Problem set 8, due November 16.

(1) Prove that (as stated in class) the direct image of O_{𝔸^{1}\{0}} under the map z ↦ z^{n} is the direct sum of D-modules O_{i} = (O, ∇ = d+\frac{idt}{nt}), i = 0, … n−1.

(2) Let j : 𝔸^{1} → ℙ^{1} be the embedding. Let p(x) be any non-constant polynomial and let M(e^{p}) denote the D_{𝔸^{1}}-module generated by the function e^{p}.

(a) Prove that 𝔻(M(e^{p})) ≃ M(e^{−p}).

(b) Show that the natural map j_{!}(M(e^{p})) → j_{∗}(M(e^{p})) is an isomorphism.

(Hint: prove that j_{∗}(M(e^{p})) is irreducible and then use (a)).

(3) Let j : 𝔸^{1} \{0} → 𝔸^{1} be the embedding, and π : 𝔸^{1} → pt be the projection. Show that π_{∗}j_{!}(O) = 0.

(4) (a) Let the additive group G_{a} = 𝔸^{1} act on a smooth variety X (you can assume that X is affine). Show that G_{a} equivariant D-modules form a full subcategory in all D-modules on X. Prove that this subcategory is closed under subquotients and extensions.

(b) Which of the properties listed in the previous problem fail if the additive group G_{a} is replaced by the multiplicative group G_{m}? (Find a counterexample).

(a) Let Ξ = Ξ_{1} be the D-module on 𝔸^{1} defined in the previous problem set. Let T be its intermediate extension to ℙ^{1}. Show that pr_{∗}(T) = 0, where pr : ℙ^{1} → pt.

(b) Consider the action of G_{a} on ℙ^{1} given by t : z ↦ \frac{z}{1+tz}. Show that T is an object of the category of G_{a} equivariant D-modules on ℙ^{1}, which is both injective and projective. (Hint: to check an object is projective, it is enough to see that Ext^{1} from it to any irreducible vanishes. Use previous homework problems to classify irreducibles, and apply part (a) to calculate Ext^{1})

(c) Find another indecomposable injective object in this category, and another indecomposable projective.

(d) Show that the sum of the two projective objects you’ve constructed is a projective generator for the category of G_{a} equivariant D-modules on ℙ^{1}. Use this to construct an equivalence between this category and modules over an explicit 5-dimensional k-algebra.

(5) Let X be an affine smooth variety. Consider the regular bi-module^{1} for D(X), and apply to it the equivalence between right and left D-modules to get a D(X) ⊗ D(X) = D(X × X) module. Describe the corresponding D-module on X × X geometrically.

1Recall that a regular bimodule for an associative ring R is identified with R as an abelian group, and the action of R ⊗ R^{op} is given by r_{1} ⊗ r_{2} : x ↦ r_{1}xr_{2}.