Problem set 6, due by the end of semester.

(1) Prove that any projective object in the category of k-vector spaces on ℝ^{n} is zero.

(2) Let X be the union of two copies of ℝ^{2} intersecting at a point. Compute cohomology of the stalk of the dualizing complex 𝕍_{X} (with coefficients in a field k).

(3) Let G be a finite group acting on a finite dimensional vector space V over a field k; assume that characteristic of k is prime to |G|. Prove that the ring R = Sym(V)^{G} is Cohen-Macualay.

Show that it is also Gorenstein provided that every element of G acts in V by a transformation of determinant 1.

Give an example of a finite G acting on V such that R = Sym(V)^{G} is not Gorenstein (prove that it is not).

(4) Let X = S^{2} be the two-dimensional sphere; fix a point x ∈ X. D be the full subcategory in D^{b}(Sh_{X}) whose objects are complexes whose cohomology sheaves have finite dimensional stalks, and are (locally) constant on X \{x}. Describe the Grothendieck group K(D).

(5) Classify indecomposable injective modules over R = k[x] (where k is a

field).

[You should obtain one module for each maximal prime ideal].

Write down a (right) resolution for the free R module.

Use this resolution to redo the computation of Ext_{R}^{·}(M, R) for a finite dimensional R-module M.

(6) Let R be a ring which is finite dimensional over a field k. Assume R is nilpotent, i.e. it has a codimension 1 ideal consisting of nilpotent elements. Show that R is Gorenstein iff any two non-zero ideals in R have a non-zero intersection.