Math 212 Homework 3 Due March 2nd, 2009
1. Let Abe an abelian category with enough injectives and enough projectives. Show that
ExtAis balanced; that is, given objects Aand B, an injective resolution B→I•and a
projective resolution P•→A, show that
H∗HomA(P•, B)∼
=H∗HomA(A, I•).
2. Let C•• be a first quadrant double cochain complex (in an abelian category) such that all
rows are acyclic complexes. Prove that the total complex Tot(C••) is acyclic.
3. Let H⊆Gbe a normal subgroup. Show that M7→ MHdefines an additive functor from
G-modules to G/H-modules that has an exact left adjoint.
4. Let C•• be a first quadrant double cochain complex of vector spaces over a field k, and
let (En, dn) be one of the associated spectral sequences. Suppose that for some r, the vector
space Lp,q Ep,q
ris finite dimensional. Prove that the vector space LnHn(Tot(C••)) is finite
dimensional as well and that
X
p,q
(−1)p+qdimkEp,q
r=X
n
(−1)ndimkHn(Tot(C••)).
5. Let Ep,q
2=⇒Hp+qbe a (bounded convergent) first quadrant cohomological spectral
sequence. Show that there is an exact sequence
0→E1,0
2→H1→E0,1
2→E2,0
2→H2.