Problem set 4, due October 12

(1) In this problem set all schemes are assumed to be affine.

Let X be an algebraic variety, Δ : X → X × X the diagonal embedding. Let J denote the ideal of Δ(X) in O_{X×X} and let X^{(n)} denote the closed subscheme of X × X corresponding to the ideal J^{n}. Thus O_{X^{(n)}} has two structures of an O_{X}-module.

Show that (D_{X})_{≤n} = Hom_{O_{X}} (O_{X^{(n)}}, O_{X}) where the action of O_{X} on O_{X} is via any of the above two actions. Note that we don’t need to X to be smooth.

(2) In this problem we are going to give an equivalent description of the category M(D_{X}) via crystals (this notion was invented by Grothendieck for characteristic p applications; the notion of crystals that we are going to use is a modification of Grothendieck definition). The advantage of this approach is that in this case we will not need to treat the case of singular varieties separately.

In the sequel we denote by M(D_{X}) the category of D-modules on X (as defined in class); for singular X this is not the same as the category of D_{X}-modules.

Let η : A → B be a homomorphism of commutative algebras such that B is finite over A. Recall that in this case we have the functor η^{!} : A−mod → B − mod defined as

η^{!}(M) = Hom_{A}(B, M).

Let X be a scheme over our base field k (the definitions below makes sense (and are interesting) when k has arbitrary characteristic but we shall consider only the case when it has characteritic 0). Recall that a nilpotent extension of X is a closed embedding i : X → Y where Y is another scheme and the ideal of X in Y is nilpotent. If Y and Z are two nilpotent extensions of X we say that η : Y → Z is a morphism of extensions if it is a morphism of schemes which is equal to identity on X.

A !-crystal on X is a collection of the following data:

(a) An O_{Y} -module M_{Y} for every nilpotent extension Y of X.

(b) An isomorphism α_{η} : M_{Y} ≃ η^{!}M_{Z} for every finite map of extensions

η : Y → Z. This data should satisfy the following compatibility condition: for every

chain Y →^{η} Z →^{ρ} W of finite morphisms of nilpotent extensions of X we have α_{η∘ρ} = α_{η} ∘ α_{ρ}.

We denote by Crys(X) the category of !-crystals on X.

Let X be a (not necessarily smooth) algebraic variety. M ∈ M(D_{X}) (defined via right modules). Let also X → Y be a nilpotent extension of X. We may imbed Y into some smooth variety Z. In this case M gives rise to a D_{Z}-module M_{Z} on Z supported on X. Define M_{Y} to be the set of all elements of M_{Z} which are scheme-theoretically supported on Y.

(a) Show that the collection {M_{Y} } has a natural structure of a !-crystal. Thus we obtain a functor F : M(D_{X}) → Crys(X). We claim that this functor is an equivalence of categories. The rest of the problem is devoted to the proof of this statement.

(b) Let π_{1}^{n}, π_{2}^{n} : X^{(n)} → X be the natural projections. Let M be an O_{X}-module. Use problem 1 to show that to give a structure of a right D_{X}-module on M is the same as to specify a collection of isomorphisms β_{n} : (π_{1}^{n})^{!}M ≃ (π_{2}^{n})^{!}M which satisfy a compatibility condition (write down the compatibility condition explicitly) and such that β_{1} = id. Note that if M = M_{X} for some crystal as above then we may construct the isomorphisms β_{n} in the following way: from the definition of !- crystals we should have isomorphisms α_{π_{i}^{n}} : (π_{i}^{n})^{!}M → M_{X^{(n)}} for i = 1, 2. Thus we define β_{n} = α_{π_{2}^{n}}^{−1}∘α_{π_{1}^{n}}. Hence we get a functor G from Crys(X) to the category of right D_{X}-modules. Note that composing F and G we see that the algebra D_{X} acts on every M ∈ M(D_{X}).

(c) Assume that X is smooth. Show that the functors F and G between M(D_{X}) and Crys(X) are mutually inverse. Thus in this case M(D_{X}) ≃ Crys(X).

(d) Deduce from c) that F is an equivalence of categories for arbitrary X.