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Formal-local structure of the Hilbert scheme of points on three-dimensional complex affine space around… Hsu, Ting Chen Leo 2016

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Formal-local Structure of the Hilbert Scheme of Points onThree-Dimensional Complex Affine Space around SpecialMonomial IdealsbyTing Chen Leo Hsua thesis submitted in partial fulfillmentof the requirements for the degree ofMASTER OF SCIENCEinthe faculty of graduate and postdoctoral studies(Mathematics)The University of British Columbia(Vancouver)April 2016c© Ting Chen Leo Hsu, 2016AbstractWe show that the formal completion of the Hilbert scheme of points in C3 at subschemes carvedout by powers of the maximal ideal corresponding to the origin is given as the critical locus ofa homogeneous cubic function. In particular, the Hilbert scheme is formal-locally a cone aroundthese distinguished points.iiPrefaceThe research in this thesis is conducted by Ting Chen Leo Hsu, under the guidance of ProfessorJim Bryan.The content of chapter Chapters 4 and 5 draws from the formalisms developed by M. Kontsevich,V. Hinich, and M. Manetti.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Moduli problems and deformation theory . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Main result: vanishing theorem on the local structure of the Hilbert scheme of points 41.3 Outline of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Notations and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Hilbert scheme of points in C3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Moduli space of ideals H[n] and Haiman’s affine-local coordinates . . . . . . . . . . . 72.1.1 Cotangent spaces in Haiman coordinates . . . . . . . . . . . . . . . . . . . . . 102.2 Moduli space of framed modules M[n] . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.1 Tangent and cotangent spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Isomorphism M[n] → H[n] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Induced maps on cotangents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5 Path equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.5.1 At monomial ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Differential graded Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.1 Why DGLAs? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 DG-vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3 Algebra objects and DGLAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.4 The deformation functor associated with a DGLA . . . . . . . . . . . . . . . . . . . 273.5 DGLA of abstract modules lP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.5.1 Cyclic structure on lP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.6 DGLA of framed modules gp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29iv4 Coalgebras as formal stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.1 Cofree cocommutative coalgebras as formal pointed manifolds . . . . . . . . . . . . . 334.2 Maps of cofree coalgebras as formal maps . . . . . . . . . . . . . . . . . . . . . . . . 374.3 Coderivations as vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.4 Tangent spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.5 Formal pointed DG-manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.6 Tangent complexes and tangent quasi-isomorphisms . . . . . . . . . . . . . . . . . . 414.7 The deformation functor associated with a formal pointed DG-manifold . . . . . . . 415 L∞-algebras and homotopy classification of formal pointed DG-manifolds . . . 425.1 Homotopy classification of formal pointed DG-manifolds . . . . . . . . . . . . . . . . 425.2 L∞-algebras as local models of formal pointed DG-manifolds . . . . . . . . . . . . . 435.3 L∞-algebras as homotopy Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 455.4 Explicit minimal model of a DGLA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.4.1 Pseudoinverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.4.2 Oriented binary trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.4.3 Explicit formula for minimal model . . . . . . . . . . . . . . . . . . . . . . . . 485.4.4 Characterization of pseudoinverses . . . . . . . . . . . . . . . . . . . . . . . . 495.5 The deformation functor associated with an L∞-algebra . . . . . . . . . . . . . . . . 496 Proof of main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.1 Weights on gp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526.2 Weights on cotangent spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.3 Weights on minimal models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.4 Weights on pyramid ideals and proof of main theorem . . . . . . . . . . . . . . . . . 54Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56vAcknowledgmentsThank you Jim, for your consistent patience and support throughout the entirity of my studies,and also for allowing me the freedom to explore wonderfully exotic topics on my own. Thanks toyou, my journey in mathland has been a very fullfilling one.Thank you Mom and Dad, for never needing any reason to believe in things I do, other than simplythat they matter to me. You never stop amazing me with how much you love me.Thank you Max, for being a kind person, and an understanding brother. You are a source ofinsights and the cause of many reflections. I will always treasure you.Thank you Brenda, for always being here for me, both to impart much-needed sanity, and to serveas walking proof that there exists at least one human being who is completely unfazed in the eventof proofs falling through. But above all, thank you for loving me.Thank you Julian, for showing me that writing is supposed to be difficult. It was a pleasure to bedaily thesis buddy in Paris. It was like a LAN party, but much more rigorous.Thank you Oleg, for infecting me with your contagious excitement about research. Our nightlydiscussions in Barcelona provided me with much-needed refreshments for the brain.Thanks to all my friends, who learned in a reasonably short time that there is never a good timeto ask about how my thesis is going.viChapter 1Introduction1.1 Moduli problems and deformation theoryWhen studying algebraic and geometric objects, one is often not only interested in single objects,but families of them over some parameter objects. These families should be continuous in theappropriate sense relative to the parameter objects. Furthermore, we’d like to consider familiesonly up to some kind of equivalence.To formalize this notion, suppose the parameter objects form a category B. A functor M :Bop → Set taking a parameter object b to the set M (b) of equivalence classes of families overb is called a moduli problem. Encoded in the functoriality of M is also the data of a set mapM (b) → M (b′) for each b′ → b in B. We should think of b′ → b as a change of parameter,and M (b) → M (b′) as a reparametrization of (equivalence classes of) families via the change ofparameter.When a moduli problem is representable, i.e. if M : Bop → Set is naturally isomorphicto HomB(−,M) for some M ∈ B, then we call M the moduli space associated with M . ByYoneda’s Lemma, two moduli spaces representing the same moduli problem are isomorphic (andthe isomorphism is canonical if we fix the natural isomophisms Hom(−,M) ∼=M ).Remark 1.1.1. In what follows, we will assume B = Sch is the category of schemes over a basefield k.An example of a moduli problem in algebraic geometry is the Hilbert functor parameterizingsubschemes of a fixed scheme X:Definition 1.1.2 (Hilbert functor). Let Sch denote the category of schemes, and fix a schemeX ∈ Sch. The Hilbert functorH X : Schop → Set takes a scheme S to families of closed subschemesof X over S. I.e. H X(S) is the set of closed subschemes of X × S, flat over S. More explicitly,H X(S) is the set of closed embedding Z ↪→ X × S such that the composition with the projectionZ ↪→ X × S → S1is flat, quotiented by isomorphisms Z → Z ′ commuting with the embeddings in the obvious way.Proposition 1.1.3 (Gro¨thendieck). For a projective C-scheme X, the Hilbert functor H X isrepresentable by a projective scheme HX if we restrict ourselves to families of subschemes of a fixedHilbert polynomial.Corollary 1.1.4. The closed-points of HX correspond exactly to the subschemes of X.Proof. Hom(SpecC,HX) ∼=H X(SpecC) is just the set of subschemes of X × SpecC ∼= X.For this paper, we will be interested in the Hilbert scheme of n points (i.e. 0-dimensionalsubschemes of length n) on C3, the three-dimensional complex affine space. We denote this schemeby H[n]. Concretely, H[n] has underlying setProposition 1.1.5 (Hilbert scheme of n points in C3). The closed points of H[n] areH[n] = {I ⊂ C[x1, x2, x3] | C[x1, x2, x3]/I ∼= Cn as a C-vector space}.Proof. Corollary 1.1.4 tells us that the closed points of H[n] are 0-dimensional subschemes Z ⊂ C3with the constant Hilbert polynomial P (Z) = n. But P (Z) = dimH0(OZ) = dimC(C[x1, x2, x3]/I(Z)).We will give an explicit description of the scheme structure of H[n], due to Mark Haiman [5], inSection 2.1. For this paper, we would like to study the formal-local structure of H[n] around special0-dimensional subschemes where all the points are concentrated at the origin in C3. These specialsubschemes correspond to ideals in O(C3) = C[x1, x2, x3] generated by monomials in x, y, z, andwe will call them monomial ideals.Through the functorial description of H[n] in Definition 1.1.2, the formal-local structure of H[n]is exactly the restriction of the functor Hom(−,H[n]) to “small” subschemes. Algebraically, weunderstand “small“ to mean spectrums of Artinian rings over C.Definition 1.1.6. A formal moduli problem is a functor from CAlgsmk → Set, where CAlgsmk denotesthe category of commutative Artianian rings over k.Remark 1.1.7. We could have defined formal moduli problems as functors (Schsm)op → Set, whereSchsm = (CAlgsmk )op, in line with the definition of moduli problems above, but defining the way wedid saves us writing many Specs.Construction 1.1.8. Let M be a moduli problem, and x ∈ M (k) a closed point, the formalcompletion of M at x is the formal moduli problem Mx : CAlgsmk → Set defined byMx(A) =M (A)×M (A/mA) {x},2for A ∈ CAlgsmk an Artinian ring with unique maximimal ideal mA.Proposition 1.1.9. In the case that M is represented by scheme M , the formal completion ofMx in Construction 1.1.8 is represented by the formal completion Mˆx in the category of formalschemes.Proof. Let A ∈ CAlgsmk be an Artinian ring, and pt ∈ Spec(A) the unique closed point correspond-ing to the maximal ideal mA ⊂ A. Then Spec(A) ∼= Spf(A), and we haveMx(A) ∼= HomSch∗((Spec(A), pt), (M,x))∼= HomfSch∗((Spf(A),pt), (Mx, x))∼= HomfSch∗((Spf(A),pt), (Mˆx, x))where the last isomorphism is the universal properties of the formal completion of schemes.Remark 1.1.10. Unpacking Construction 1.1.8, we see that, in particular, if M is represented byM , and x ∈M is a closed point, then the tangent space of M at x, given byHomSch∗((Spec(k[]/(2), pt), (M,x))is simply the first-order deformations of x. Here k[]/(2) is the Artinian ring of dual numbers.Similarly, the n-th order structure of M at x, given by HomSch∗((Spec(k[]/(n+1)),pt), (M,x)),classifies higher order deformations of x.We would like to answer questions such asQuestion 1.1.11. Given a first-order deformation χ ∈ Mx(k[]/(2)), when can we lift it to ann-th order deformation? Concretely, this is asking about the image of the mapMx(k[](n+1))→Mx(k[](2))induced by the obvious map k[](n+1)→ k[](2).The general answers to this kind of questions is given by (formal) deformation theory, whichprovides a systematic method of studying formal-local structures of moduli spaces. It turns out thata satisfactory answer for Question 1.1.11 exists only when one generalizes the definition of formalmoduli functors to the derived algebraic geometry setting. Many different, but closely related,approaches exist in literature. They all involve upgrading the domain and/or codomain of thethe formal moduli functor to account for higher categorical, or derived structures. Lurie [10] andToe¨n-Vezzosi [17] take a global approach and define their moduli problems (not just formal modulifunctors) as functors from simplicial rings to simplical sets, while Manetti [12] takes a local approachand define his formal moduli problems to be functors from differential graded artinian rings to sets.3Kontsevich [8] and Hinich [6] also provide their own local models for derived deformation theory,and we will discuss those in more details in later sections. A detailed survey of the equivalencebetween these approaches is given in [14].The common feature of all these formulations of formal moduli problems is that, when k is ofcharacteristic 0, they are completely described by their tangent spaces – for each of these approaches,there is a natural way to define the (derived) tangent space of a moduli problem as an L∞-algebra,or as a differential graded Lie algebra (DGLA) up to equivalence. (See Remark 1.1.13). One canthen show that this tangent construction is an equivalence of categories.Remark 1.1.12. Being able to talk about tangent spaces to moduli problems is indicative of howthese moduli problems are in fact generalized spaces (derived stacks). In reality, very few moduliproblems are actually representable, but that doesn’t prevent us from being able to do geometrywith them. With the appropriate definitions using the right axioms, moduli problems will supportgeometric notions such as sheaves and cohomology. Thus, for the rest of the discussion we will notdistinguish between moduli spaces and moduli problems.Remark 1.1.13. To be more precise, it’s actually the tangent space of the loop space of the formalmoduli spaces that has a natural Lie structure, but the loop space construction is an equivalenceof categories, so we can always recover the original space from its loop space, up to equivalence).For more detail, see Lurie [10] or Section 3.1.1.2 Main result: vanishing theorem on the local structure of theHilbert scheme of pointsThe Hilbert scheme of points on smooth surfaces is known to be smooth and irreducible [3]. How-ever, for threefolds, this is no longer true. In fact, H[n] is already singular for n = 4; for large n, itis reducible. For a general introduction to the subject, see [13].Remark 2.2.8 will show that H[n] can be described as the critical locus of a homogeneous cubicon a smooth quasi-projective variety of dimension 2n2+n. In particular, it is given by homogeneousquadratic equations.Our goal in this paper is to investigate the formal-local structure of the Hilbert scheme. Inparticular, we will show that around distinguished points corresponding to monomial ideals of theform (x1, x2, x3)N , the formal neighbourhoods are also given by the critical locus of a homogeneouscubic. In particular, the Hilbert scheme is formal-locally a cone around these points.The result, as stated below in Theorem 1.2.1, actually depends on a conjecture (Conjec-ture 3.6.4); however, even though the particular form of the theorem could change if the conjectureis not exactly correct, we would still have an interesting description of the formal-local neighbour-hood of the Hilbert scheme around these points. We have thus decided to include the conjecturein the belief that it would allow us to present the main result with greater geometric clarity.4Theorem 1.2.1 (Main result). Let I = (x1, x2, x3)N ⊂ C[x1, x2, x3], and [I] ∈ H[n] the correspond-ing point in the Hilbert scheme of points in C3, where n = 16N(N + 1)(N + 2) is the C-dimensionof C[x1, x2, x3]/I. Then there is a formal power series f : T[I]H[n] → C on the tangent space at [I],such that the formal completion of H[n] at x is given as the critical locus of f . I.e.,Hˆ[n][I] ∼= {df = 0} ⊂ T[I]H[n]as formal schemes. Moreover, there exist coordinates on T[I]H[n] on which f is a homogeneouscubic, so that Hˆ[n][I] is a cone.Remark 1.2.2. In Theorem 1.2.1, T[I]H[n] is a formal scheme in the following way: any C-vectorspace V is naturally the formal spectrum of the formal power series ring k[[V ∨]], which has under-lying C-vector space∏k≥0(V∨)k.The case of N = 2 was already shown in [2], where f is constructed explicitly as a Pfaffian.The proof of Theorem 1.2.1 will be presented at the end of the paper in Section 6.4. We presenthere a sketch:Remark 1.2.3. We will utilize the equivalence between formal deformation functors and DGLA/L∞-algebras discussed in Section 1.1. To be more precise, we will replace LHS and RHS of the equationin Theorem 1.2.1 by the functors they represent. Then(1) write the LHS as the deformation functor associated with a DGLA g; and(2) write the RHS as the deformation functor associated with an L∞-algebra H quasi-isomorphicto g.Since quasi-isomorphisms of DGLA/L∞-algebras induce isomorphisms of deformation functors, weare done.Item (2) will can be broken down further:(2a) H will be obtained using the minimal model theorem, a standard tool in the homotopytheory of L∞-algebras that allows us to explicitly compute the cohomology of an DGLA as aquasi-isomorphic L∞-algebra with trivial differential.(2b) Using a (partial) cyclic structure on H, we will show that its Maurer-Cartan locus is givenby df = 0 for some formal series f : H1 → C.(2c) We can clearly choose f to have no constant term; f has no linear term by construction(because the vanishing locus of df must include the origin in H1, which corresponds to thetrivial deformation, which always exists).(2d) The trivial differential condition on H ensures that f has no quadratic term.(2e) Finally, by studying H as a representation under a torus action, we will show that f has noquadric terms or above.5Item (2e) is the only place where we will need to restrict to ideals of the form (x1, x2, x3)N for someN > 0. The torus action is defined at any monomial ideal, but our proof will need the specificweight decomposition of H at ideals of the form (x1, x2, x3)N .1.3 Outline of the paperIn Chapter 2, we describe the scheme structure of H[n] following Haiman [5]. Then we describe anexplicit isomorphism to an alternative description of the Hilbert scheme of points as a GIT quotientthat better allows us to study the formal structures using DGLAs. Much of this work is so that wecan accomplish Item (2e) in Remark 1.2.3 by explicit computation in Haiman’s coordinates.In Chapter 3 we define DGLAs, and describe the DGLA g that captures the formal-local struc-ture of the Hilbert scheme of points H[n] around monomial ideals ((1) in Remark 1.2.3). We willdescribe the cyclic structure on g that will remain partially in H.In Chapters 4 and 5, we define coalgebraic approaches to formal moduli problems used byHinich [6] and Kontsevich [8]. These sections mainly serve to build the foundation required todiscuss minimimal models of DGLAs. Although these sections also provide an introduction tocoalgebraic approaches and their homotopy theories, the details will not be important for provingthe main result of the paper.Finally, in Chapter 6, we apply the minimal model to g to construct H (Item (2a) in Re-mark 1.2.3), and study its decomposition as a representation of the torus group (Item (2e) inRemark 1.2.3). We will tie everything together in the proof of Theorem 1.2.1 in Section 6.4.1.4 Notations and conventionsIµ is the monomial ideal corresponding to partition µ (defined in Definition 2.1.7).pµ the framed module corresponding to partition µ (defined in Definition 2.5.3).(gµ, dµ, [ ]µ) = (gpµ , dgpµ , [ ]gpµ ). (Notation 6.0.6).δ(P ) = 1 whenever the proposition P is true, and 0 otherwise; δab = δ(a = b).h, u ∈ µ, whereas r denote a general triple of nonnegative integers.ei = (δij)j=1,2,3. E.g. e2 = (0, 1, 0).(X, v) ∈ End(V )⊕3 ⊕ V denotes a general framed modulep = (P, vP ) ∈ End(V )⊕3 ⊕ V denotes a fixed framed module(X, v) ∈ End(V )⊕3 ⊕ V denotes a vector in the tangent space TpM[n] of M[n] at somep = [P, vP ].XˆS denotes the formal completion of X along closed subscheme S. Xˆx = Xˆ{x} for x a closedpoint in X.6Chapter 2Hilbert scheme of points in C32.1 Moduli space of ideals H[n] and Haiman’s affine-localcoordinatesThis section follows Haiman’s paper [5], except we are in dimension three instead of two. A similardiscussion in all dimensions is given in Hubregtse [7].Let R = C[x1, x2, x3] be the polynomial ring in three variables.As described in Corollary 1.1.4, the Hilbert scheme of n points in C3 has the underlying setthe set of ideals I ⊂ R = C[x1, x2, x3] such that underlying C-vector space of the quotient R/I isn-dimensional.Remark 2.1.1. The scheme structure on this set is inherited by embedding these ideals into aGrassmanian, as shown in Proposition 2.1 of [5]. Geometrically, the Hilbert scheme parameterizeszero-dimensional subschemes Spec(C[x1, x2, x3]/I) ⊂ C3 of length n. This technique in general isdue to Gro¨thendieck in [4].In his paper, Haiman describes the scheme structure explicitly by covering the Hilbert schemeby open affine subschemes Hµ indexed by 3-dimensional partitions µ of size n, which we will definesoon. He then provides explicit affine coordinate functions crh, and gave the equations carving outHµ as a affine subscheme of Spec(C[crh]). To construct the open cover, he first observes that givenan ideal I in our Hilbert scheme, the underlying C-vector space of the quotient C[x, y, z]/I alwayshas a basis (possibly not unique) Bµ that corresponds to some 3-dimensional partition µ of sizen. For a fixed µ, he then defines Hµ to be the set of ideals I ∈ H[n] such that Bµ is a basis forC[x, y, z]/I. We summarize [5] here:Definition 2.1.2. A 3D partition (or simply a partition) is a subset µ ⊂ (Z≥0)3 satisfying theclosure condition: if h ∈ µ and u  h then u ∈ µ, where  is the partial order on triples:(u1, u2, u3)  (h1, h2, h3) iff ui ≤ hi for i = 1, 2, 3.7The size of a partition is the size of the set.Notation 2.1.3. We will always denote triples of integers in Z3≥0 by the letters h, u and r. Of these,h and u will strictly be used to denote those inside a given partition µ, whereas r is generally assumedto be any triple. Despite having a convention, we will still explicitly spell out the constraints outwhere appropriate.Notation 2.1.4. Define e1 = (1, 0, 0), e2 = (0, 1, 0), and e3 = (0, 0, 1).Notation 2.1.5. We will often use multi-index notation and write xr = xr11 xr22 xr33 for r = (r1, r2, r3) ∈Z3≥0. In particular, xei = xi.Definition 2.1.6. Given a parition µ, define Bµ to be the set{xh : h ∈ µ} ⊂ C[x, y, z], where weused the multi-index notation in Notation 2.1.5.Definition 2.1.7. Given a parition µ, define Iµ to be the ideal (xr)r/∈µ ⊂ C[x, y, z] generated bymonomials not in Bµ. We call Iµ the monomial ideal corresponding to the partition µ.Lemma 2.1.8. Let Hµ to be the set of all ideals I ∈ H[n] such that Bµ is a basis for C[x, y, z]/I.Then Hµ covers H[n]. (We suppress the dependence on n to prevent clutter).Proof. Analog of Proposition 2.1 of [5] in dimension 3.We can exhibit Hµ as an explicit affine scheme as follows:Definition 2.1.9. On Hµ, construct functions crh defined asxr =∑h∈µcrh(I)xh mod Ifor I ∈ Hµ, r ∈ Z3≥0 and h ∈ µ.Immediately, we see that:Proposition 2.1.10. The functions satisfy the equationscr+eih =∑u∈µcrucu+eih for r /∈ µ, h ∈ µ, i = 1, 2, 3; andcrh = δrh for r, h ∈ µ, i = 1, 2, 3,which we will call the Haiman equations.8Proof. Multiplying Definition 2.1.9 by xi, we havexr+ei =∑u∈µcru(I)xu+ei=∑u,h∈µcru(I) cu+eih (I)xhFrom which the first set of equations follow. The second set of equations are obvious.Remark 2.1.11. The ideal I ∈ Hµ given by crh(I) = δ(r ∈ µ) δrh is precisely the monomial ideal Iµcorresponding to µ. Here δ(r ∈ µ) = 1 if r ∈ µ, or 0 if r /∈ µ.The following proposition says that the functions crh along with the equations they satisfyembeds Hµ into the affine space SpecC[crh]. It also gives the gluing between Hµ and Hµ˜ in termsof these explicit affine coordinates. Because of this, we will refer to the functions crh as Haimancoordinates.Proposition 2.1.12.(1) (Hµ,O|Hµ) ∼= Spec C[crh](Haiman eqns) , whose structure sheaf we will call Oµ.(2) Under the isomorphism of Item (1), O(Hµ ∩ Hµ˜) = O|Hµ(Hµ ∩ Hµ˜) is the localization ofO(Hµ) by powers of det(ch˜h). We will write ((c−1)hh˜) for the inverse of (ch˜h) in O|Hµ(Hµ ∩Hµ˜).(3) The canonical isomophisms O|Hµ(Hµ ∩Hµ˜)→ O|Hµ˜(Hµ ∩Hµ˜) identifying the two represen-tation of O(Hµ ∩Hµ˜) above is(C[crh](Haiman eqns))det ch˜h→ C[c˜rh˜](Haiman eqns)det c˜hh˜crh 7→∑u˜∈µ˜c˜ru˜(c˜−1)u˜h.Proof. Proof of Item (1) is the content of Proposition 2.3.1. of [7].To see Item (2), we show that the condition defining Hµ ∩ Hµ˜ in Hµ is precisely that (ch˜h) isan invertible matrix. Suppose I ∈ Hµ ∩ Hµ˜, then (c˜hh˜(I)) is the inverse to (ch˜h(I)). Conversely, if(ch˜h(I)) is invertible for I ∈ Hµ, we show that the xh˜ span R/I, so that I ∈ Hµ˜ also. The equation∑h˜∈µ˜ah˜xh˜ = 0 mod Iimplies ∑h˜∈µ˜∑h∈µah˜ch˜h xh = 0 mod I9since xh˜ =∑h∈µ ch˜h xh. By linear independence of the xh,∑h∈µ ah˜ch˜h = 0. Multiplying by theinverse of ch˜h, we get ah˜ = 0 for each h˜.Item (3) can be shown by following how Haiman embeds his affine opens into the Grassmanian.We omit the proof here.2.1.1 Cotangent spaces in Haiman coordinatesThe Haiman coordinates also gives us a description for the cotangent space of H[n] at an idealI ∈ Hµ:Proposition 2.1.13.T∨I H[n] =〈dcrh ; r /∈ µ, h ∈ µ〉linearized Haiman equations,where the linearized Haiman equations aredcr+eih =∑u∈µ(dcrucu+eih (I) + cru(I)dcu+eih).Corollary 2.1.14. In particular, at the monomial ideal Iµ ∈ Hµ, the linear Haiman equations aredcr+eih = dcrh−ei ,for r /∈ µ and h ∈ µ, where we have adopted the convention that dcrh = 0 unless h ∈ µ.Proof. At a monomial ideal, crh = δ(r ∈ µ) δrh.Remark 2.1.15. Interpreting dcrh graphically as arrows going from r outside to h inside of thepartition, the elements of the cotangent space at a monomial ideal can be interpreted as arrowsup to translational equivalence provided by the linear Haiman equations Corollary 2.1.14, with theconvention that arrows with heads outside the partitions (including the negative octants) are zero.Remark 2.1.16. Clearly the arrows with tails just outside of µ span T∨IµH[n].2.2 Moduli space of framed modules M[n]In this section we give another description of the Hilbert scheme, in terms of a subscheme of amatrix variety quotiented out by an action of the general linear group. First notice that:Remark 2.2.1. Again let R = C[x1, x2, x3]. Instead of parametrizing ideals, the Hilbert schemecan instead be thought of as parameterizing n-dimensional quotients M of R along with a mapR → M whose image generates R. A map φ : R → M is called a framing, and the pair (M,φ) iscalled a framed module. Framed R-module M and M ′ are identified under isomorphisms M →M ′10that commute with the framings. Framed modules and ideals are equivalent under the equivalence(M,φ) 7→ kerφ and I 7→ (R/I,R→ R/I).Remark 2.2.1 gives us an alternative description of the Hilbert scheme, one that will give riseto a concrete DGLA that we use to prove our vanishing result around special monomial ideals. Inthis section we expand on that idea.Noting that the R-module structure of R/I is completely determined by how x1, x2, and x3multiply, we can reconstruct it from its underlying n-dimensional C-vector space V , a distinguishedvector C → V corresponding to 1 ∈ R/I, along with three pairwise-commuting endomorphismsXi : V → V . Finally, we quotient out by simultaneous conjugation by isomorphisms V ∼= V ′commuting with the four maps.Remark 2.2.2. Since all n-dimensional vector spaces are isomorphic, we could have fixed a repre-sentative V , and just quotient by Aut(V ) ∼= Gl(V ).We now follow Section 1.2 of [16] to construct our moduli space. The various claims in theconstruction are proven in Lemma 1.2.1 and 1.2.2 of [16]. First, some notations:Notation 2.2.3. For r = (r1, r2, r3) ∈ (Z≥0)3, and X = (X1, X2, X3) ∈ End(V )⊕3, we writeXr = Xr11 Xr22 Xr33 .Construction 2.2.4. Fix V an n-dimensional C-vector space. Let Gl(V ) act onX = {(X1, X2, X3, v) ∈ End(V )⊕3 ⊕ V : [Xi, Xj ] = 0}by g·(Xi, v) = (gXig−1, gv) for g ∈ Gl(V ). Consider the linearization of the Gl(V ) action associatedwith the character χ : Gl(V )→ C∗ given by the determinent. Then the locus of semi-stable pointsand the locus of stable points coincide, and are given by the condition that the set{Xrv : r ∈ Z3≥0}spans V . GIT then gives us a quasi-projective geometric quotient:M[n] = {(X1, X2, X3, v) ∈ End(V )⊕3 ⊕ V : [Xi, Xj ] = 0}//χGl(V )={(X1, X2, X3, v) ∈ End(V )⊕3 ⊕ V : [Xi, Xj ] = 0, Xrv spans V}/Gl(V )Remark 2.2.5. Note that the Gl(V )-action in Construction 2.2.4 is indeed well defined since thecommutation condition[Xi, Xj ] = 0 (2.1)and the spanning condition are both invariant under the Gl(V )-action on End(V )⊕3 ⊕ V .11Proposition 2.2.6. The Gl(V )-action of Construction 2.2.4 is free.Proof. Suppose g ∈ Gl(V ) fixes (X, v), i.e. [Xi, g] = 0 and gv = v. Then gXrv = Xrgv = Xrv. Inparticular, since {Xrv} spans V , g must be the identity.Remark 2.2.7. Explicitly, GIT constructs M[n] as M[n] = ProjS where S is the graded-ringS =⊕n≥0O(X )Gl(V ),χn .Here the notation RG,φ denotes the G-covariant elements of R transforming under g ∈ Gl(V ) asx 7→ φ(g)x. Note thatO(X ) = C[End(V )∨⊕3 ⊕ V ∨]commutation rel..Remark 2.2.8. The equations [Xi, Xj ] = 0 can also be written as the derivatives of the homoge-neous cubicf = tr([X1, X2]X3),which is invariant under the Gl(V )-action. ThusM[n] can be written as the critical locus of f . I.e.M[n] = Zero(df) ⊂ (End(V )⊕3 ⊕ V )/Gl(V ).2.2.1 Tangent and cotangent spacesThe tangent space T[P,vP ]M[n] at [P, vP ] ∈M[n] is obtained by linearizing the equations and Gl(V )-action defining M[n]:Proposition 2.2.9.T[P,vP ]M[n] ∼={(X, v) ∈ End(V )⊕3 ⊕ V | [Pi, Xj ] = [Pj , Xi] for i < j}{([P, α], αvP ) | α ∈ End(V )} ,where [X,α] is defined by [X,α]i := [Xi, α]. Here we wrote the tangent vectors with an underlineto prevent confusion with the notation for a point in X . They are both elements of End(V )⊕3⊕V ,though, and that’s where the commutators are defined.The cotangent space at [P, vP ] isProposition 2.2.10.T ∨[P,vP ]M[n] ∼={` ∈ End(V )∨⊕3 ⊕ V ∨ | `([P, α], αvP ) = 0 for all α ∈ End(V )}〈linearized commutation relations〉 .12The linearized commutation relations are defined in Definition 2.2.12 after we introduce somenotations.Notation 2.2.11. We will define dXi : End(V )⊕3 → End(V ) by dXi(X) = Xi, and define[dXi, Xj ] : End(V )⊕3 → End(V ) by [dXi, Xj ](X) = [dXi(X), Xj ] = [Xi, Xj ]. We need this seem-ingly redundant notation because dXi is conceptually better understood as a map T End(V )⊕3 →End(V ) produced by canonically identifying the tangent space T End(V ) with End(V ). We willuse it to remind ourselves that it takes a vector in the tangent space as input.Definition 2.2.12. The linearized commutation relations mentioned in Proposition 2.2.10 areφ([dXi, Pj ]) = φ([dXj , Pi])for i < j, and φ ∈ End(V )∨. Note that these indeed satisfy the requirement `([P, α], αvP ) = 0 since[[P, α]i, Pj ]− [[P, α]j , Pi] = [[Pj , Pi], α] = 0.2.3 Isomorphism M[n] → H[n]In [16, Prop 1.2.2], Szendr¨oi shows that moduli space of framed modules constructed in Section 2.2is canonically isomorphic to the Hilbert scheme (Proposition 1.1.5). In particular, we get anisomorphism pulling back the Haiman coordinates in Section 2.1 to the matrix coordinates. Inthis section, we describe this isomorphism explicitly, so it can be leveraged to prove the vanishingresults in Chapter 6.Definition 2.3.1. For each parition µ, and (X, v) ∈ End(V )⊕3 ⊕ V , define Bµ(X, v) to be the set{Xhv : h ∈ µ}.Definition 2.3.2. Then for each partition µ, we define set Mµ to be [X, v] ∈ M[n] such thatBµ(X, v) is a basis for V (note it’s enough that it spans V ).Remark 2.3.3. If we fix a basis of V , the condition that Bµ(X, v) spans V is equivalent to thecondition det J(Bµ(X, v)) 6= 0, where J(v1, . . . , vn) is the n × n matrix with columns v1, . . . , vnwritten in the basis. We will fix an arbitrary ordering on Bµ(X, v) for this; the particular orderinghas no significance on the following discussion.Remark 2.3.4. Since J(Bµ(g · (X, v))) = gJ(Bµ(X, v)) for g ∈ Gl(V ), we see thatdet J(Bµ(X, v)) ∈ O(X )Gl(V ),χ,the degree 1 part of the graded-ring in Remark 2.2.7.13Remark 2.3.5. We see that Mµ is exactly the distinguished affine open of M with function ringgiven by the homogeneous localizationO(Mµ) ∼= S(det J(Bµ(X,v))),where S is the graded-ring of Remark 2.2.7. Explicitly, O(Mµ) consists of fractions of the formf(det J(Bµ(X,v)))n, with f ∈ O(X )Gl(V ),χn .Definition 2.3.6. If B is a basis of V , v ∈ V and b ∈ B, we define coeffB(v, b) ∈ C to be theunique coefficient of b in the expansion of v in the basis B, so thatv =∑b∈BcoeffB(v, b)b.Lemma 2.3.7. If A ∈ End(V ), and v,B, b as in Definition 2.3.6, thencoeffB(Av, b) =∑b′∈BcoeffB(v, b′) coeffB(Ab′, b)Proof. Applying Definition 2.3.6 we getAv =∑b∈BcoeffB(v, b)Ab and Ab =∑b∈BcoeffB(Ab, b) b.The result follows.Definition 2.3.8. Let h ∈ µ and r ∈ Z3≥0, define coeffrh ∈ O(Mµ) bycoeffrh(X, v) = coeffBµ(Xrv,Xhv).(Note we have suppressed the dependence on µ).Remark 2.3.9. This is indeed an element of O(Mµ) since one can showcoeffrh(X, v) =det(J(Bµ(X, v) with the column Xhv replaced by Xrv)det(J(Bµ(X, v))),with the numerator and denominator both living in O(X )Gl(V ),χ.Definition 2.3.10. Define maps µ :Mµ → Hµ of affine schemes by the dual map∗µ : O(Hµ)→ O(Mµ)crh 7→ coeffrh .14Lemma 2.3.11. This is indeed well-defined since1. The coeffrh are Gl(V )-invariant.2. the Haiman equations get mapped to zero.Proof. 1. Left-multiplying the equation defining coeffrh by g ∈ Gl(V ), we getgXrv =∑h∈µcoeffrh(X, v) gXhv,x which showscoeffrh(g · (X, v)) = coeffBµ(g·(X,v))(gXrv, gXhv) = coeffrh(X, v).2. Applying Lemma 2.3.7 with A = Xi for i = 1, 2, 3, v = Xrv, and b = Xhv, we see that thecoeffrh satisfiescoeffr+eih =∑h∈µcoeffru coeffu+eih .Proposition 2.3.12. The maps µ in Definition 2.3.10 glue to a map  :M[n] → H[n].Proof. We must check that the µ’s agree on overlapsMµ∩Mµ˜, or equivalently, that the followingdiagram commutesMµ ∩Mµ˜//Mµ˜µ˜""Mµµ%%Hµ˜Hµ // H[n].This is the same as the existence of a map Mµ ∩Mµ˜ → Hµ ∩Hµ˜ such thatMµ ∩Mµ˜&&//Mµ˜µ˜$$Mµµ&&Hµ ∩Hµ˜// Hµ˜Hµ15commutes, or dually, a map O(Hµ ∩Hν)→ O(Mµ ∩Mν) commutingO(Mµ ∩Mµ˜) O(Mµ˜)ooO(Mµ)OOO(Hµ ∩Hµ˜)hhO(Hµ˜)oo∗µ˜ffO(Hµ)OO∗µhh.Using Proposition 2.1.12, we obtain a spacecraft:O(Mµ ∩Mµ˜) O(Mµ˜)ooO(Mµ)OOOµ˜(Hµ ∩Hµ˜)∗µ˜,µllOµ˜(Hµ˜)oo∗µ˜kkOµ(Hµ ∩Hµ˜)∗µ,µ˜aa∼66Oµ(Hµ)OO∗µaawhere Oµ(Hµ ∩Hµ˜)→ Oµ˜(Hµ ∩Hµ˜) is the isomorphism from Proposition 2.1.12.We now define ∗µ,µ˜ : Oµ(Hµ∩Hµ˜)→ O(Mµ∩Mµ˜) to be the unique map induced by O(Hµ)→O(Mµ) → O(Mµ ∩ Mµ˜) under the universal property of localization. ∗µ,µ˜ and ∗µ˜,µ then bydefinition commutes the left and right square, respectively. To use the universal property, we onlyneed to show that det(crh) ∈ O(Hµ) is mapped to a unit in O(Mµ ∩Mµ˜):det(ch˜h) 7→ det((coeffBµ)h˜h) =1det((coeffBµ˜)hh˜),where the last equality follows from Lemma 2.3.13.It then remains to show that the middle triangle commutes. The right side takescrh 7→∑h˜∈µ˜c˜rh˜(c˜−1)h˜h 7→∑h˜∈µ˜(coeffBµ˜)rh˜(coeff−1Bµ˜)h˜h,whereas the left side takes crh 7→ (coeffBµ)rh. The two are equal in O(Mµ ∩Mµ˜) by Lemma 2.3.13.16Lemma 2.3.13. In O(Mµ ∩Mµ˜),∑h∈µ(coeffBµ)rh(coeffBµ˜)hh˜= (coeffBµ˜)rh˜.In particular, ∑h∈µ(coeffBµ)h˜h(coeffBµ˜)hh˜= δh˜h .Note that of course the equations with h and h˜ interchanged also holds.Proof.Xrv =∑h∈µ(coeffBµ)rh(X, v)Xhv =∑h∈µ∑h˜∈µ˜(coeffBµ)rh(X, v)(coeffBµ˜)hh˜(X, v)X h˜v.2.4 Induced maps on cotangentsIn this section we study the linear isomorphism between the cotangent spaces at p ∈M[n] and thecorresponding point (p) ∈ H[n].Remark 2.4.1. Suppose p ∈M[n]. The pullback map ] : OH[n] → OM[n] induces a linear isomor-phism of cotangent spaces T∨(p)H[n] → T∨pM[n] given bydcrh 7→ d coeffrhAt p = [P, vP ], we have a natural spanning set of T∨pM[n] defined using Bµ(P, vP ):Definition 2.4.2. Fixing [P, vP ] ∈Mµ, we writedXhui = coeffBµ(P,vP )(dXiPuv, P hv)dvh = coeffBµ(P,vP )(dv, Phv).We will adopt the convention that dXrr′i = 0 if either r 6∈ µ or r′ 6∈ µ, and similarly, dvr = 0 ifr 6∈ µ.Remark 2.4.3. The dXhui and dvh for h, u ∈ µ span T∨[X,v]M[n].The goal of this section is to write the image of the cotangent map in Remark 2.4.1 as linearcombinations of the natural basis elements dXhui and dvh of Definition 2.4.2. This will be donethrough several steps:17(1) Calculate d coeffrh explicitly as a section in the cotangent bundle ofM[n] over the open patchMµ. This is done by taking the derivatives of the distinguished elements coeffrh ∈ O(Mµ)(Proposition 2.4.7).(2) Simplify the expression from Item (1) by introducing interim notation dXhγ (Proposition 2.4.9).(3) Write dXhγ as linear combinations of our basis elements dXhui and dvh (Lemma 2.4.13).Before starting on Item (1), we need to set up some notations.Definition 2.4.4. A path is a sequence γ = (γ1, . . . , γk) where γ` = ej for some j = 1, 2, 3. Ageneralized path is a sequence γ = (γ1, . . . , γk) where γ` = ±ej for some j = 1, 2, 3. We write| γ | = k for the length of the path. Also, if r = ∑ γ := ∑` γ`, we say that γ is a path to r.Definition 2.4.5. We will call γr the canonical path to r defined asγr = (e1, . . . , e1︸ ︷︷ ︸r1 times, e2, . . . , e2︸ ︷︷ ︸r2 times, e3, . . . , e3︸ ︷︷ ︸r3 times).Notation 2.4.6. If we have a path γ, and a sequence (A1, . . . , Ak) of length k = | γ |, we write(A1, . . . , Ak)γ = Aγkk · · ·Aγ11 whenever the product makes sense. Note the reversed order.Proposition 2.4.7.d coeffrh(X, v) =| γr |∑j=1coeffBµ(X,v)((X, . . . , dX︸︷︷︸jth, . . . , X)γrv,Xhv) + coeffBµ(X,v)(Xrdv,Xhv)−∑h′∈µcoeffrh′(X, v)·| γh′ |∑j=1coeffBµ(X,v)((X, . . . , dX︸︷︷︸jth, . . . , X)γh′ , Xhv) + coeffBµ(X,v)(Xh′dv,Xhv)Proof. DifferentiatingXrv =∑h∈µcoeffrh(X, v)Xhvgives usd(Xrv) =∑h∈µd coeffrh(X, v)Xhv + coeffrh(X, v) d(Xhv)| γr |∑j=1(X, . . . , dX︸︷︷︸jth, . . . , X)γrv +Xrdv=∑h∈µd coeffrh(X, v)Xhv + coeffrh(X, v)| γh |∑j=1(X, . . . , dX︸︷︷︸jth, . . . , X)γh +Xhdv18Substituting in(X, . . . , dX︸︷︷︸jth, . . . , X)γh =∑h′∈µcoeffBµ(X,v)((X, . . . , dX︸︷︷︸jth, . . . , X)γh , Xh′v)Xhdv =∑h′∈µcoeffBµ(X,v)(Xhdv,Xh′v),reindexing, and factorizing out the Xhv term, we see that the proposition follows from the definitionofcoeffBµ(X,v)| γr |∑j=1(X, . . . , dX︸︷︷︸jth, . . . , X)γrv +Xrdv,Xhvafter applying linearity in the first argument.We can simplify the large formula in Proposition 2.4.7 by introducing new notations.Notation 2.4.8. Implicitly fix [X, v] ∈Mµ. Then for each h ∈ µ and path γ, we define dXhγ anddvhγ in T∨[X,v]M[n] asdXhγ(X, v) =∑j=1,...,| γ |coeffBµ(X,v)((X, . . . , dX︸︷︷︸jth term, . . . , X)γv,Xhv)dvhγ(X, v) = coeffBµ(X,v)(X∑γdv,Xhv)Proposition 2.4.7 then readsProposition 2.4.9.d coeffrh = dXhγr + dvhγr −∑h′∈µcoeffrh′(dXhγh′ + dvhγh′ ).Proof. Simple rewrite of Proposition 2.4.7 using Notation 2.4.8.Corollary 2.4.10. d coeffrh = 0 if r ∈ µ.Remark 2.4.11. It’s easy to check that the dual vector d coeffrh(P, vP ) lives in the cotangent spaceT∨pMµ over a point p ∈Mµ. Indeed, it vanishes on tangent vectors corresponding to infinitessimalactions of Gl(V ).Remark 2.4.12. It can now be explicitly checked that the linear Haiman equations are mappedto zero under the cotangent map of Remark 2.4.1.To complete the goal of this section, we can now write dXhγ and dvhγ in terms of the naturalspanning set in Definition 2.4.2:19Lemma 2.4.13.dXhγ =| γ |∑j=1∑h′,h′′∈µcoeff∑j−1`=1 γ`h′ (dXγj )h′′ h′ coeffh′′+∑| γ |`=j+1 γ`hdvhγ =∑h′∈µdvh′coeffh′+∑γh .Proof.dXhγ(X, v) =| γ |∑j=1coeffBµ(X,v)((X, . . . , dX︸︷︷︸jth term, . . . , X)γv,Xhv)=| γ |∑j=1coeffBµ(X,v)(X∑| γ |`=j+1 γ`dXγjX∑j−1`=1 γ` , v,Xhv)from which the lemma follows directly by applying Lemma 2.3.7 twice.2.5 Path equivalenceRecall that the elements of the cotangent space are identified under the linear commutation relationsin Definition 2.2.12. We’d like to further understand the equivalence classes of the expression ford coeffrh in Proposition 2.4.9. It turns out that the dXhγ ’s where the γ’s end at the same points areidentified, so that we could have used any path to r in Proposition 2.4.9.Lemma 2.5.1. If γ and γ′ have the same endpoints, then dXhγ + dvhγ and dXhγ′ + dvhγ′ differby an element in Definition 2.2.12.Proof. Since they have the same endpoints, γ and γ′ must have the same length `, and furthermore,they must be related by a permutation σ ∈ S`. Decompose σ into a product of involutions τiswitching i and i+ 1. It then suffices to show that for any path ρ of length ` and involution τi,dXhρ − dXh(τiρ)is an element in Definition 2.2.12. We will do this by showing that∑k((X, . . . , dX︸︷︷︸kth, . . . , X)ρ − (X, . . . , dX︸︷︷︸kth, . . . , X)τiρ)= A([dXρi+1 , Xρi ]− [dXρi , Xρi+1 ])20for some A ∈ End(V ), so that applying coeffBµ(X,v)(−, Xhv) gives us the lemma. To this end,∑k((X, . . . , dX︸︷︷︸kth, . . . , X)ρ − (X, . . . , dX︸︷︷︸kth, . . . , X)τiρ)=∑k 6=i,i+1((X, . . . , dX︸︷︷︸kth, . . . , X)ρ − (X, . . . , dX︸︷︷︸kth, . . . , X)τiρ)+ (X, . . . , dX︸︷︷︸ith, . . . , X)ρ − (X, . . . , dX︸︷︷︸ith, . . . , X)τiρ+ (X, . . . , dX︸︷︷︸i+1th, . . . , X)ρ − (X, . . . , dX︸︷︷︸i+1th, . . . , X)τiρ=Xρ` · · · ([dXρi+1 , Xρi ]− [dXρi , Xρi+1 ]) · · ·Xρ1 .Corollary 2.5.2. In Propositions 2.4.7 and 2.4.9, we may replace γr by any path to r, γh′ by anypath to h′, and still get the same section in the cotangent bundle over Mµ.Proof. Apply Lemma 2.5.1 to Proposition 2.4.9.2.5.1 At monomial idealsAt monomial ideals, we can make several important simplifications to the result calculated inthe previous section. The result is crucial to the proof of the main theorem in Section 6.4 as itwill ultimately allow us to calculate the weight structure of the cotangent spaces of M[n] as arepresentation of the torus action, by transferring it to the cotangent spaces of H[n].Definition 2.5.3. Let µ be a partition, we define pµ = [Pµ, vµ] ∈ Mµ by coeffrh(Pµ, vµ) = δ(r ∈µ)δrh. It’s easy to see that (pµ) = Iµ, the monomial ideal corresponding to µ. We will thus call pµthe framed module corresponding to µ.Lemma 2.5.4. At pµ, Proposition 2.4.9 simplifies tod coeffrh = δ(r 6∈ µ) dXhγr .Proof. coeffrh = δ(r ∈ µ) δrh at pµ, so Proposition 2.4.9 gives usd coeffrh = δ(r 6∈ µ)(dXhγr + dvhγr)However, if r 6∈ µ, Xr = 0 for our monomial ideal, so δ(r 6∈ µ) dvhγr = 0.21Lemma 2.5.5. At pµ, dXhγ becomesdXhγ =| γ |∑j=1(dXγj )(h−∑| γ |`=j+1 γ`) (∑j−1`=1 γ`)Proof. coeffrh = δ(r ∈ µ) δrh at pµ, sodXhγ =| γ |∑j=1∑h′,h′′∈µδ(∑j−1`=1γ` ∈ µ)δ∑j−1`=1 γ`h′ (dXγj )h′′h′δ(h′′ +∑| γ |`=j+1γ` ∈ µ)δh′′+∑| γ |`=j+1γ`h=| γ |∑j=1δ(∑j−1`=1γ` ∈ µ)(dXγj )(h−∑| γ |`=j+1 γ`)(∑j−1`=1 γ`).The result follows because of our convention for out-of-scope indices in Definition 2.4.2.The following corollaries provide more intuition for the cotangent spaces at monomial ideals.They are not needed to prove our main result.Corollary 2.5.6. At pµ, if γ = γ′ + γout where γout is entirely outside of µ (defined below), thendXhγ = dX(h−∑γout)γ′ .whenever h −∑ γout ∈ µ (and h ∈ µ of course). By “entirely outside”, we mean that ∑ γ′ +∑n`=0(γout)` /∈ µ for all 1 ≤ n < | γout |.Proof. The summands in Lemma 2.5.5 vanish for j > | γ′ |, sodXhγ =| γ′ |∑j=1(dXγj )(h−∑| γ |`=j+1 γ`) (∑j−1`=1 γ`)=| γ′ |∑j=1(dXγ′j )(h−∑γout−∑| γ′ |`=j+1 γ′`) (∑j−1`=1 γ′`)= dX(h−∑γout)γ′ if h−∑γout ∈ µ.Corollary 2.5.7. At pµ, let h ∈ µ. If r, r′ 6∈ µ are related by a generalized path (see Definition 2.4.4)γ (i.e. if r′ − r = ∑ γ) such that h+∑j`=1 γ` ∈ µ for all 1 ≤ j ≤ | γ |, thend coeffrh = d coeffr′h+∑γ .Proof. We break γ into a sequence of paths which contains only +ej and paths that contains only−ej . Then the result follows from Corollary 2.5.6.22Chapter 3Differential graded Lie algebrasDGLAs will be defined over Sections 3.2 and 3.3. Section 3.5 will introduce the DGLA of abstractmodules, and finally, Section 3.6 will introduce the DGLA of framed modules, i.e. the tangentDGLA of (the loop space of) the moduli space of framed modules (Construction 2.2.4).3.1 Why DGLAs?We would like to give some intuition to motivate working with DGLAs before defining them, as themachinery can seem a little arcane without some background on current research in deformationtheory. As mentioned in Section 1.1, there exists many ways in the literature for formalizing thenotion of a formal deformation problems to the derived setting. Some work with global derivedstacks, and take their formal completion at a point (Lurie and Toe¨n-Vezzosi), whereas some directlyformulate their stacks in the formal-local setting (Kontsevich, Hinich, and Manetti). Regardless ofthe approaches, however, the result is some category-theoretic structure “Moduli” of formal derivedstacks, however they are defined. The central result of deformation theory then says that, in eachcase, whenever k of characteristics 0, there is an equivalence of some form between Moduli andLiedgk , the category of differential graded Lie algebras over k.For example, Lurie [10] works in the setting of ∞-categories, where he defines Moduli to bethe∞-subcategory of Fun(CAlgsmk ,Spaces) carved out by some Schlessinger-type conditions on thefunctors. Here Spaces is the ∞-category of spaces whose objects are simplicial sets. The relevanceof DGLAs then comes from the fact that, over a field k of characteristics zero, one can constructan equivalence of ∞-categories between Moduli and Liek, where Liek is the underlying ∞-categoryof the ordinary category Liedgk .The geometric intuition behind the equivalence is that, given a formal moduli stack X ∈ Moduli,we can look at its tangent complex TX at its unique k-point X(k). In general, this tangent complexwill have the structure of a k-module spectrum, which is the derived version of a k-vector space.Under the Dold-Kan correspondence, these are equivalent to chain complexes of k-vector spaces.However, the tangent complex TΩX of the loop space (again at the unique point) ΩX is evenmore interesting, and admits a natural Lie structure. This is because ΩX has a natural ∞-group23structure given by concatenation of loops (up to coherent homotopy); a∞-version of Lie’s theoremthen gives us an equivalence between the formal ∞-group and the tangent complex at the identity.In general, the Lie structure on TΩX can be concretely realized as an L∞-algebraic structure on thechain complex, but up to equivalence, this can be further identified with a differential graded Liealgebra (with a possibly different underlying chain complex). Since in general, the original spaceX can be recovered from ΩX up to homotopy, the above discussion produces a map X 7→ TΩXthat gives an equivalence Moduli→ Liek.With the above intuition in mind, we go on to introduce differential grade Lie algebras anddiscuss some of their basic properties in the next few sections.3.2 DG-vector spacesNotation 3.2.1. Let ModDGk denote the symmetric monoidal category of DG k-vector spaces whoseobjects are cochain complexes· · · d−2−−→ V −1 d−1−−→ V 0 d0−→ V 1 d1−→ · · ·and whose morphisms are cochain maps. The symmetric monoidal structure is given by the usualtensor product defined by(V ⊗W )n :=⊕n=p+qV p ⊗k V q,along with the symmetry isomorphism V ⊗W ∼= W ⊗V is given by the direct sum of isomorphismsV p ⊗k W q ∼= W q ⊗k V p weighted by a factor of (−1)pq.Remark 3.2.2. In the literature, one usually find DG- as a prefix for differential graded thingswith an underlying cochain complex, where the differential has degree 1, and dg- for those withunderlying chain complexes, where the differentials are of degree −1.Definition 3.2.3. A map V • → W • of DG-vector spaces that induce isomorphisms H•(V ) →H•(W ) on cohomology groups is called a quasi-isomorphism.Notation 3.2.4. If V is a graded vector space, let V [n] denote V with gradings shifted by n, sothat V [n]p = V p+n. Let V ∨ denote the k-linear dual of V , defined by (V ∨)p := Homk(V −p, k). Ingeneral, x ∈ V p for some p will be called a homogeneous element of degree p, and we will writex := p. We sometimes omit the bar if the element appears as an exponent, so that for example(−1)x = (−1)x.We now introduce notations for graded-symmetrizations and antisymmetrizations that will beused in Section 3.6 through Chapter 5.24Notation 3.2.5. Let V be a Z-graded vector space, we denote the graded-symmetric product by·, and the graded-antisymmetric product by ∧.We have two actions of the symmetric group Sn on V⊗n:Notation 3.2.6. Let v1, . . . , vn ∈ V .Let (σ) := (σ, v1, . . . , vn) ∈ {1,−1} be such that vσ1  · · ·  vσn = (σ) v1  · · ·  vn, and letχ(σ) be defined the same way for the antisymmetric product ∧.We denote the symmetrization map by ιsn : Vn → V ⊗n and the quotient map by pisn : V ⊗n →V n. Explicitly, ιn is induced by the map1n!∑σ∈Snσ· : V ⊗n → V ⊗n,where σ ∈ Sn acts on V ⊗n byσ · (v1 ⊗ · · · ⊗ vn) = (σ, v1, . . . , vn) vσ1 ⊗ · · · ⊗ vσn.Note that ιsn is a section of pisn.Similarly, we denote by ιαn : V∧n → V ⊗n and piαn : V ∧n → V n their antisymmetric counterparts,induced by the actionσ · (v1 ⊗ · · · ⊗ vn) = χ(σ, v1, . . . , vn) vσ1 ⊗ · · · ⊗ vσn.When no confusion is possible, we will write either the symmetric or antisymmetric maps as simplyιn and pin. We also write Πn for the composition ιnpin (or Πsn and Παn when ambiguous). Lastly, iff1, . . . , fn are maps V →W , we writef1  · · ·  fn = pisn(f1 ⊗ · · · ⊗ fn) : V ⊗n →Wnf1 ∧ · · · ∧ fn = piαn(f1 ⊗ · · · ⊗ fn) : V ⊗n →W∧n.3.3 Algebra objects and DGLAsWe will define differential graded Lie algebras using the symmetric monoidal structure on ModDGk .Algebra objects of a general symmetric monoidal category are defined as follows:Definition 3.3.1 (algebras). Let (C ,⊗) be a symmetric monoidal category. An (unital) associativealgebra object in C is an object A ∈ C together with a unit 1 : 1 → A and a multiplication map25µ : A⊗A→ A that satisfiesA⊗A⊗Aµ⊗idid⊗µ// A⊗AµA⊗A µ // AA⊗AµAid⊗1;;id // A Aidoo1⊗idccA morphism between algebra objects of C is a morphism φ : A→ B in C commuting with themultiplication maps:A⊗A //φ⊗φAφB ⊗B // BThe category of algebra objects of C together with their morphisms will be denoted by Alg(C ).Remark 3.3.2. One can verify that the diagrams in Definition 3.3.1 correspond to the familiaraxioms of unit, associativity, and homomorphisms.Remark 3.3.3. Note that we have implicitly identified various iterated tensor products with theircanonical representatives using the appropriate canonical isomorphisms included as part of the dataof symmetric monoidal categories.In analogy to associative algebras, one can define Lie algebra objects of a symmetric monoidalcategory in a diagram-theoretic way. We leave the details to the reader.Notation 3.3.4. Define LieDGk to be the category of Lie algebra objects of ModDGk . These will becalled differential graded Lie algebras over k, or DGLAs short.Remark 3.3.5. Concretely, a DGLA g over k is a cochain complex (g•, d) of k-modules equippedwith a Lie bracket [ ] : g⊗ g→ g of degree 0 satisfying(1) [x, y] = −(−1)xy[y, x](2) [z, [x, y]] = [[z, x], y] + (−1)zx[x, [z, y]](3) d[x, y] = [dx, y] + (−1)x[x, dy]for homogeneous x, y, z. A map of DGLA is then a map of the underlying chain complexes com-muting with the brackets.Note Item (3) is equivalent to requiring that d be a derivation of degree 1 with respect to thebracket, and Item (2) says any homogeneous element z defines a derivation [z,−] : gp → gp+z ofdegree z.26Remark 3.3.6. We can also view [ ] as a map g∧ g→ g of degree 0, satifisfying Items (2) and (3)of Remark 3.3.5.3.4 The deformation functor associated with a DGLAIn this section we will describe how to get a moduli functor from a DGLA, following [11].Construction 3.4.1. The Maurer-Cartan locus of a nilpotent DGLA g is the subset of g1MC(g) ={x ∈ g1 : dx+ 12[x, x] = 0}.We define an action of exp g0 on g1 byeα ? x = x+∑n≥01(n+ 1)![α,−]n([α, x]− dα).This action is stable on MC(g), and we call this the gauge action. Therefore we can defineDef(g) =MC(g)gauge action of exp(g0).Construction 3.4.2. We define a functor LieDGk → Moduli by associating with a DGLA g a functorDefg : CAlgsmk → Set given byDefg(A) = Def(g⊗mA),where mA is the unique maximal ideal in A, and g⊗mA is the DGLA with bracket and differential[x⊗ r, y ⊗ s] = [x, y]⊗ rsd(x⊗ r) = dx⊗ r.Note g ⊗ mA is nilpotent since mA is nilpotent. When a deformation problem X ∼= Defg for someDGLA g, we say that g controls the deformation problem X.3.5 DGLA of abstract modules lPNotation 3.5.1. We write T∨ = T∨0 C3 the cotangent space at the origin, so thatT∨ = C〈dx1, dx2, dx3〉.Definition 3.5.2. Let P ∈ End(V )⊕3. We associate with P a DGLA lP that has underlying chain27complexl•P = ∧•T∨ ⊗ End(V )and bracket[ω ⊗X,ω′ ⊗X ′] = (ω ∧ ω′)⊗ [X,X ′].The differential isd(ω ⊗X) = [∑idxi ⊗ Pi, ω ⊗X].Note l1P∼= End(V )⊕3, and we will often write a general element simply as a triple of matrices.3.5.1 Cyclic structure on lPDefinition 3.5.3. A choice of an isomorphism vol : T∧3 → C gives lP a cyclic structure. Wesometimes view this as a map of graded vector spaces vol : T∧• → C[3], with the map being zeroon all other components.Remark 3.5.4. A cyclic structure on lP is equivalent to a choice of a volume form in T∧3.Proposition 3.5.5. A cyclic structure induces a nondegenerate bilinear pairingT∧k × T∧3−k −→ C(ω, ω′) 7−→ vol(ω ∧ ω′),which can be extended to a pairing on lPκ : lkP × l3−kP −→ Cκ(ω ⊗X,ω′ ⊗X ′) = vol(ω ∧ ω′) tr(XX ′).The pairing is equivalent to an isomorphismlkP −→(l3−kP)∨x 7→ κ(x,−).We can also view the pairing as a map κ : l•P ⊗ l•P → C[3].Proof. κ is nondegenerate because T∧k⊗T∧3−k → C and (A,B) 7→ tr(AB) are both nondegenerate.28Proposition 3.5.6.κ(dx, y) = −(−1)xκ(x, dy)for all x, y ∈ lP .Proof. Writing x = ω ⊗X and y = η ⊗ Y , P = ∑ dxi ⊗ Pi, we haveκ(dx, y) = κ([∑idxi ⊗ Pi, ω ⊗X], η ⊗ Y )=∑iκ(dxi ∧ ω ⊗ [Pi, X], η ⊗ Y )=∑ivol(dxi ∧ ω ∧ η) tr([Pi, X]Y )= −(−1)1·x∑ivol(ω ∧ dxi ∧ η) tr(X[Pi, Y ])= −(−1)xκ(x, dy).Note tr([Pi, X]Y ) = tr(X[Pi, Y ]) follows from tr([Pi, XY ]) = 0.Proposition 3.5.7. The paring on on lP induces a nondegenerate pairing on H•(lP ).Proof. The pairing on H(lP ) is obtained by first restricting κ to ker d•⊗ker d•, then showing that theresulting map vanishes on im d•⊗ker d•+ker d•⊗ im d•. This follows easily from Proposition 3.5.6.The fact that the pairing is nondegenerate follows from Serre’s duality applied to the structuresheaf.3.6 DGLA of framed modules gpWe are now ready to define the DGLA that controls the formal deformation problem of M[n] at apoint [P, vP ] ∈M[n].Definition 3.6.1. Let p = (P, vP ) ∈ End(V )⊕3 ⊕ V be so that [p] ∈ M[n]. We define the DGLAgp to have underlying chain complexgp = lP ⊕ V [−1],29and bracket [ ]gp : gp ∧ gp → gp defined through the universal properties of direct sum, like sogp ∧ gp(pil∧pil)ια2 //2(pi0pil⊗piV )ια2[ ]g&&lP ∧ lP[ ]l""l0P ⊗ V [−1]m&&gppiVpil// lPV [−1],where pi0 : lP → l0P is the projection onto l0P (viewed as a chain complex concentrated at degree0), and m : l0P ⊗ V [−1]→ V [−1] ∼= (End(V )⊗ V → V )[−1] is the shifted multiplication map. Fornotation of antisymmetrization, see Notation 3.2.6.The differential is defined asdg = [p,−]g.where (P, vp) ∈ g1p.When no confusion would arise, we will simply write d and [ ] for dg and [ ]g, respectively.Remark 3.6.2. Almost directly by definition, pil : gp → lP is a map of DGLAs. In particular, it isa fibration.Lemma 3.6.3. Definition 3.6.1 indeed gives a well-defined DGLA.Proof. We only need to check that the bracket satisfy the Jacobi identity in Remark 3.3.5. This isequivalent to[ ]g ([ ]gpiα2 ∧ id) ια3 = 0.See Notation 3.2.6 for notations. Expanding using our definition[ ]g = ([ ]l(pil ∧ pil)⊕ 2m(pi0pil ⊗ piV ))ια2 ,we have[ ]g ([ ]gpiα2 ∧ id) ια3 =([ ]l(pil ∧ pil)⊕2m(pi0pil ⊗ piV ))Πα2(([ ]l(pil ∧ pil)⊕ 2m(pi0pil ⊗ piV ))Πα2 ⊗ id)ια330Distributing over the first direct sum, the first part is[ ]l(pil ∧ pil)Πα2(([ ]l(pil ∧ pil)⊕ 2m(pi0pil ⊗ piV ))Πα2 ⊗ id)ια3= [ ]l(pil ∧ pil)(([ ]l(pil ∧ pil)⊕ 2m(pi0pil ⊗ piV ))Πα2 ⊗ id)ια3= [ ]l(pil([ ]l(pil ∧ pil)⊕ 2m(pi0pil ⊗ piV ))Πα2 ∧ pil id)ια3= [ ]l([ ]l(pil ∧ pil)Πα2 ∧ pil)ια3= [ ]l([ ]lpiα2 ∧ id)ια3pil = 0since [ ]l satisfies the Jacobi identity. The second part is2m(pi0pil ⊗ piV )Πα2(([ ]l(pil ∧ pil)⊕ 2m(pi0pil ⊗ piV ))Πα2 ⊗ id)ια3= m(pi0pil ⊗ piV )(([ ]l(pil ∧ pil)⊕ 2m(pi0pil ⊗ piV ))Πα2 ⊗ id)ια3−m(pi0pil ⊗ piV )(id⊗([ ]l(pil ∧ pil)⊕ 2m(pi0pil ⊗ piV ))Πα2)σ312ια3= m(pi0[ ]l(pil ∧ pil)Πα2 ⊗ piV)ια3 −m(pi0pil ⊗ 2m(pi0pil ⊗ piV )Πα2)ια3 .Butpi0[ ]lpiα2 = [ ]00(pi0 ⊗ pi0) = 2µΠα2 (pi0 ⊗ pi0) = 2µ(pi0 ⊗ pi0)Πα2 ,Hence the first term ism(pi0[ ]l(pil ∧ pil)Πα2 ⊗ piV)ια3 = m(pi0[ ]lpiα2 (pil ⊗ pil)Πα2 ⊗ piV)ια3= m(2µ(pi0 ⊗ pi0)Πα2 (pil ⊗ pil)Πα2 ⊗ piV)ια3= 2m(µ(pi0pil ⊗ pi0pil)Πα2 ⊗ piV)ια3= 2m(µ⊗ id)(pi0pil ⊗ pi0pil ⊗ piV )(Πα2 ⊗ id)ια3= 2m(id⊗m)(pi0pil ⊗ pi0pil ⊗ piV )(id⊗Πα2 )ια3= 2m(pi0pil ⊗m(pi0pil ⊗ piV )Πα2)ια3 ,and thus cancels out with the second term.As mentioned in Section 1.2, the following result is a conjecture. However, we have goodreasons to believe that it is true. For example, it specializes to an isomorphism on the tangentspace.Conjecture 3.6.4. The DGLA gp controls the formal deformation problem of M[n] at [p]. I.e.Defgp∼= HomfSch∗((Spf −, pt), (Mˆ[n][p] , [p])).31Remark 3.6.5. Letting A = k[]/(2) be the ring of dual numbers, we see that Defgp = ker d1/d0g0,which agrees with the tangent space of Proposition 2.2.9.Remark 3.6.6. Unpacking Definition 3.6.1, we see that H1(gp) is canonically isomorphic to thetangent space of M[n] at p in Proposition 2.2.9.Proposition 3.6.7.0→ V [−1]→ gp pil−→ lP → 0is a short exact sequence of complexes.Proof. pil is a map of DGLAs, which is a chain map on the underlying chain complexes. The kernelof this map is V [−1].Proposition 3.6.8. There are canonical isomorphisms of vector spacesH0(gp) ∼= 0Hk(gp) ∼= Hk(lP ), k = 1, 2, 3.Proof. Follows directly from the long exact sequence on cohomology induced by the short exactsequence in Proposition 3.6.7.32Chapter 4Coalgebras as formal stacksIn this section, we introduce the theory of coalgebras, and explain how to view them as formalstacks. The purpose of this section is twofold: it aims to1. provide a short introduction of the machinery behind Kontsevich and Hinich’s approach toformal derived stacks; and to2. set up the necessary vocabulary to discuss L∞-algebras in Chapter 5.Even though the results discussed in this section is useful for understanding and motivating thesubsequent section on L∞-algebras, they will not be directly needed in the proof of Theorem 1.2.1.Readers only interested in the proof of Theorem 1.2.1 can skip directly to the definition of L∞-algebras (Definition 5.3.6) in Section 5.3.4.1 Cofree cocommutative coalgebras as formal pointedmanifoldsWe start by defining coalgebra objects in a general symmetric monoidal category C . In [6], Hinichshowed that, when C is the the category of Z-graded vector spaces, we obtain good models forformal stacks.The definition of a coalgebra object is simply the category-theoretic dual of that of an algebraobject (Definition 3.3.1).Definition 4.1.1 (coalgebras). Let (C ,⊗) be a symmetric monoidal category. A (coassociative)coalgebra object in C is an object C ∈ C together with a comultiplication map ∆ : C → C ⊗ Cthat satisfies the coassociativity condition:C∆∆ // C ⊗ C∆⊗idC ⊗ C id⊗∆// C ⊗ C ⊗ CA morphism between coalgebra objects of C is a morphism F : C → D in C commuting with33comultiplication maps:C //FC ⊗ CF⊗FD // D ⊗D.The category of coalgebra objects of C together with their morphisms will be denoted byCoAlg(C ).Definition 4.1.2 (counital coalgebras). A counital coalgebra in C is a coalgebra C ∈ CoAlg(C )with a counit  : C → 1 satisfyingC Cidoo∆id // CC ⊗ C⊗id;;id⊗cc .A morphism between counital coalgebras must further commute with the counit maps:C //F1D?? .If, furthermore, the comultiplication of a coalgebra is invariant under the symmetry isomorphismof the symmetric monoidal category, the coalgebra is said to be cocommutative:Definition 4.1.3 (cocommutative coalgebras). Let CoCAlg(C ) denote the full subcategory spannedby objects of CoAlg(C ) whose comultiplication satisfy the further diagramC∆∆%%C ⊗ C τ // C ⊗ C,where τ denotes the symmetry isomorphism of the symmetric monoidal category C . Furthermore,let CoCAlg/1(C ) denote counital cocommutative coalgebra objects in the symmetric monoidalcategory C with terminal object 1.A subclass of cocommutative coalgebras – conilpotent coaugmented coalgebras – is of particularinterest to deformation theory:Notation 4.1.4. For n ≥ 2, denote by ∆(n) : C → C⊗n the iterated comultiplications defined34inductively by∆(n) := (∆⊗ id⊗n)∆(n−1) and ∆(2) := ∆.Definition 4.1.5 (conilpotent and locally conilpotent). Let C be a cocommutative coalgebra. Wesay C is conilpotent if ∆(n) vanishes for n  0. We say C is locally conilpotent if for any c ∈ C,∆(n)(c) = 0 for n 0.Remark 4.1.6. For coalgebras of finite length (i.e. Artin objects in C ), local conilpotency andconilpotency coincide.Definition 4.1.7 (coaugmented (counital) coalgebra). A coaugmented (counital) coalgebra is acounital coalgebra (C,∆, ), together with a map of counital coalgebras 1 : k → C, where k is thecoalgebra with comultipliation 1 7→ 1⊗ 1 and counit 1 7→ 0. In particular, the coaugmentation is asection of the counit map (i.e. 1 = idk).We denote the category of coaugmented counital coalgebra objects in C by CoCAlg1/1(C ).A conilpotent coaugmented coalgebra is naturally interpreted as a pointed formal stacks inHinich, with the coaugmentation picking out the base point, and the conilpotency guaranteeingformal properties (he calls these unital coalgebras in place of coaugmented conilpotent coalgebras).Kontsevich uses a slightly different presentation for pointed formal stacks, which he calls formalpointed manifolds. These are cofree coalgebras without counits (note the definition of cofree includesconilpotency). We discuss these next.Definition 4.1.8 (reduced symmetric coalgebra). Given V ∈ C , we can construct the reduced sym-metric coalgebra Sc(V ) =⊕n≥1 Vn, which is a coalgebra (without counit) with comultiplication∆c given on each V n for n ≥ 1 by∆c|V n =n−1∑k=1(nk)(ιkpisk ⊗ ιn−kpisn−k)ιsn,where ιk (without the superscript) is the inclusion Vk ↪→ Sc(V ).See Notation 3.2.6 for definitions of ι and pi.Definition 4.1.9 (cofree). Let C be a conilpotent coalgebra. We say C is cofree if it is isomorphicto the reduced symmetric coalgebra Sc(V ) of some V ∈ C .Proposition 4.1.10. When C = ModZk is the category of graded vector spaces, the functor V 7→Sc(V ) is right adjoint to the forgetful functor from conilpotent coalgebras to the category of gradedvector spaces. I.e. they are truly cofree in the category-theoretic sense. In general, withoutconilpotency, the right adjoint is much more difficult to describe.Proof. See [12, Proposition 5.5].35Definition 4.1.11. A formal pointed manifold in C is a cofree (hence conilpotent) coalgebra objectof C (without counit). A morphism of formal pointed manifolds, also called a formal map, is simplya coalgebra map. The category of formal pointed manifolds along with formal maps is then a fullsubcategory FormalPtManifold ⊂ CoCAlg(C ).See Remark 4.1.15 for another description of formal pointed manifolds.The notions of coalgebras without counits and coaugmented coalgebras with counits are in factequivalent (Remark 4.1.13). Under this equivalence, reduced symmetric coalgebras get mapped tosymmetric coalgebras, defined in Definition 4.1.14. Kontsevich’s formal pointed manifolds thus cor-respond to formal stacks in Hinich that happen to be symmetric. We now describe this equivalenceexplicitly.Construction 4.1.12. Given a coalgebra (C,∆) without counits, we can counitize it to get acoaugmented counital coalgebra (Cˆ, ∆ˆ, , 1) where Cˆ := k⊕C, ∆ˆ = ∆+1⊗ id + id⊗1,  : k⊕C → kis the projection with kernel C, and 1 : k → k ⊕ C is the inclusion.Conversely, if we are given a coaugmented counital coalgebra (C,∆, , 1), the reduced noncouni-tal coalgebra (C,∆) is given by C = ker  and ∆ = ∆ − 1 ⊗ id− id⊗1. One should check that ∆indeed defines a map C → C ⊗ C when restricted to C ⊂ C, and that it is a comultipication.Remark 4.1.13. The pair counitization and reduction gives an equivalence of category between thecategory of coalgebras in C without counits and the category of coaugmented coalgebras in C withcounits. Furthermore, this equivalence is well-defined when restricted to conilpotent coalgebras.Definition 4.1.14 (symmetric coalgebra). The counitization of the reduced symmetric coalgebraSc(V ) in Definition 4.1.8 is the symmetric coalgebra Sc(V ) =⊕n≥0 Vn, which is a counitalcoalgebra with comultiplication ∆c : Sc(V )→ Sc(V )⊗ Sc(V ) given by∆c|V n =n∑k=0(nk)(ιkpisk ⊗ ιn−kpisn−k)ιsn,where ιk is the inclusion Vk ↪→ Sc(V ). Explicitly, this is∆c(1) = 1⊗ 1 and∆c(v1  · · ·  vn) =∑σ∈Snn∑k=01k!(n− k)!(σ) (vσ(1)  · · ·  vσ(k))⊗ (vσ(k+1)  · · ·  vσ(n)).Note that k now ranges from 0 to n, and we have to interpret vσ(1)· · ·vσ(0) and vσ(n+1)· · ·vσ(n)as 1.Remark 4.1.15. Remark 4.1.13 allows us to identify formal pointed manifolds of Definition 4.1.11with symmetric coalgebras, and formal maps with maps of coalgebras preserving counits and coaug-mentations. For the rest of the paper, we will freely interchange between these two descriptions.In fact, since we will mainly be working with cofree coaugmented coalgebras, we simply call these36formal pointed manifolds.4.2 Maps of cofree coalgebras as formal mapsRecall from Definition 4.1.2 that a map f : C → D between formal pointed manifolds are just a mapof the underlying counital coalgebras – i.e. one that commute with comultiplications and counits.The following proposition shows that these maps can be interpreted as formal series preserving thebase points.For the rest of the paper, we will assume that C = ModZk is the category of Z-graded k-vectorspaces along with the tensor product over k. However, all of our discussion in Sections 4.2 to 4.4applies to any k-linear symmetric monoidal category C such that Sc : C → CoCAlg(C ) is right-adjoint to the forgetful functor. This includes the category of ordinary vector spaces, or Z2-gradedvector spaces, and so on. For a more comprehensive discussion of these categories, see [9, Chapter2]. For this reason, we will keep the notation general, and only focus on the case C = ModZk fromSection 4.5 onward.Proposition 4.2.1. Reduction, followed by composition with the corestriction Sc(W ) → W , in-duces an isomorphism of sets:HomCoCAlgconil1/1 (C )(Sc(V ), Sc(W ))∼−→ HomC (Sc(V ),W )f 7→ pi1f,whose inverse is given byf 7→ f˘ :=∑n≥11n!fn∆(n).Proof. To get the isomorphism, compose Remark 4.1.13 with the universal property of the cofreefunctor to getHomCoCAlgconil1/1 (C )(Sc(V ), Sc(W ))∼−→ HomCoCAlgconil(C )(Sc(V ), Sc(W ))∼−→ HomC (Sc(V ),W ).For proof of the explicit form of the inverse, see [12, Prop. 5.5].Notation 4.2.2. SinceHomC (Sc(V ), V ) ∼=∏k≥1HomC (Vk, V ),we will often write elements of HomC (Sc(V ), V ) as (fk)k≥1, and write pi1f = (f11 , f12 , f13 , · · · ) forf ∈ HomCoCAlg(C )(Sc(V ), Sc(V )).37Lemma 4.2.3. In the notation of Notation 4.2.2, f˘ is given explicitly on V n byf˘(v1  · · ·  vn) =∑k≥1∑i1,...,in≥1i1+···+ik=nn!k!i1! · · · in! (fi1pii1  · · ·  fikpiik) ιnfor f = (f1, f2, . . . ).Proof. See [15, Proof of Proposition 1.2].Remark 4.2.4. Morphisms of formal pointed manifolds capture our intuition regarding formalpower series. In particular, if f : Sc(V ) → Sc(W ) is a map, we can interpret f1k : V k → W inProposition 4.2.1 and Notation 4.2.2 as the degree-k Taylor coefficient of f .To see this, recall that a homogeneous polynomial of degree k is a (nonlinear) function f˜ : V →W satisfying f˜(λv) = λkf˜(v) for all λ ∈ k and v ∈ V . An elementary result usually under thename of polarization formula tells us that homogenous polynomials of degree p are are in bijectionwith symmetric k-multilinear forms V × · · · × V → W , i.e. maps V k → W . The correspondenceis given by{homo. poly. V →W of degree k} ←→ Homk(V k,W )f˜ 7−→ f(v1  · · ·  vk) := 1λ!∂∂λ1· · · ∂∂λkf˜(λ1v1 + · · ·+ λkvk)f˜(v) := f(v, . . . , v)←− [ f.It can be verified that compositions of morphisms of formal pointed manifolds do in fact corre-spond to compositions of polynomials.4.3 Coderivations as vector fieldsWe have seen that coalgebra morphisms can serve as formal maps between formal pointed manifolds.In this section, we define vector fields on formal pointed manifolds as derivations on the coalgebra.Definition 4.3.1. A coderivation along a map f : C → D of coalgebras is a linear map Q : C → DsatisfyingC //QC ⊗ Cf⊗Q+Q⊗fD // D ⊗D.A coderivation along id : C → C is simply called a coderivation of C. We will denote the object38in C classifying derivations along f : C → D by Coder(C,D; f) ∈ C , and the object classifyingderivations of C by Coder(C) := Coder(C,C; id).Remark 4.3.2. The composition of two coderivations Q ∈ Coder(C,D; f) and R ∈ Coder(D,E; g)does not in general give a derivation in Coder(C,E; gf). However, on Coder(C), the commutator[R,Q] = RQ− τ(R⊗Q) of two derivations does define a map Coder(C)∧2 → Coder(C). It is easilychecked that this map satisfy the Jacobi identity, and thus gives Coder(C) the natural structure ofa Lie object in C . Here τ : Coder(C)⊗2 → Coder(C)⊗2 is the symmetry isomorphism coming fromthe symmetric monoidal structure of C .Definition 4.3.3. A vector field on a formal pointed manifold M is simply a coderivation on M .Vector fields on formal pointed manifolds, just like formal maps between them, are describedby their Taylor expansions, once we fix isomorphisms to Sc(V ):Proposition 4.3.4. Composition with the corestriction Sc(V ) → V induces an isomorphism inModk:Coder(Sc(V ))∼−→ HomC (Sc(V ), V )Q 7→ pi1Q,whose inverse map is given byq 7→ qˆ :=∑n≥0(q  idn−1)∆(n).Explicitly, qˆ is given on Sn(V ) byqˆ(v1  · · ·  vn) =∑σ∈Snn∑k=01k!(n− k)!(σ, v1, . . . , vn) q(vσ(1)  · · ·  vσ(k)) vσ(k+1)  · · ·  vσ(n).Note that in the above formula we interpret a string v1  · · ·  v` of length ` = 0 to mean 1.Notation 4.3.5. SinceHomC (Sc(V ), V ) ∼=∏k≥0HomC (Vk, V ),we will often write elements of HomC (Sc(V ), V ) as (qk)k≥0, and write pi1Q = (Q10, Q11, Q12, Q13, · · · )for Q ∈ Coder(Sc(V )).Using the isomorphism in Proposition 4.3.4, we can transfer the Lie structure of Coder(Sc(V ))onto∏HomC (Vk, V ). When C = ModZk is the category of graded vector spaces, the symmetryisomorphism is simply k-multiplication by (−1)k` on weights (k, `), so the bracket is then givenexplicitly in components by the Nijenhuis-Richardson bracket:39Definition 4.3.6 (Nijenhuis-Richardson bracket).Homk(Vk, V )×Homk(V `, V )→ Homk(V k+`−1, V )(q, r) 7→ [q, r]NR := qrˆ − (−1)k` rqˆ.4.4 Tangent spacesDefinition 4.4.1. The tangent space to a formal pointed stack (C,∆, , 1) is the object TC :=ker ∆ ∈ C , where ∆ is the comultiplication of the reduction (C,∆).Proposition 4.4.2. If f : C → D is a morphism of formal pointed manifold, then f restricts to amap TC → TD on tangent spaces. The same thing is true for coderivations Q : C → D along f .Proof. Since ∆Df = (f ⊗ f)∆C , f maps ker ∆C to ker ∆D. And similarly for coderivations.Corollary 4.4.3. Let C be a formal pointed manifold. An isomorphism C ∼= Sc(V ) for someV ∈ C induces an isomorphism of tangent space TC ∼= V .Proof. Let D = Sc(V ) in Proposition 4.4.2, and note that TD = ker ∆c = V .Remark 4.4.4. Fixing isomorphisms C ∼= Sc(V ) and D ∼= Sc(W ) in Proposition 4.4.2, the inducedlinear map on tangents is just the first Taylor coefficient f11 : V →W (See Notation 4.2.2).4.5 Formal pointed DG-manifoldsSo far we have been working with the notion of formal pointed manifolds in a general symmetricmonoidal category C . For the rest of the paper, we will focus on the case when C = ModZk is thecategory of graded vector spaces along with the tensor product.Definition 4.5.1. A formal pointed DG-manifold is a pair (C,Q) where C is a formal pointedZ-graded manifold (i.e. a formal pointed manifold (Definition 4.1.11) in the category of Z-gradedvector spaces), and Q is a vector field on C of degree 1 vanishing at the marked point (i.e. Q(1)=0),and satisfying Q2 = 12 [Q,Q] = 0.We will mostly be dealing with formal pointed DG-manifolds in this paper, as opposed tothe more general conilpotent DG-coalgebras (C,Q) used by Hinich. However, later on in Con-struction 4.7.1, we will be building deformation functors from DG-coalgebras, just like we did forDGLAs. Since this works for any DG-coalgebras, so we’ll define them here:Definition 4.5.2. A DG-coalgebra is a pair (C,Q) where C is a coaugmented (counital) coalgebraand Q is a coderivation of degree 1 satisfying Q(1) = 0 and Q2 = 0.404.6 Tangent complexes and tangent quasi-isomorphismsThrough Proposition 4.4.2, a formal pointed DG-manifold (C,Q) induces a graded vector space TCand a linear map Q1 of degree 1. Since Q21 = 0, (TC , Q1) is in fact a cochain complex.Definition 4.6.1. The tangent complex of a formal pointed DG-manifold (C,Q) is the cochaincomplex (TC , Q1) induced by Proposition 4.4.2.Remark 4.6.2. Again through Proposition 4.4.2, a map of formal pointed DG-manifold inducescochain map of their tangent complexes at the marked points.Definition 4.6.3. A formal map inducing a quasi-isomorphism of tangent complexes is called atangent quasi-isomorphism, or t-qis in short.4.7 The deformation functor associated with a formal pointedDG-manifoldRecall that in Section 3.4, we constructed deformation functors from DGLAs. This section will dothe same for formal pointed DG-manifolds instead of a DGLA. In fact, the construction works forconilpotent DG-coalgebras (C,Q) (i.e. Hinich’s formal stacks) in general.Construction 4.7.1. Let C = (C,Q) be a conilpotent DG-coalgebra. We define the Maurer-Cartan functor associated with C to beMCC : CAlgsmk → SetA 7→ HomCoCAlgk/k(A∨, C),where A∨ is the dual coalgebra of A, conilpotent because A is nilpotent.We would like to define the deformation functor DefC : CAlgsmk → Set associated with C bytaking A to MCC(A) quotiented out by x0 ∼ x1 iff there is x(t, dt) ∈ MCC(1⊕mA ⊗ k[t, dt]) suchthat x(0, 0) = x0 and x(1, 0) = x1. Here k[t, dt] denotes the DG-algebra of differential forms on theaffine line Ak. However, 1⊕mA⊗k[t, dt] is not strictly speaking in CAlgsmk , so the precise definitionsrequire a bit more work. The interested can consult [12], where this situation is remedied by writing1⊕mA ⊗ k[t, dt] as a union of Artinian algebras. DefC is then shown to be a deformation functor.41Chapter 5L∞-algebras and homotopyclassification of formal pointedDG-manifoldsRecall from Section 4.5, a formal pointed DG-manifold is a formal pointed manifold C in thecategory of Z-graded vector spaces, along with a vector field Q vanishing at the marked point.In this section, we show that an isomorphism C ∼= Sc(V ) allows us to transfer the DG-manifoldstructure on C to an L∞-algebra structure on V . An L∞-structure, defined in Section 5.3, is anatural generalization of a DGLA structure to include brackets of higher arity. Before we get to it,however, we’d like to motivate L∞-algebras through a slightly different angle – as the solution tothe problem of classifying formal pointed DG-manifolds up to homotopy.In Section 5.1, we outline the problem of homotopy classification of formal pointed DG-manifolds.In Section 5.2, we define L∞[1]-algebras, which are used in literature to solve the problem of Sec-tion 5.1. The actual proofs will be outside the scope of this paper, as our goal in Sections 4.1 to 5.2is simply to motivate L∞-algebras in Section 5.3, in the context of the broader research landscape.We will thus defer the various proofs in Section 5.2 to [9].5.1 Homotopy classification of formal pointed DG-manifoldsRecall that a map of formal pointed DG-manifolds inducing a quasi-isomorphism of tangent com-plexes is called a tangent quasi-isomorphism (Definition 4.6.3). The most important result regard-ing the category of formal pointed DG-manifolds is that the tangent quasi-isomorphisms can beinverted. I.e.Proposition 5.1.1. If there is a t-qis C → D, then there is a t-qis D → C.Proof. Follows from Corollary 5.2.7.Thus, tangent quasi-isomorphisms gives a homotopy equivalence relation between formal pointedDG-manifolds. We would like to classify formal pointed DG-manifolds up to t-qis, and through42doing so, we will also gives a constructive proof of Proposition 5.1.1. The strategy is to decomposean arbitrary formal pointed DG-manifold into a direct product of minimal and linear contractibleones. The isomorphism classes of minimal formal pointed DG-manifolds will turn out to character-ize our homotopy equivalence classes. In other words, we will be able to decide if two formal pointedDG-manifolds are tangent quasi-isomorphic by checking if their minimal models are isomorphic.We now proceed to the necessary definitions.Definition 5.1.2. A formal pointed DG-manifold is minimal if its tangent complex (Definition 4.6.1)has the trivial differential. It is linear contractible if it is isomorphic to the symmetric coalgebra ofits tangent complex and if the tangent complex is contractible.Proposition 5.1.3. Any formal pointed DG-manifold is isomorphic to the direct product of aminimal one and a linear contractible one. (The direct product of formal manifolds is just tensorproduct of coalgebras).Proof. Follows from Proposition 5.2.6.Definition 5.1.4. The minimal factor in Proposition 5.1.3 is called a minimal model.Proposition 5.1.5. The minimal models of a formal pointed DG-manifold is unique up to isomor-phism. Morever, the isomorphism classes of mimimal formal pointed DG-manifolds coincide witht-qis classes of formal pointed DG-manifolds.Proof. Follows from Corollaries 5.2.8 and 5.2.9.5.2 L∞-algebras as local models of formal pointed DG-manifoldsThe decomposition of a formal pointed DG-manifold into minimal and linear contractible partsis most naturally understood by giving local models to these formal pointed DG-manifolds, calledL∞[1]-algebras. In this section, we define L∞[1]-algebras essentially as formal pointed DG-manifoldsC with a choice of “local affine coordinates” given by a graded vector space V and an isomorphismC ∼= Sc(V ). The Q2 = 0 condition can then be understood as algebraic equations satisfied bypolynomials on V .Definition 5.2.1. An L∞[1]-algebra over k is a pair (V,Q) where V ∈ ModZk , andQ ∈ Coder(Sc(V ))is a coderivation of degree +1 on Sc(V ) such that Q(1) = 0 and Q2 = 12 [Q,Q] = 0.A morphism f : (V,Q) → (W,R) of L∞[1]-algebras (also called a L∞[1]-morphism) is a mapf : Sc(V ) → Sc(W ) of coaugmented counital coalgebras (i.e. a morphism of coalgebras satisfyingf(1) = 1) commuting with the coderivations: fQ = Rf .The category of L∞[1]-algebras over k together with L∞[1]-morphisms will be denoted L∞[1]Algk.43Remark 5.2.2. An L∞[1]-algebra (V,Q) is just a formal pointed DG-manifold (C,Q) with a choiceof isomorphism C ∼= Sc(V ). We think of this as a choice of local affine coordinates on the formalpointed DG-manifold.Remark 5.2.3. Under the isomorphism of Proposition 4.3.4, we can rewrite the conditionQ2 = 0 interms of the Nijenhuis-Richardson brackets (Definition 4.3.6): an L∞[1]-algebra is then a sequence(V, q1, q2, · · · ) where the collection of degree 1 maps qk := Q1k ∈ Homk(V k, V ) satisfyn−1∑k=1[qk, qn−k]NR = 0for every n ≥ 1. Notice that we have not included a q0 term, as the requirement Q(1) = 0 translatesto q0 = Q10 = 0.Notation 5.2.4. If (V, q1, q2, . . . ) ∈ L∞[1]Algk (i.e. qk satisfy the condition in Remark 5.2.3), thenwe write the associated formal pointed DG-manifold given by Proposition 4.3.4 as Sc(V, q1, q2, . . . ).Remark 5.2.5. Definition 5.1.2 can now be made explicit:(M,Q) is minimal if (M,Q) ∼= Sc(V, 0, q2, q3, . . . ) for some (V, 0, q2, q3, . . . ) ∈ L∞[1]Algk; (M,Q)is linear contractible if (M,Q) ∼= Sc(V, q1, 0, 0, . . . ) for some (V, q1, 0, 0, . . . ) ∈ L∞[1]Algk, and(V, q1) has trivial cohomology. For this reason, we will call these types of L∞[1]-algebras minimaland linear contractible, respectively.Proposition 5.2.6. Any (V, q1, q2, . . . ) ∈ L∞[1]Algk can be decomposed into a direct sum ofminimal and linear contractible ones.Proof. See [9, Section 3.1]. The main idea is to construct the minimal model by induction on thedegree of the Taylor expansion. The base of the induction is the decomposition of the complex(V, q1) into the direct sum of a complex with vanishing differential and a complex with trivialcohomology.Corollary 5.2.7. If there is a quasi-isomorphism V →W between two L∞[1]-algebras, then thereis a quasi-isomorphism W → V .Proof. See [9, Corollary 3.0.11]. The key idea is that a quasi-isomorphism between two minimalL∞-algebras is an isomorphism through the inverse mapping theorem, because it induces an iso-morphism on the space of generators. Letting Vm and Wm denote minimal models of V and W inthe decomposition of Proposition 5.2.6, we then get inclusions Vm ↪→ V , Wm ↪→W and projectionsV → Vm, W →Wm, which are quasi-isomorphisms since the linear contractible factors have trivialcohomology. Then the compositionVm ↪→ V →W →Wm44defines a quasi-isomorphism Vm → Wm, which is an isomorphism by the above discussion. It isnow easy to see that the compositionW →Wm → Vm ↪→ Vgives the desired quasi-isomorphism W → V .The proof of Corollary 5.2.7 also gives us:Corollary 5.2.8. The minimal model of a L∞[1]-algebra is unique up to isomorphism.Corollary 5.2.9. Isomorphism classes of minimal L∞[1]-algebras coincide with quasi-isomorphismclasses.5.3 L∞-algebras as homotopy Lie algebrasIn Section 5.2, we motivated L∞[1]-algebras as a solution to the homotopy classification problemof Section 5.1. In this section, we show that they can also be seen as a generalization of DGLAswhere Jacobi’s identity is only satisfied up to “coherent homotopy” given by brackets of higherarity.To get graded-antisymmetric structures instead of graded-symmetric ones, we need to shift thedegrees of the underlying Z-graded vector spaces by 1.Construction 5.3.1. The shift functor defined in Notation 3.2.4 induces an isomorphism of gradedvector spaces(V [1])n ∼= V ∧n[n].Thus an L∞[1]-algebra (V, q1, q2, · · · ) with qk : V k → V [1] is equivalently a sequence (W, [ ]1, [ ]2, · · · )where W := V [−1], and [ ]k : W∧k →W [2−k] is produced as follows: we get a map W∧k[k]→W [2]from the compositionW∧k[k] ∼= (W [1])k = V k → V [1] = W [2],which is the same as a map W∧k →W [2− k].The equation satisfied by the qk’s in Remark 5.2.3 can then be translated into a generalizedversion of the Jacobi identity for DGLAs.Proposition 5.3.2 (generalized Jacobi identities). Let (V, qk) be an L∞[1]-algebra. Construc-tion 5.3.1 then gives us (W, [ ]k) such that for each n ≥ 1,∑σ∈Sn∑k+`=n+1(−1)k(`−1)χ(σ,w1, · · · , wk))[[wσ(1), · · · , wσ(k)]k, wσ(k+1), · · · , wσ(n)]` = 045for all w1, . . . , wn ∈W .Remark 5.3.3. The n = 1 equation gives ([ ]1)2 = 0, making (W, [ ]1, 0, · · · ) into a chain complexof k-vector spaces.Remark 5.3.4. The n = 2 equation gives[[v, w]2]1 = [[v]1, w]2 + (−1)v[v, [w]1]2.Remark 5.3.5. The n = 3 equation gives[w1, [w2, w3]2]2 = [[w1, w2]2, w3]2 + (−1)w1w2 [w2, [w1, w3]2]2+ [[w1, w2, w3]3]1 + [[w1]1, w2, w3]3 + (−1)w1 [w1, [w2]1, w3]3 + (−1)w1+w2 [w1, w2, [w3]1]3which is the ordinary Jacobi identity when [ ]3 vanishes. Roughly, this says that the ordinary Jacobiidentity is “satisfied up to a homotopy controlled by [ ]3”.Definition 5.3.6 (L∞-algebra). We will call a Z-graded k-vector space W together with graded-antisymmetric chain maps [ ]k : W∧n → W [2 − k] satisfying the equations of Proposition 5.3.2 anL∞-algebra over k. The category of L∞-algebras over k with L∞-morphisms will be denoted byL∞Algk.Remark 5.3.7. Remarks 5.3.3 to 5.3.5 show that a DGLA (g, d, [ ]) can be seen as an L∞-algebra(g, d, [ ], 0, 0, . . . ). It can be checked that any map of DGLAs is then an L∞-morphism. However,the converse is not true: there are in general more L∞-morphisms between two DGLAs than thereare DGLA morphisms. In other words, the inclusion of Liedgk into L∞Algk is not full.5.4 Explicit minimal model of a DGLAIn [9], Kontsevich and Soibelman gives a inductive construction of minimal models for a generalL∞-algebra. In the case of DGLAs, Schuhmacher provided in [15] an explicit construction usingformulas involving pseudoinverses and trees. We summarize Schuhmacher’s results in this section.5.4.1 PseudoinversesDefinition 5.4.1. Given a chain complex (V •, d), a linear map η : V → V of degree 1 is apseudoinverse to the differential d ifηdη = η and dηd = d.η is square-zero if furthermore η2 = 0. We will usually implicitly assume our pseudoinverses are46square-zero.Proposition 5.4.2. Given (V, d) a chain complex, a pseudoinverse η is equivalent to the data ofits kernel ker η ⊂ V and image im η ⊂ V (both graded vector subspaces). More precisely, given Kand I that are linear complements of im d and ker d, respectively, there is a unique pseudoinversewith ker η = K and im η = I.Proof. Theorem 10 of Chapter 5 in a textbook by Ben-Israel and Greville [1].The following remark shows that a Hermitian structure on the complex can automatically giveus a pseudoinverse. However, we don’t actually use this in the paper.Remark 5.4.3. Let (V, d, 〈 , 〉) be a chain complex over C endowed with a Hermitian metric 〈 , 〉,then defines a pseudoinverse through Proposition 5.4.2 by taking K and I to be the orthogonalcomplement of im d and ker d, respectively.Proposition 5.4.4. The pseudoinverse of Remark 5.4.3 is square-zero.Proof. Since ker d ⊂ im d, we have ker η = (im d)⊥ ⊂ (ker d)⊥ = im η.5.4.2 Oriented binary treesDefinition 5.4.5. An oriented binary tree is a binary tree with an ordering on the nodes and leaves.We will introduce them by examples, as the rigourous definition is rather dry, and unnecessarilytechnical for our purpose. Writing OTree(n) be the set of oriented binary tree with n leaves, welist out the elements of OTree(n) for n = 1, . . . , 4 in Examples 5.4.6 to 5.4.9.Example 5.4.6. OTree(1) has a single element: the tree consisting of only the root and no leaves.Example 5.4.7. OTree(2) has a single element00.1 0.2.The numbers on the nodes and indicate their ordering.Example 5.4.8. OTree(3) consists of two elements:00.10.11 0.12 0.200.10.20.21 0.22 .47Example 5.4.9. OTree(4) has five elements:00.10.110.111 0.112 0.12 0.200.10.110.120.121 0.122 0.200.10.11 0.120.20.21 0.2200.10.20.210.211 0.212 0.2200.10.20.210.220.221 0.222 .Remark 5.4.10. Notice that a binary tree T ∈ OTree(n) has exactly n− 1 (non-leaf) nodes.Definition 5.4.11. Given T ∈ OTree(n) and N a node or leaf of T , let wT (N) be the number oftimes one must choose left in order to reach N from the root. In other words, it is the number of1’s in the number labeling the node/leaf in the examples.Then if L1, . . . , Ln are the leaves of T ∈ OTree(n), we define e(T ) = (−1)wT (L1)+···+wT (Ln).Example 5.4.12. The unique element of OTree(1) has e = 1. The unique element of OTree(2)has e = −1. The first tree of Example 5.4.8 has e = −1, and the second has e = 1.Construction 5.4.13. Given a binary tree T ∈ OTree(n), and binary operators o1, . . . , on−1 :V ⊗ V → V , we define a n-ary operator T (o1, . . . , on−1) : V ⊗n → V in the following way:T (o1, . . . , on−1)(v1 ⊗ · · · ⊗ vn) ∈ V is the expression obtained by replacing the n − 1 orderednodes of T by o1, . . . , on−1 and the n ordered leaves by v1, . . . , vn, in order.Example 5.4.14. If T is the second tree in Example 5.4.9, thenT (o1, . . . , o3)(v1 ⊗ · · · ⊗ v4) = o1(o2(v1 ⊗ o3(v2 ⊗ v3))⊗ v4),as obtained from the treeo1o2v1o3v2 v3 v4 .Remark 5.4.15. Oriented trees, along with e(T ) and T (o1, . . . , on−1) are defined in full rigour inSchuhmacher [15]. For our purpose, the above informal presentation will suffice.5.4.3 Explicit formula for minimal modelProposition 5.4.16. Given a chain complex (V •, d) along with a square-zero pseudoinverse η,[d, η] := dη + ηd is idempotent with kernel ker d ∩ ker η.Proof. Evidently ker d ∩ ker η ⊂ ker[d, η]. Since η[d, η] = η and d[d, η] = d,ker η ∩ ker d = ker η[d, η] ∩ ker d[d, η] ⊃ ker[d, η] ∩ ker[d, η] = ker[d, η].48Notation 5.4.17. Let H := ker d∩ker η, and denote by ιH : H → V and pH : V → H the inclusionand the projection through 1− [d, η], respectively.Construction 5.4.18. Given a DGLA (g, d, [ ]) along with a square-zero pseudoinverse η, theminimal model induced by η is the minimal L∞-algebra (H, 0, b2, b3, b4, . . . ) where H = ker d∩ker η,andbk =(−12)k−1 ∑T ∈OTree(k)e(T ) T (pH [ ], η[ ], . . . , η[ ])ιαk ιHfor k ≥ 2, where OTree(k) is the set of oriented trees with k leaves, e(T ) = ±1, and T (o1, . . . , ok)is obtained by replacing the nodes of the tree T with binary operators o1, . . . , ok.The minimal model (H, b•) admits an L∞-morphism φ : (H, b•) → (g, d, [ , ]) to the originalDGLA, given byφ1k = −(−12)k−1 ∑T ∈OTree(k)e(T ) T (η[ ], . . . , η[ ])ιαk ιHfor k ≥ 1.Proposition 5.4.19. The minimal model defined in Construction 5.4.18 is indeed an L∞-algebraand φ is an L∞-morphism. Moreover, it is a quasi-isomorphism (i.e. φ induces isomorphisms oncohomology groups).Proof. See [15, Theorem 4.4 and 4.5].Remark 5.4.20. Since H has the zero differential, it is canonically isomorphic to its cohomology.Hence the quasi-isomorphism φ : H → g induces an isomorphism H• → H•(g).Remark 5.4.21. Pseudoinverses is one way, but not the only way to obtain minimal models ofDGLAs. The proof of the minimal model theorem in [9, Section 3.1] shows that there are manyfreedoms quantified by higher cohomology classes.5.4.4 Characterization of pseudoinverses5.5 The deformation functor associated with an L∞-algebraDefinition 5.5.1. Let V = (V, [ ]1, [ ]2, . . . ) ∈ L∞Algk be an L∞-algebra, and C = Sc(V [1])its associated symmetric coalgebra. We simply define MCV := MCC and DefV := DefC (cf.Construction 4.7.1).49Proposition 5.5.2. Explicitly, MCV (A) is the set of x ∈ V 1 ⊗ mA satisfying the generalizedMaurer-Cartan equation[x]1 +12![x, x]2 +13![x, x, x]3 + · · · = 0.And DefV (A) is MCV (A) quotiented out by x0 ∼ x1 iff for some a(t) polynomial in t with valuesin V 0 ⊗mA, there exists a solution x(t) to the differential equationx′(t) = [α(t)]1 + [α(t), x(t)]2 +12![α(t), x(t), x(t)]3 + · · ·such that x(0) = x0 and x(1) = x1.Proof. See discussion in [9, Chapter 3, Section 4.2] or Manetti [12, Lemma 5.11].Comparing Proposition 5.5.2 with Construction 3.4.1, we see that the deformation functorassociated with an L∞[1]-algebra is just the straightforward generalization of that for a DGLA.Thus:Corollary 5.5.3. If g ∈ LieDGk is a DGLA viewd as an L∞[1]-algebra, then the two definitions ofMCg and Defg in Definition 5.5.1 and Construction 3.4.1 agree.The following proposition exihibits MCV as a formal scheme of zeros of Q:Proposition 5.5.4. Let (V, [ ]•) ∈ L∞Algk and A ∈ CAlgsmk . Viewing V 1 as the formal schemewith function ring O(V 1) = k[[V 1∨]] ∼= ∏k≥0 V 1∨k and Q := ∑ 1k! [ ]k as a function in O(V 1), wehave that MCV (A) is precisely the A-points of the zero locus Zero(Q) ⊂ V 1. I.e.MCV ∼= HomfSch∗((Spf −,pt), (Zero(Q), 0)).Proof. Let A be an Artinian ring, Q naturally extends to QA : V1 ⊗mA → V 2 ⊗mA. ThenHomfSch∗((Spec(A),pt), (Zero(Q), 0))∼={φ ∈ HomCAlgtopk (C[[V1∨]], A) : φ(Q) = 0, φ((V 1∨)) ⊂ mA}∼= {φ ∈ Homk(V 1∨,mA) : ΦV 1∨,A(ιφ)(Q) = 0}∼= {x ∈ V 1 ⊗mA : QA(x) = 0}∼= MCV (A)by Proposition 5.5.2. Here CAlgtopk denotes the category of topological rings over k, Φ−,− denotesthe binatural isormophism HomCAlgCtop(C[[−]],−) ∼= Homk(−, U−) coming from the adjunctionC[[−]] a U , and ι : mA → A is the inclusion homomorphism.50Chapter 6Proof of main theoremIn this section, we will apply the machinery developed in Chapters 3 to 5 to the cotangent spacesof the Hilbert scheme studied in Chapter 2.Recall that, given p ∈ M[n] representing a framed module, we can construct a DGLA gp thatcontrols the formal deformation problem ofM[n] at p (Conjecture 3.6.4). In particular, H1(gp) willbe isomorphic to the tangent space TpM[n] (Remark 3.6.6). If we then choose a pseudo-inverse η,the minimal model constructed in Construction 5.4.18 produces a minimal L∞-algebra (Hp, 0, b•)that also controlsM[n] around p. Moreover, Proposition 5.5.4 tells us that the brackets b• directlytranslate to the equations defining our formal defomation problem as a formal subscheme of TpM[n].The goal of this section is to prove Proposition 6.4.3, which says that, in the special case wherep = pµ for a µ that we will call a “pyramid partition”, these equations will be quadratic with theright choice of η. We re-state the proposition here:Definition 6.0.5 (Pyramid ideals and partitions). A monomial ideal Iµ is a pyramid ideal ifIµ = (x, y, z)N for some N > 0. In this case, we call µ a pyramid partition.Proposition (Restatement of Proposition 6.4.3). Let µ be a pyramid partition. Then (gpµ , dpµ , [ ]pµ)is quasi-isomorphic to an L∞-algebra (Hpµ , 0, b•) withbk|(H1pµ )∧k = 0 : (H1pµ)∧k → H2pµ .for k > 2.Since H1p∼= H1(gp) ∼= TpM[n], Proposition 6.4.3 would immediately prove Theorem 1.2.1.Our strategy for proving Proposition 6.4.3 is to put a weight structure on the DGLAs andtangent spaces associated with the deformation problem. As we will see, this is only possible whenwe restrict to monomial ideal/framed modules corresponding to a partition µ. To simplify things,we will introduce new notations:51Notation 6.0.6. Let pµ be the framed module corresponding to a partition µ. We write (gµ, dµ, [ ]µ)for (gpµ , dgpµ , [ ]gpµ ).In Sections 6.1 to 6.3, we endow gµ, the cotangent spaces H[n] andM[n], and the minimal modelof Construction 5.4.18 with weight structures. Ultimately, these weight structures exist becauseof the torus action on H[n]. We will not, however, describe the action itself, as it suffices for ourpurpose to directly define the weights on the vector spaces, and check that the results obtained sofar respect the weight structures.In Section 6.4, we study the induced weight structure on the minimal models (Hµ, 0, b•) of gµobtained with pseudoinverses η compatible with its weight structure. The weight structure places arestriction on where b• can be nonzero, allowing us to prove the vanishing result (Proposition 6.4.3).Finally, we end the section with the proof of Theorem 1.2.1.6.1 Weights on gpDefinition 6.1.1. Let p = (P, vP ). We will define the weights on l•P = ∧•T∨ ⊗ End(V ) andgp = lP ⊕ V [1] by induction on their components:• The distinguished basis elements dxi ∈ T∨ will be given weights −ei.• P hvP ∈ V will be given weight h.• End(V ) will be given weights induced by the isomorphism End(V ) ∼= V ⊗ V ∨.• Weights will be additive under tensoring and wedging.The length of a weight w ∈ Z3 is ∑iwi ∈ Z.Notation 6.1.2. We will write weights 0 = (0, 0, 0) and 1 = (1, 1, 1).Definition 6.1.3. We say a weighted-space W is concentrated in a set S of weights if the weight-homogeneous elements in W have weights in S.Remark 6.1.4. The pairing lkP ⊗ l3−kP → C in Proposition 3.5.5 is weight-homogeneous of weight1. Thus the isomorphisms lkP → (l3−kP )∨ are also of weight 1.The following results are specific to framed modules corresponding to a partition µ. Recall thenotation of Notation 6.0.6.Corollary 6.1.5. pµ =∑i dxi ⊗ (Pµ)i + vµ ∈ g1µ is an element of weight 0.Proof. w(pµ) =∑i(w(dxi)+w((Pµ)i))+w(vµ). But w((Pµ)i) = ei, and w(vµ) = 0 since vµ = P0µvµ.Therefore w(pµ) =∑i(−ei + ei) + 0 = 0.52Proposition 6.1.6. The differential dµ and bracket [ ]µ of gµ are weight-homogeneous of weight 0.Proof. It suffices to show that [ ]µ is weight-homogeneous of weight 0, because dµ = [pµ,−]µ, and pµis weight-0 (Corollary 6.1.5). Following its definition in Definition 3.6.1, we see that [ ]p is actuallyweight-homogeneous of weight 0 for any p, not just for ones of the form pµ.Corollary 6.1.7. The cohomlogy H•(gµ) is a weighted graded vector space.Proof. dµ is weight 0 (Proposition 6.1.6).6.2 Weights on cotangent spacesDefinition 6.2.1. At monomial ideals Iµ ∈ H[n], we give weights r − h to the dual basis vectordcrh ∈ T∨IµH[n] in Proposition 2.1.13. This is compatible with the linear Haiman equations inCorollary 2.1.14.Definition 6.2.2. At the cotangent spaces T∨pµM[n] at pµ, we give weight u−h to dXhu and weight−h to dvh in Definition 2.4.2.Remark 6.2.3. The weights on T∨pµM[n] (Definition 6.2.2) are compatible with the weights on gµ(Definition 6.1.1) under the identification H1(gµ) ∼= TpµM[n] in Remark 3.6.6.Proposition 6.2.4. At a monomial ideal Iµ, the cotangent isomorphism T∨IµH[n] → T∨pµM[n] ofRemark 2.4.1 is a weighted isomorphism, in the weights of Definitions 6.2.1 and 6.2.2.Proof. By definition, dcrh maps to d coeffrh, which also has weight r− h by Lemmas 2.5.4 and 2.5.5.6.3 Weights on minimal modelsProposition 6.3.1. We can pick η a pseudoinverse (in the sense of 5.4.1) to dµ that is weight-homogeneous of weight 0.Proof. d = dµ can be decomposed into a collection of maps dα : gα → gα[−1] indexed by weight α.The task of finding a weight-0 pseudoinverse to d is then equivalent to finding a pseudoinverse ηαfor each dα.Proposition 6.3.2. If η is a weight-homogeneous pseudoinverse of weight 0, then H := ker η∩ker din Construction 5.4.18 will be a weighted subspace of gµ. Moreover, the isomorphism H• → H•(gµ)of Remark 5.4.20 will be an isomorphism of weighted chain complexes.Proof. It’s easy to check that kernels of weight-homogeneous maps are weighted subspaces, andthat intersection of weighted subspaces is again weighted. The isomorphism of Remark 5.4.20 issimply the inclusion Hµ → ker dµ followed by the projection ker dµ → H(gµ), both of which areweight-0 maps.536.4 Weights on pyramid ideals and proof of main theoremWe are ready to specialize to pyramid ideals. Proposition 6.4.1 together with Corollary 6.4.2 willshow that H1 and H2 are concentrated in weights of length −1 and −2, respectively, and so dueto weight considerations, the higher brackets vanishes on H1 (Proposition 6.4.3). Finally, we endthis section with the proof of Theorem 1.2.1.Proposition 6.4.1. If µ is a pyramid partition, the cotangent space T∨IµH[n] and T∨pµM[n] areconcentrated in weights of length 1.Proof. At a pyramid ideal, all arrows (Remark 2.1.15) are of the form dcrr−ei , where r is just outsideof the partition. Thus their weights are all (r − ei)− r = −ei for some i. Proposition 6.2.4 lets ustransfer the result from H[n] to M[n].Corollary 6.4.2. If µ is a pyramid partition, and η as above, then H1µ = ker η1 ∩ ker d1µ is concen-trated in weights of length −1. Moreover, H2µ is concentrated in weights of length −2.Proof. Proposition 6.4.1 tells us that the tangent space TpµM[n] is concentrated in weights oflength −1. The result for H1µ follows since H1µ ∼= H1(gµ) ∼= TpµM[n] as weighted vector spaces(Proposition 6.3.2 and Remark 6.2.3). The result for H2µ is then an immediate consequence ofRemark 6.1.4, since a weight-homogeneous element x ∈ H1∨µ of weight w(x) must be paired to aweight-homogeneous element y ∈ H2µ of weight w(y) = −1 − w(x). Its length would therefore be∑w(y) = −∑ 1−∑w(x) = −3− (−1) = −2.Proposition 6.4.3. Let µ be a pyramid partition, η any weight-0 pseudoinverse of dµ, and (Hµ, 0, b•)the induced minimal model. Then for k ≥ 3,bk|(H1µ)∧k = 0 : (H1µ)∧k → H2µ.Proof. Since H1µ is concentrated in weights of length −1 (Corollary 6.4.2), and dµ, η, [ ]µ are allweight 0 (Proposition 6.1.6), the image of bk|(H1µ)∧k is concentrated in weights of length −k. Butsince H2µ is concentrated in weights of length −2 (Corollary 6.4.2), this forces bk|(H1µ)∧k to be trivialunless k = 2.We are now ready to prove the main theorem of the paper.Proof of Theorem 1.2.1. Let Iµ be a monomial ideal as in the theorem, µ its corresponding pyramidpartition, and pµ = (C[x, y, z]/Iµ, 1) the corresponding framed module.The quasi-isomorphism Hµ → gµ induces an isomorphism DefHµ ∼= Defgµ of deformation func-tors. Conjecture 3.6.4 tells usDefgµ∼= HomfSch∗((Spf −,pt), (Mˆ[n]pµ , pµ)) ∼= HomfSch∗((Spf −,pt), (Hˆ[n]Iµ , Iµ)).54On the other hand, DefHµ = MCHµ since H0µ∼= 0. Our vanishing result (Proposition 6.4.3)applied to Proposition 5.5.4 (with (V,Q) = (Hµ, 0, b2, 0, . . . )[1]) then tells usDefHµ = MCHµ∼= HomfSch∗((Spf −, pt), (Zero(b2), 0),where Zero(b2) is considered as a formal subscheme of H1µ viewed as a formal scheme. But H1µ∼=TpµM[n] ∼= TIµH[n], so chaining the above equations together, we have the required resultHˆ[n]Iµ ∼= Zero(b2) ⊂ TIµH[n].To finish the proof, we note that b2 = df , where f : TIµH[n] → C is the homogeneous cubicdefined on x ∈ TIµH[n] ∼= H1µ byf(x) := κ(x, b2(x)).Remark 6.4.4. The vanishing result of Proposition 6.4.3 allowed us to prove the geometric resultin Theorem 1.2.1. A derived geometer might ask if the derived version of Theorem 1.2.1 holds:i.e. if the DGLA gµ is formal. A formal DGLA is one that is quasi-isomorphic to its cohomology,viewed as a DGLA with zero differential and the induced bracket. Equivalent, g is formal if wecould find a minimal model H of g on which all higher brackets vanish.If we could prove that, in Proposition 6.4.3, the entire bracket bk vanishes for k > 2, gµ wouldbe formal. Since Hµ (as in Proposition 6.4.3) is nonzero only at degree 1, 2, and 3. The only othercomponent of bk that could be nonzero isbk|(H1µ)∧k−1∧H2µ : (H1µ)∧k−1 ∧H2µ → H3µ.Unfortunately, our arguments by weights does not force this map to vanish: H0µ(lPµ)∼= V isconcentrated in weights µ, so H3µ(gµ)∼= H3µ(lPµ) is concentrated in weights in −1 − µ. However,the image of the map above is in general concentrated in weights of length −k− 1, which can verywell lie in −1− µ, since −1− µ contains weights of length from −3 to −3−N inclusive, where Nis the height of the pyramid ideal µ. This means that for a pyramid partition of height N , we canonly conclude bk vanishes entirely using weight reasons alone when k > N + 2. In particular, wecan show gµ is formal using weight reasons only at the trivial case of N = 0, corresponding to anordinary point at the origin in C3.55Bibliography[1] A. Ben-Israel and T. Greville. Generalized Inverses: Theory and Applications. CMS Books inMathematics. Springer, 2003. → pages 47[2] A. Dimca and B. Szendroi. The Milnor fibre of the Pfaffian and the Hilbert scheme of fourpoints on Cˆ3. ArXiv e-prints, Apr. 2009, 0904.2419. → pages 5[3] J. Fogarty. Algebraic families on an algebraic surface. American Journal of Mathematics,90(2):511–521, 1968. → pages 4[4] A. Grothendieck. Techniques de construction et the´ore`mes d’existence en ge´ome´triealge´brique iv: Les sche´mas de hilbert. Se´minaire Bourbaki, 6:249–276, 1960. → pages 7[5] M. Haiman. t, q-Catalan numbers and the Hilbert scheme. Discrete Mathematics,193(1–3):201–224, 1998. → pages 2, 6, 7, 8[6] V. Hinich. DG coalgebras as formal stacks. Journal of Pure and Applied Algebra,162(2–3):209–250, 8 2001. → pages 4, 6, 33[7] M. E. Huibregtse. The cotangent space at a monomial ideal of the Hilbert scheme of pointsof an affine space. ArXiv e-prints, June 2005, arXiv:math/0506575v1. → pages 7, 9[8] M. Kontsevich. Deformation quantization of poisson manifolds. Letters in MathematicalPhysics, 66(3):157–216, Dec 2003. → pages 4, 6[9] M. Kontsevich and Y. Soibelman. Deformation theory. Book in preparation, 2002. → pages37, 42, 44, 46, 49, 50[10] J. Lurie. Derived Algebraic Geometry X: Formal moduli problems. Preprint available atmath.harvard.edu/˜lurie/papers/DAG-X.pdf, 2011. → pages 3, 4, 23[11] M. Manetti. Deformation theory via differential graded Lie algebras. Seminari di GeometriaAlgebrica 1998-99 Scuola Normale Superiore, 1999, arXiv:math/0507284. → pages 27[12] M. Manetti. Extended deformation functors. International Mathematics Research Notices,2002(14):719–756, 01 2002. → pages 3, 35, 37, 41, 50[13] E. Miller and B. Sturmfels. Hilbert schemes of points, pages 355–378. Springer New York,New York, NY, 2005. → pages 4[14] J. P. Pridham. Unifying derived deformation theories. Advances in Mathematics,224(3):772–826, 6 2010. → pages 456[15] F. Schuhmacher. Deformation of L∞-algebras. ArXiv e-prints, 2004, arXiv:math/0405485.→ pages 38, 46, 48, 49[16] B. Szendroi. Non-commutative Donaldson-Thomas theory and the conifold. ArXiv e-prints,May 2007, 0705.3419. → pages 11, 13[17] B. Toen and G. Vezzosi. Homotopical Algebraic Geometry II: geometric stacks andapplications. American Mathematical Society, 2008, math/0404373. → pages 357

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