20 Cotangent complex In these lectures,1 we will motivate the cotangent complex by recalling some standard examples deformation theory. The basic principle is that the existence and number of deformations of an object is controlled by some cohomology theory. The model is that H 0 parametrizes “infinitesimal” automorphisms, H 1 parametrizes first-order deformations, and H 2 parametrizes obstructions to extending first-order deformations. We then turn to constructing the cotangent complex, although we instead focus on a dual problem of constructing “higher derived functors” for derivations. (Consequently, what we do will have no logical relevance in subsequent work in positive characteristic as we require, but it has some psychological/cultural value.) This will rest on simplicial algebras and a method of defining a notion of derived functors when the category is not abelian. We then discuss some of the properties of these derived functors and a few illustrative examples. Daniel Litt will continue with an overview of the properties of the cotangent complex and several examples of using it to analyze questions in deformation theory in a form we will actually need. 20.1 Motivation: Deformation theory We first will give two examples of the principle that a cohomology theory controls the deformation theory of an object. Recall that we should think of H 0 as parameterizing “infinitesimal” automorphisms, H 1 as parametrizing first-order deformations, and H 2 as parametrizing obstructions to extending first-order deformations. Example 20.1.1 The first example is analyzing deformations of representations. Let Γ be a discrete group and ρ : Γ → GLn (C) be a representation. A first-order deformation of ρ is a representation ρ : Γ → GLn (C[]/2 ) that reduces to ρ modulo . In other words, it is a function such that ρ (γ) = ρ(γ)(1 + M (γ)) for M : Γ → GLn (C). Writing out the condition to be a homomorphism, we see that M is forced to satisfy a cocycle condition that makes it represent a class in H 1 (Γ, Mn (C)). Here γ ∈ Γ acts on the Lie algebra Lie(GLn (C)) = Mn (C) through conjugation by ρ(γ). Modifying ρ via conjugation (which preserves the reduction to ρ) changes the cocycle by a coboundary. Next, observe that H 0 (Γ, Mn (C)) = Mn (C)γ , which is just End(ρ). Thus the H 0 does record infinitesimal automorphisms. Finally consider the question of lifting to order 2 . Given a deformation corresponding to [M ] ∈ H 1 (Γ, Mn (C)), define ob : H 1 (Γ, Mn (C)) → H 2 (γ, Mn (C)) to send [M ] to [M ] ∪ [M ] ∈ H 2 (Γ, Mn (C) ⊗ Mn (C)) and then map this class to H 2 (Γ, Mn (C)) via map induced by the commutator. A second-order deformation is possible if and only if ob([M ]) = 0. Example 20.1.2 We next consider deformations of a smooth variety X/k. Let Tx denote the tangent sheaf. The deformation theory is controlled by H ∗ (X, TX ). • H 0 (X, TX ) is the global vector fields, which one can think of as infinitesimal automorphisms of the variety. 1 March 2 and 9, 2015. Talk by Akshay Venkatesh, notes live-texed by Tony Feng, edited by Jeremy Booher. 1 e → Speck[]/2 where the fiber over 0 recovers • A first order deformation of X is a flat family X X. There is a bijection o n first-order ∼ H 1 (X, TX ) − → deformations . of X To make a deformation from a cohomology class, write X as a union of affines Ui = SpecAi . Then e is a union of SpecAi []/2 , but glued in a non-trivial way. This gluing is an automorphism of X the k[]-algebra Aij []/2 that is the identity modulo . Such a map takes the form a + b 7→ a + (δa + b) where δ is a k-linear derivation of Aij (this is pretty much the definition of derivation). A derivation is a section of the tangent sheaf, so we need a section of the tangent sheaf on overlaps, satisfying some cocycle condition, and this is exactly an element of H 1 (X, TX ). It is true, but not obvious, that if X is smooth then this map is a bijection. The smoothness is used in arguing that deformations of the affines is trivial. If X is not smooth, then open affines can deform non-trivially, so we have to replace TX be a complex of sheaves H• , such that H0 = TX . Then the deformations will be parametrized by H ∗ (X, H• ) (in the sense of hypercohomology). Roughly speaking, there will be two contributions to the first-order deformations: one from H 1 (X, TX ), which handles “gluing” of the affine deformations, and one from H 0 (X, H1 ), which handles deformations on the affines themselves. It turns out to be easier to construct the dual complex, which is the cotangent complex. One should think of it as a derived version of the cotangent sheaf. • The obstruction map H 1 (X, TX ) → H 2 (X, TX ) sends α to α ^ α ∈ H 2 (X, TX ⊗ TX ), then to H 2 (X, TX ) via the map induced on cohomology by the Lie bracket TX ⊗ TX → TX . Again, a second order deformation exists if and only if the obstruction class is zero. Remark 20.1.3 H ∗ (X, TX ) has the structure of a graded Lie algebra, i.e. we have H i (X, TX ) × H j (X, TX ) → H i+j (X, TX ) (via cup product and Lie bracket). Remark 20.1.4 How do you know when you can lift from k[]/2 to k[]/3 ? In order to answer this in characteristic 0, you need to lift the graded Lie algebra structure to cochains. This runs into a story of differential graded lie algebras (of no help for characteristic p). But an important point is that nothing matters beyond H 2 as far as the deformation theory is concerned. Unless the ring you’re deforming over is itself a DGLA, then they don’t have such a geometric interpretation. The situation of lifting only to k[]/2 is also misleading because there is a section k → k[]/2 . This means first order deformations always exist. When doing mixed-characteristic stuff as we will have to do (or even in deformation theory for Galois representations), one has to work with “first-order” situations like Z/p2 Z → Z/pZ that have no ring-theoretic section, in which case the way we think locally about the picture even in the smooth case is necessarily a bit more sophisticated than over the dual numbers. In particular, the set of deformations is generally just a torsor over H 1 , and so might also be empty. This already comes up in the deformation theory of smooth schemes for lifting proper flat curves to characteristic 0 as in SGA1. Also, for more general deformation situations keeping track of flatness in the deformation process requires some work. For example, in the smooth case it is flat deformations (equivalently, smooth deformations) of the affines that are trivial (not just “deformations” in some very general sense). 2 Example 20.1.5 Let X be a projective smooth curve of genus g ≥ 2. Then H 1 (X, TX ) has dimension 3g − 3 and H 2 (X, TX ) = 0, because X is a curve. This means that there are 3g − 3 parameters for first-order deformation, and beyond that there are no obstructions. That corresponds to Mg being smooth of dimension 3g − 3 at the point corresponding to X. Example 20.1.6 Let X be an abelian variety of dimension g over a field k. As the tangent sheaf of an abelian variety is trivial, h1 (X, TX ) = g 2 and h2 (X, TX ) = g g2 . However, the obstruction map is actually zero, as the Lie algebra of an abelian variety is abelian. So deformations always exist. Note however that the limit over successive deformations can not necessarily be made algebraic. Remark 20.1.7 There are quite a few details hidden in the above sketch. The biggest is an important theorem of Grothendieck that over an affine base, an infinitesimal deformation of an abelian scheme is automatically an abelian scheme. Furthermore, the infinitesimal automorphisms as a scheme and as an abelian scheme agree. A discussion of these results can be found in [1, Section 6.3]. Thus in this case there is no difference between considering deformations of the scheme and deformations of the scheme together with additional structure. Usually adding additional structure makes the deformation theory more complicated. It is also good to distinguish the moduli space (which is a global notion) from the formal deformation ring (which is a local notion). The formal deformation ring is what the deformation ring in the preceding example is analyzing. It is formally smooth of relative dimension g 2 over the Cohen ring of k. If instead we require the deformations to come with the the additional data of a polarization then (with hard work in characteristic p for polarizations of degree divisible by p) the formal deformation ring is always of dimension g(g + 1)/2. In residue characteristic p > 0, this deformation ring is not smooth if p divides the degree of the polarization. The advantage of this deformation problem is that the formal deformation (obtained as the inverse limit) can be algebraized. This is related to why the correct moduli space for abelian varieties must include the data of a polarization (of fixed degree d2 ). In the next section, we will attach to a scheme X a cotangent complex which is a complex of sheaves on X. We’ll do the affine version: given a map of rings A → B, we will construct LB/A ∈ D(B − Mod), and the general cotangent complex is the sheaf version of this. (Usually you do this by repeating the construction with sheaves. If you want to do it by gluing, then you have to be careful since gluing in derived categories is a can of worms.) This LB/A is a “derived functor of Ω1 .” 20.2 Non-abelian derived functors. If C is an abelian category and F : C → AbGrp is a functor, say right-exact. Then we can define higher derived functors of F by replacing M ∈ C by a “free resolution” (we are being intentionally vague about what this means) . . . → Pn → Pn−1 → . . . → P0 → M → 0 and then we apply F and take homology. There is a way of doing this if C is not an abelian category, in such a way that it agrees with the above construction when C is abelian, using simplicial resolutions. For full generality, see [2]. After we set this up, we’ll apply this to the functor given by derivations over some given ring in order to get the cotangent complex. 20.3 Simplicial sets Simplicial sets are an abstraction of a familiar example from algebraic topology. 3 Example 20.3.1 A model example is the chain chain complex of a topological space X. If ∆n denotes the standard n-simple, then we define Pn = Z[Maps(∆n , X)], the free abelian group on maps from ∆n → X. The collection of sets Maps(∆n , X) has various face maps, corresponding to the ways of including ∆n−1 ,→ ∆n . 1 0 → . . . Maps(∆2 , X) → → Maps(∆ , X) ⇒ Maps(∆ , X) There are also degeneracy maps coming from maps ∆n → ∆n−1 . This gives a combinatorial model of homotopy theory. Definition 20.3.2 A simplicial set is a collection of sets {Σn } together with, for each n, n + 1 face maps di : Σn → Σn−1 and n + 1 degeneracy maps si : Σn → Σn+1 satisfying the following axioms: 1. If i < j, we have di dj = dj−1 di . 2. If i < j, we have di sj = sj−1 di . 3. If i = j or i = j + 1, we have di sj = id. 4. if i > j + 1, we have di sj = sj di−1 . 5. If i ≤ j, we have si sj = sj+1 si . These are exactly the axioms the face and degeneracy maps in the above example satisfy. It is also possible to rephrase this in terms of a simplicial set being a contravariant functor out of the simplex category: the category of linearly ordered finite sets with order preserving maps. A simplicial object in a category C is the same, but with Σn and the maps being objects and morphisms in C. Equivalently, it is a contravariant functor from the simplex category to C. 20.4 Outline of the Construction Recall that given a map of rings A → B, we can form the Kähler differentials Ω1B/A , which is a B-module having the property that HomB (Ω1B/A , N ) = DerA (B, N ). To extend this to a complex, the natural attempt is to derive (in the non-abelian sense) either the functor B 7→ Ω1B/A or B 7→ DerA (B, N ). Instead of taking a free (or projective) resolution as one would normally do, in this case we use a “free simplicial resolution.” Let C be the category of augmented A-algebras, i.e. A-algebras C equipped with augmentation map C → B. For a fixed B-module N , we can consider the functor on C C 7→ DerA (C, N ). Alternatively, in the Ω1 formulation we would consider C 7→ Ω1C/A ⊗C B. We will discuss how to define (non-abelian) derived functors, which will yield DeriA (B, N ). Taking i = 0, we recover Der0A (B, N ) = DerA (B, N ). Taking i = 1, we can interpret Der1A (B, N ) = ExalcommA (B, N ), where these are (square zero) algebra extensions e → B → 0. 0→N →B Remark 20.4.1 This is called André Quillen cohomology. 4 Deriving Ω1B/A instead, one gets LB/A ∈ DB−mod . This is related to DeriA by the derived version of the above equivalence: Exti (LB/A , N ) = DeriA (B, N ). Although the LB/A is more invariant (because it doesn’t depend on the N ), we’ll work with the other thing because it’s easier to think about. 20.5 Construction To define DeriA (B, N ) we replace B by a “free simplicial resolution”, apply DerA (−, N ), and take H ∗ . We’ll now elaborate on what the terms in this statement mean. What is a simplicial A-algebra? It is a simplicial object in the category of A−algebras in the sense discussed in 20.3. This means in every degree n ≥ 0, one has an A-algebra Σn , and a collection of maps modeled on the structure of the set of singular n-simplices of topological spaces: • face maps . . . Σ1 ⇒ Σ0 and • degeneracy maps . . . Σ2 ⇔ Σ1 ← Σ0 Definition 20.5.1 A free simplicial A-algebra is a simplicial A−algebra such that for each n there is a set Sn such that Σn = A[Sn ] and such that the degeneracy maps sends Sn into Sn+1 . Remark 20.5.2 Requiring the compatibility between the generators and the degeneracy maps leads to a good mapping property for free simplicial A-algebras. To explain “resolution,” we’ll give a definition which is obviously not “the right definition” but works in this case. Given a simplicial A−algebra, take the alternating sum of the face maps gives an A-algebra chain complex . . . → Σ3 → Σ2 → Σ1 → Σ0 . To be a resolution of B means that this complex is exact except in degree 0, and that Σ0 → B descends to an isomorphism Σ0 /Im(Σ1 ) → B. Remark 20.5.3 The correct definition would not use additive structure of our category. Quillen does this in more generality. We can now define DeriA (B, N ). Definition 20.5.4 Pick a free simplicial resolution {Σi } for B. Apply DerA (−, N ) and take the alternating sum of the face maps to form the chain complex . . . ← DerA (Σ2 , N ) ← DerA (Σ1 , N ) ← DerA (Σ0 , N ). The cohomology of this chain complex is Der∗A (B, N ). In order to globalize this construction, one has to glue. That ties into the question of how DeriA (B, N ) depends on choices. You can show that any two choices of resolutions differ by homotopy. But it’s better than that, and you actually need more: the choices are a “contractible” space, in some sense any two homotopies differ up to homotopy, etc. Remark 20.5.5 Another way this is done in the literature is to choice a canonical resolution. 5 20.6 Formal properties We now record some formal properties, and mention with the corresponding statements for the cotangent complex. Daniel Litt will work with the cotangent version next time. 1. If A → B is smooth, then DeriA (B, N ) = 0 for i > 0. (On the other side of the picture, H i LB/A = 0 for i > 0.) 2. If B = A/I, then Der0 (B, N ) = 0, and Der1A (B, N ) = Hom(I/I 2 , N ). Moreover, if I is generated by a regular sequence, then Deri = 0 for i > 1. (On the cotangent side, LB/A = I/I 2 [1].) 3. If B, B 0 are A-algebras, then DeriA (B ⊗A B 0 , N ) = DeriA (B, N )⊕DeriA (B 0 , N ) if TorkA (B, B 0 ) = 0 for k > 0. Remark 20.6.1 The equality DeriA (B ⊗A B 0 , N ) = DeriA (B, N ) ⊕ DeriA (B 0 , N ) always holds in the category of simplicial algebras, but one needs the “flatness” condition to relate this to usual algebras. So far, the proofs are fairly trivial. For instance, for (3) one chooses free resolutions of B, B 0 . Tensoring them over A gives a free resolution of B ⊗A B 0 4. If A → B → C, and N is a C-module, then there is a sequence 0 → DerB (C, N ) → DerA (C, N ) → DerA (B, N ) → Der1B (C, N ) → . . . The corresponding statement for the cotangent complex is that there is a distinguished triangle LB/A ⊗ C → LC/A → LC/B → e be a free resolution of B. Proof Sketch. Let B e ?B /B A /C e be a free resolution of C over B: e Then let C e @C e ?B /B A /C e → C. e As B and B e have free simplicial resolutions B, e Now consider the simplicial maps A → B e N ) = DeriA (B, N ). Likewise, DeriA (C, e N ) = DeriA (C, N ). The non-trivial thing one has DeriA (B, e N ) agrees with DeriB (C, N ). This eventually reduces to a statement to check is that DeriBe (C, about derivations. 6 20.7 Examples Let us now do a small calculation with our bare hands. For notational simplicity, our notation assumes at various points that there are finitely many generators - this is not crucial. Let B be a (finitely generated) A-algebra. Pick generators b1 , . . . , bn for B. Then we choose Σ0 = A[x1 , . . . , xn ] → B sending xi 7→ bi . What about Σ1 ? The degeneracy map must send x1 , . . . xn to generators of Σ1 which we will denote by x1 , . . . , xn . Let J be the kernel of Σ0 → B, i.e. B = A[x1 , . . . , xn ]/J. Let f1 , . . . , fs generate J as an ideal of A[x1 , . . . , xn ]. Then we define Σ1 = A[x1 , . . . , xn ; y1 , . . . , ys ]. We have to say what the maps d0 and d1 are. They both send xi to xi , but d1 yi = fi and d0 yi = 0. What about Σ2 ? It must include generators xi (degeneracies of xi ), and also two different degeneracies of the yi . We also need to adjoint additional elements zi such that one face map takes zi to generators of the kernel of Σ1 → Σ0 and all the others kill it. If J = (f1 , . . . , fs ) with the fi being a regular sequence, then we don’t need to add any zi . In this case, the chain complex has an especially simple form. Denoting C = A[x1 , . . . , xn ], it is → . . . C[y10 , . . . , ys0 , y11 , . . . , ys1 ] → → C[y1 , . . . , ys ] ⇒ C. Let ∂i be the alternating sum of the face maps, making this a chain complex. Example 20.7.1 Suppose J = (f ). Then ker ∂0 = {p ∈ C[y1 , . . . , ys ] : p(f ) = p(0)}. Example 20.7.2 Suppose B = A/I where I is generated by a regular sequence f1 , . . . , fs . Then the resolution looks like → A[y10 , . . . , ys0 , y11 , . . . , ys1 ] → → A[y1 , . . . , ys ] ⇒ A. Taking DerA (−, N ), one gets 0 N 3s ← N 2s ← − N s ← 0. To see the left map is zero, simply remember that two of the face maps A[y10 , . . . , ys0 , y11 , . . . , ys1 ] → A[y1 , . . . , ys ] send yij to yi , the other to 0. We can also unwind to see that DeriA (B, N ) = 0 for i > 1. This shows us property 2 in the Formal Property section. References [1] D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, third ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906 (95m:14012) [2] Daniel Quillen, On the (co-) homology of commutative rings, Applications of Categorical Algebra (Proc. Sympos. Pure Math., Vol. XVII, New York, 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 65–87. MR 0257068 (41 #1722) 7

© Copyright 2021