u03b1)=n,\u03b1\u2229\u03bb=0s(n,k)=\u2211\u03b1:\u03b1\u2229\u03bb=0s(b(\u03b1),k).\u0002A.3 Labelled partitions and integer partitionsA labelled partition of u is a partition \u03bb\u00eeu where the order of elements of \u03bb is impor-tant. We denote a labelled partition by \u03bb\u00ee` u. If we forget the labelling the resultingpartition is called the associated unlabelled partition to \u03bb. Note that given a labelledpartition \u03bb there are b(\u03bb)! other labelled partitions with the same associated unla-belled partition as that of \u03bb. If \u03bb,\u00b5 \u00ee` u are two labelled partitions we adapt thenotation \u00b5(\u03bb,\u00b5) to denote the Mobius number of the associated unlabelled partitionsof \u03bb and \u00b5.Recall that given an integer t, an integer partition is a sequence (si)i\u22651 of integerssuch that\u2211i\u22651 isi = t. To a set partition \u03bb \u00ee t we may therefore associate an integerpartition \u03bb ` t defined as \u03bb = (\u03bb1,\u03bb2, . . .) if \u03bb has \u03bbi elements of size i. We call \u03bb theinteger partition type of \u03bb. We define the length of \u03bb as b(\u03bb)=\u2211i\u03bbi. So the length ofthe integer type of \u03bb is identical to the number of blocks of \u03bb.104Appendix BSplitting covers of gerbesLet X\u2192 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 \u00e9tale locallyneutral. After passing to a dense open on X, we may assume there exists a finite \u00e9talecovering X \u2192X, such that the pullback of X is a neutral gerbes over X, written asX|X \u009b BXGfor some X-group schemeG. We say X is a trivializing cover for X. We may also assumethat X \u2192 X is a Galois cover [48, Proposition 5.3.9] with Galois group \u0393 = Aut(X\/X).Then by classification of gerbes there exists a homomorphism\u03d5 : \u0393 \u2192Aut(BG)= [AutG\/G]such that X\u009bX\u00d7\u0393 ,\u03d5 BG. We may assume \u03d5 is injective by passing to an intermediatecover:Xker\u03d5--------------------\u2192 X\u02dc \u0393\/ker\u03d5-------------------------------\u2192X.Note that X\u02dc is now the minimal trivializing Galois cover of X\u2192X.Definition B.1. The above covering X\u02dc \u2192X is called a splitting cover of X over X.Lemma B.2. Given a Galois covering X \u2192 X and \u0393 = Aut(X\/X), an action of \u0393 on BGthat results X as a form of BG on X is unique up to 2-isomorphisms of stacks.Proof. Let \u03d5 : \u0393 \u2192Aut(BG) and \u03d5\u2032 : \u0393 \u2192Aut(BG) be two such actions. IfX\u02dc\u00d7\u0393 ,\u03d5 BG \u009b X\u02dc\u00d7\u0393 ,\u03c8 BG105as gerbes over X, then from classification of gerbes we can construct a natural trans-formation between the functors \u03d5 and \u03c8. \u0002Lemma B.3. Splitting covers are unique up to isomorphisms of stacks.Proof. Let X1 and X2 be two splitting covers with corresponding \u03d5 : \u0393 \u2192Aut(BG) and\u03d5\u2032 : \u0393 \u2032\u2192Aut(BG). We can create a product trivializing cover X\u02dc\u2192X with \u2206=Aut(X\u02dc\/X)a new group of automorphisms over X. Then by the previous lemma there is a naturaltransformation between the two composition functors\u0393 \u0016 v))\u000b\u0013\u220688 88&& &&Aut(BG)\u0393 \u2032 ( \b 55both corresponding to trivializing covers of X. Therefore there exists a natural trans-formation between \u03d5 and \u03d5\u2032. This proves uniqueness. \u0002We 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 \u2192 X be a smooth morphism of schemes. If X\u02dc \u2192 X is the splittingcover for X, then Y\u02dc = Y \u00d7X X\u02dc \u2192 Y is a union\u2294Y\u02dci of splitting covers of X|Y \u2192 Y .BY\u02dcG\u000f\u000f \/\/ Y\u02dci\u00d7\u0393i BG\"\"\u000f\u000f\u2294Y\u02dci \/\/\u000f\u000fY\u000f\u000fBX\u02dcG\/\/!!X##X\u02dc \/\/ XProof. Let \u0393 be the Galois group of X\u02dc \u2192X and Y\u02dci a connected component of Y\u02dc . Let \u0393ibe the stabilizer of Y\u02dc as a subgroup of Aut(Y\u02dc \/Y). All \u0393i are conjugate to each otherand all Y\u02dci are isomorphic to each other. Note that since X is the form of BG twistedby \u03d5 : \u0393 \u2192Aut(BG), thenX|Y \u009b Y\u02dci\u00d7\u0393i,\u03d5i Aut(BG).where \u03d5i is the restriction of the homomorphism \u03d5 to the subgroup \u0393i. Since \u03d5 isinjective, so is \u03d5i, which implies that Y\u02dci is the splitting cover for X|Y for any i. \u0002106","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/hasType":[{"value":"Thesis\/Dissertation","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#dateIssued":[{"value":"2016-09","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt":[{"value":"10.14288\/1.0302066","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/language":[{"value":"eng","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeDiscipline":[{"value":"Mathematics","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/provider":[{"value":"Vancouver : University of British Columbia Library","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/publisher":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/rights":[{"value":"Attribution-NonCommercial-NoDerivatives 4.0 International","type":"literal","lang":"*"}],"https:\/\/open.library.ubc.ca\/terms#rightsURI":[{"value":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/","type":"literal","lang":"*"}],"https:\/\/open.library.ubc.ca\/terms#scholarLevel":[{"value":"Graduate","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/title":[{"value":"The inertia operator and Hall algebra of algebraic stacks","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/type":[{"value":"Text","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#identifierURI":[{"value":"http:\/\/hdl.handle.net\/2429\/58120","type":"literal","lang":"en"}]}}~~