Math 750 -- Homological algebra -- Homeworks
Homework 1, due Tuesday, February 27:
a) Give an example of a bounded chain complex in Ch(Z-mod) which is acyclic but not split exact. (So both conditions of the Theorem we proved in class are needed.)
b) Prove that any acyclic complex of vector spaces over a field is split exact. (Do not assume that the complex is bounded!)
c) Prove that the full subcategory F of Ch^-(R-Mod) consisting of bounded above complexes of free R-modules satisfies the properties below:
-- the cone of a morphism between two objects in F is again in F;
-- if G : R-Mod -> A is an additive functor to an abelian category A, then G takes quasi-isomorphisms in F to quasi-isomorphisms in Ch(A);
-- if M is an R-module, a resolution of M is an object C of F together with a map C -> M which is a quasi-isomorphism. Prove that maps M -> N of R-modules lift to maps of resolutions of the modules, uniquely up to homotopy.
d) Let f: M' -> M be an injective map of R-modules, and let M" denote the cokernel of f. Show that cone(f) is quasi-isomorphic to M". Are they always isomorphic in HoCh(R-mod)?