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The Cotangent Complex) INTEGRAL p-ADIC HODGE THEORY, TALK 5 (THE COTANGENT COMPLEX) KEVIN BUZZARD (NOTES BY JAMES NEWTON) 1. Differentials Suppose f : X ! Y is a map of varieties over a field k. f X Y k Then we have an exact sequence (see e.g. Hartshorne) ∗ 1 1 1 f ΩY=k ! ΩX=k ! ΩX=Y ! 0 but the first map is not in general injective (e.g. if Y has bigger dimension than X). Remark 1.1. If f is smooth, then the first map in the above exact sequence is injective, i.e. we have a short exact sequence ∗ 1 1 1 0 ! f ΩY=k ! ΩX=k ! ΩX=Y ! 0: Let’s consider the ‘opposite’ of the case where f is smooth: suppose f is a closed immersion, corresponding to an ideal sheaf I ⊂ OY . Then we have an exact sequence: 2 ∗ 1 1 I=I ! f ΩY=k ! ΩX=k ! 0: 2 Note that I=I is supported on X so we can think of it as an OX -module. Also, 1 we have ΩX=Y = 0 here. Remark 1.2. Again the first map in the above exact sequence is not in general injective. But if X=k is smooth, we do have a short exact sequence 2 ∗ 1 1 0 !I=I ! f ΩY=k ! ΩX=k ! 0: By considering these two exact sequences, we might guess that we are seeing a shadow of a cohomological long exact sequence for "derived functors of differen- 0 1 1 1 2 tials". We have H = ΩX=Y , H = 0 when f : X ! Y is smooth and H = I=I when f is a closed immersion with ideal sheaf I. 2. Brief chat about derived categories Suppose X is a noetherian scheme and F is a quasi-coherent sheaf of OX - modules. We are going to discuss two ways of computing/defining the cohomology groups Hi(X; F). We have H0(X; F) = F(X) = Γ(X; F). 1 2 2.1. 1st definition: write down an injective resolution, i.e. an exact sequence 0 !F!I0 !I1 !··· with each In injective (we can work in either the category of OX -modules or in the category of quasi-coherent sheaves). When we apply the left exact functor Γ(X; −) we get a complex 0 ! Γ(X; F) ! Γ(X; I0) ! Γ(X; I1) !··· 0 By left exactness, we know that H (X; F) = Γ(X; F) = ker (Γ(X; I0) ! Γ(X; I1)). In general, we define the sheaf cohomology groups i i H (X; F) = H ([Γ(X; I0) ! Γ(X; I1) !··· ]) to be the cohomology groups of the complex Γ(X; I0) ! Γ(X; I1) !··· Standard results show that the cohomology groups Hi(X; F) do not depend on the choice of injective resolution. But we can actually come up with a canonical object earlier in the process of defining sheaf cohomology. The complex of OX (X)- modules Γ(X; I0) ! Γ(X; I1) !··· certainly does depend on this choice, but it is a canonical object (call it RΓ(X; F)) of the derived category of OX (X)-modules (we start with complexes of OX (X)-modules and formally invert morphisms which induce isomorphisms on cohomology). 2.2. 2nd definition. Now we assume that our scheme X is separated, so any (finite) intersection of affine open subschemes is again affine. Fix a cover by affines Ui. Then we can consider the diagram which begins Q δ0 Q Q i F(Ui) i;j F(Ui \ Uj) i;j;k F(Ui \ Uj \ Uk) δ1 where the image of δ0 on (fi : fi 2 F(Ui)) is given by fi on each Ui \ Uj and the image of δ1 is given by fj on each Ui \ Uj. There are three (similarly defined) maps at the next step. There are also maps in the other direction: for example, we have Q one map from the second term to the first given by taking (fi;j) 2 F(Ui \ Uj) to Q (fi;i) 2 F(Ui). We have defined a cosimplicial OX (X)-module (see below). If F = OX then in fact this is a cosimplicial OX (X)-algebra. There is an associated (cochain) complex of OX (X)-modules, given by replacing the arrows δ0; δ1; : : : ; δn from degree n−1 to Pn n degree n with a single OX (X)-linear differential dn = i=0(−1) δi. This cochain complex is (a version of1) the usual Čech complex, and is isomorphic to RΓ(X; F) in the derived category. Suppose F = OX : the maps δi are ring homomorphisms, whilst the differentials dn in the Čech complex are only linear maps, so we have lost some structure by passing to the associated complex. Recall we want to define a ‘derived functor of differentials’. We need to keep track of ring structures to define differentials, so we will need to work with simplicial things, rather than just complexes of modules. 1you may be more familiar with the version where the open covering is indexed by an ordered set and intersections are indexed by i0 < i1 < ··· < in, see http://math.stanford.edu/~conrad/ papers/cech.pdf for the comparison between the two versions 3 3. Simplicial objects Let C be a category. Definition 3.1. Let ∆ be the category with objects [n] = f0; 1; : : : ; ng for n in Z≥0. The morphisms f :[n] ! [m] are given by order-preserving maps of sets, so they are determined by specifying 0 ≤ f(0) ≤ · · · ≤ f(n) ≤ m. n Important examples of morphisms are the face maps δj :[n − 1] ! [n] given by n skipping j (for 0 ≤ j ≤ n) and the degeneracy maps σj :[n] ! [n − 1] given by squishing together j and j + 1 (for 0 ≤ j ≤ n − 1). These maps generate all the morphisms in ∆. Definition 3.2. A simplicial object in C is a contravariant functor ∆ !C. (A cosimplicial object is a covariant functor...). Example 3.2.1. Let X be a simplicial complex, and order the vertices of X. Then we can define a simplicial set K: by Kn := K([n]) = f(v0; : : : ; vn): the vi are vertices of a simplex of X with v0 ≤ · · · ≤ vng Note that the vertices don’t have to be distinct, so we are allowing degenerate simplices. n Then the face map K(δj ) is given by taking (v0; : : : ; vn) to (v0;:::; v^i; : : : ; vn) n and the degeneracy map K(σj ) takes (v0; : : : ; vn−1) to (v0; : : : ; vj; vj; : : : ; vn−1). 4. Cotangent complex [JN: recall that we want to define derived functors of the differentials. Here is a more precise set-up: suppose A is a fixed (commutative, unital) ring and R is a commutative A-algebra. We are going to consider a category A − Alg=R consisting of commutative A-algebras B equipped with an A-algebra map B ! R. Now we have a functor A − Alg=R ! R − Mod 1 B 7! ΩB=A ⊗B R 1 Evaluating this functor on R gives us ΩR=A. Now we want to derive this functor and apply it to R to define the cotangent complex. As usual with derived functors, we can compute what we get when we evaluate this derived functor on R by taking a nice (simplicial) resolution of R (as an A-algebra) and applying our original functor (extended to simplicial A-algebras with a map to R) to this resolution. This is explained below, and fits into a general framework developed by Quillen to generalise derived functors beyond abelian categories] Let A be a fixed base (commutative, unital) ring. There is a forgetful functor from commutative A-algebras to sets, and this has an adjoint functor which takes a set S to the polynomial algebra with generators labelled by the elements of S, denoted A[S]. Let B be an A-algebra. We can resolve B by a simplicial A-algebra which is a polynomial algebra at each level: δ0 P • : ··· A[A[A[B]]] A[A[B]] A[B] δ1 4 The maps δ0; δ1 are determined by maps of sets A[B] ! A[B]. The first of these maps is the identity, the second is the map which takes each element f 2 A[B] to the generator [π(f)], where π : A[B] ! B is the A-algebra map induced by the identity on B. The simplicial ring P • comes with a natural map P • ! B (we can think of B as a simplicial A-algebra given by B in every degree and all maps the identity)2. Like we did for the Čech complex, we have an associated chain complex of A-modules ch(P •) and the natural map ch(P •) ! B is a (free) resolution of A-modules. 1 1 We can define a simplicial A-module ΩP •=A which is ΩP n=A at level n. In fact, 1 • ΩP •=A is a P -module. 1 Finally, to define the cotangent complex we take the tensor product ΩP •=A ⊗P • B 1 1 n which is given by ΩP n=A ⊗P n B at level n (note that ΩP n=A is a free P -module for all n, so we don’t need to worrying about taking a derived tensor product here), and define: Definition 4.1. The cotangent complex LB=A is the chain complex of B-modules 1 associated to the simplicial B-module ΩP •=A ⊗P • B. We have really defined LB=A as a complex, but it is more flexible to think of LB=A as living in the derived category of B-modules.
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