Yoshifumi Tsuchimoto
We recommend the book of Lang [1] as a good reference. The treatment here follows the book for the most part.
and for each there is a natural homomorphism
such that we obtain a long exact sequence
the 's give a commutative diagram:
such that the diagram of resolutions is exact.
be a resolution of by -acyclics. Let
be an injective resolution. Then there exists a morphism of complexes extending the identity on , and this morphism induces an isomorphism
for all .
The book of Grivel [2] is also a good reference for our future arguments.