Problem set 2, due September 28

(1) In this problem we work over a field of characteristic zero, and D(X) stands

for the ring of Grothendieck differential operators on X.

a) Let X be given by equations xy = 0 in 𝔸^{2}. What is D(X)? What is grD(X)? Is it noetherian?

b)^{∗} Do the same for the quadratic cone xy = z^{2}.

c)^{∗∗} Let X be the cubic cone x^{3} + y^{3} + z^{3} = 0. Show that D(X) is not noetherian.

(2) Let R be a commutive algebra, D = D(R). For two R-modules M and N we denote by Diff(M, N) the space of differential operators from M to N. Construct a canonical isomorphism

Hom_{D}(D ⊗_{R} M, D ⊗_{R} N) ≃ Diff(M, N).

(3) Let X be a smooth affine variety, O = O(X), D = D(X). Let also M be an O-coherent D-module. Show that if M has rank 1 over O then M is irreducible. Show that the converse is not true – construct irreducible O-coherent modules of arbitrary rank (for some X; in fact, it is possible to do it for any affine X).

(4) Let θ be a closed 1-form on a smooth variety X over a field k of characteristic p. Consider the connection ∇ = d + θ on the trivial line bundle O_{X}. Let Z ⊂ T^{∗}X be the support of this connection as a module over the p-center O(T^{∗}X).

a) Show that Z is a graph of some 1-form ω.

b) Compute ω if θ = df for some function f.

c) Compute ω if θ = \frac{df}{f} for some invertible function f.

Show that ω = θ − C(θ), where C is the map from closed 1-forms to all 1-forms, inducing the Cartier isomorphism

d) If X = 𝔸^{1}.

c)^{∗∗} X is arbitrary.