Problem set 4, due Thursday, October 20.

(1) Complete assignment 3 (see updated file for new hints, or discuss it with me if it’s still unclear what to do).

(2) Write down explicitly a Cartan-Eilenberg resolution for the two-side infinite complex of ℤ_{8} modules

· →^{4} ℤ_{8} →^{4} ℤ_{8} →^{4} · · · · · ·

(3) Let A = A − mod^{fg} where A is a Noetherian algebra, and let L ∈ A be irreducible; let B = L^{⊥} = {M ∈ A | Hom(L, M) = 0}.

(a) Show that B is pre-abelian, and describe explicitly Ker and Coker.

(b) Let P ∈ A be projective. Give a criterion for the functor Hom(P, ) to induce an exact (sending kernel to kernel and cokernel to cokernel) functor on B.

(c) Let A = k[G] where k = ℤ_{3}, G is cyclic of order 6, L = k is the trivial representation. Check B is abelian; classify (up to an isomorphism) irreducible objects in B, indecomposable projective objects. Find a ring R and an equivalence B ≅ R − mod^{fg}.

(4) Let A be a flat algebra over Z = k[x_{1}, …, x_{n}] (where k is a field), finitely generated as a module over Z. Let A be the category of A-modules which are finite dimensional over k. Check that A is an abelian category, and that A has no projective objects.

Verify that for M, N ∈ A we have Ext_{A}^{i}(M, N) = Ext_{A−mod}^{i}(M, N). [Hint: show that every finite complex of finitely generated A modules

whose cohomology is finite dimensional has a finite dimensional quasiisomorphic quotient complex (this can be shown e.g. by induction in the length of a complex). Deduce that Ext_{A−mod}^{·} is effaceable in A.]