Problem set 6, due November 2.

(1) Let j : 𝔸^{1} → ℙ^{1} be the natural 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)).

(2) Let j : 𝔾_{m} → 𝔸^{1}. Show that the direct image of j_{!}(O) to pt is 0.

(3) a) Fix λ ∈ k, and consider the D-module T_{λ} on 𝔾_{m} = 𝔸^{1} \{0}, corresponding to the trivial line bundle O_{𝔾_{m}} with connection ∇ = d + λ\frac{dt}{t}. Find out for which λ there exists an isomorphism j_{!}(T_{λ}) ≅ j_{∗}(T_{λ}), where j is the embedding of 𝔾_{m} into 𝔸^{1}.

b) More generally, let P be a polynomial in n variables, and let U ⊂ 𝔸^{n} be the complement to the zeroes of P, let j : U → 𝔸^{n} be the embedding. For λ ∈ k consider the connection ∇ = d + λ\frac{dP}{P} on the trivial line bundle on U, let P_{λ} be the corresponding D-module. Show that j_{!}(P_{λ}) ≅ j_{∗}(P_{λ}) provided that b(λ) ≠ 0, where b(λ) is the Bernstein polynomial, appearing in the proof of analytic continuation of distribution P^{λ}.

(4) Let V be a vector space. Let κ be the function on V ×V^{∗} given by κ((v, ξ)) = ξ(v). Let E_{κ} be the D-module corresponding to the trivial line bundle O_{V ×V^{∗}} with the flat connection ∇ = d + dκ.

a) Check that for a D-module M on V there exists a canonical isomorphism Φ(M) ≅ π_{2•}(π_{1}^{•}(M) ⊗_{O_{V ×V ^{∗}}} E_{κ}), where π_{1} : V × V^{∗} → V, π_{2} : V × V^{∗} → V^{∗} are the projections, and Φ is the Fourier transform.

b) For an isomorphism ϕ : R_{1} → R_{2} of associative unital rings let the graph of ϕ, Γ_{ϕ} be the R_{1} ⊗ R_{2}^{op} module, which coincides with R_{2} as an abelian group, and the action is given by r_{1} ⊗ r_{2}(r) = ϕ(r_{1})rr_{2}.

Notice that Γ_{ϕ} comes with a canonical generator 1_{R_{2}}. Check that the R_{1} ⊗ R_{2}^{op} module Γ_{ϕ} with a fixed generator determines ϕ uniquely.

c) Let F : D(V^{∗}) → D^{op}(V) be the composition of Fourier isomorphism D(V) ≅ D(V^{∗}) and the isomorphism D(V) → D^{op}(V), ∂_{x_{i}} ↦ −∂_{x_{i}}.

Show that E_{κ} ≅ Γ_{F}, so that the canonical generator 1 ∈ O_{V ×V^{∗}} = E_{κ} corresponds to the canonical generator of Γ_{F}.

d) Let now Z ⊂ V ×V^{∗} be the closed subset given by the equation κ = 1, let i : Z → V × V^{∗} be the embedding. Set π′_{1} = π_{1} ∘ i, π′_{2} = π_{2} ∘ i.

The Radon transform R is a functor D−mod(V) → D−mod(V^{∗}) defined by R(M) = π′_{2•}(π′_{1})^{•}(M).

Let e ∈ D(V) be the Euler vector field. Assume that M is the sum of generalized eigenspaces of e, and that none of the eigenvalues is an integer. Construct an isomorphism Φ(M) ≅ R(M).

[One way to do it is to use the next part of the problem].

e) Set R = i_{•}(O_{Z}).

Let e ∈ D(V), e′ ∈ D(V^{∗}) be the Euler vector fields.

Show that D(V)_{(e)} = D(V)⊗_{k[e]}k[e, (e+n)^{−1}], n ∈ ℤ is a ring. Construct an isomorphism of D(V)_{(e)} ⊗ D(V)_{(e′)}-modules:

R ⊗_{D(V ×V^{∗})} D(V)_{(e)} ⊗ D(V)_{(e′)} ≅ E_{κ} ⊗_{D(V ×V^{∗})} D(V)_{(e)} ⊗ D(V)_{(e′)}.