Math 212 Homework 2 Due February 9th, 2009
1. Compute TorZ/4
∗(Z/2,Z/2) and Ext∗
Z/4(Z/2,Z/2).
2. Suppose Ais an abelian category such that all objects are projective. Show that in this
case all objects are also injective.
3. Suppose that Ais an abelian category with enough injectives such that given any epi-
morphism I→A→0 in Awith Iinjective, Ais also injective (for example, Acould be
the category of modules over a PID). Let C•a bounded cochain complex in A. Show that
there is a homomorphism of complexes C•→H∗(C•) that induces an isomorphism on co-
homology (here the cohomology is viewed as a complex with zero differentials). Note: this
quasi-isomorphism is not natural.
4. Let Abe an abelian category with enough injectives. Assume that for all objects Aand
B, Ext1
A(A, B) = 0. Show that Ahas enough projectives.
5. Let F:A→Bbe an additive functor between abelian categories. Suppose Fhas an
exact left adjoint G:B → A. Show that for any injective I∈ A,F(I) is injective in B.
6. Let Gbe a finite group and ModGthe abelian category of (right) ZG-modules. Let
H⊆Gbe a normal subgroup. Show that the restriction functor RG
H:ModG→ModHhas
an exact left adjoint. (Try the case H={e}first.)