Confusion in Serre's Local fields book

Question

I read that for right exact functors we consider left derived functors and the resolutions that we consider are projective resolutions...

I read that for left exact functors we consider right derived functors and the resolutions that we consider are injective resolutions...

This i read in M.Scott Osborne's Basic homological algebra book in section 6.2 http://www.amazon.com/Basic-Homological-Algebra-Graduate-Mathematics/dp/038798934X

But in Serre's Local fields in chapter seven he said that $A^G$ is left exact which means he should consider injective resolutions for right derived functors but he considered Projective resolutions..

He said in section 2 of chapter 7 that :

By definition, the right derived functors of the functor $A^G$ are cohomology groups of $G$ with coefficients in $A$. Recall briefly how they are computed.

choose a resolution of the $G$- module $\mathbb{Z}$ by Projective $G$ modules

I got confused with this approach..

please let me know what i am missing..

Answer 1

What he is doing is the following.

• First, he knows there is an isomorphism $A^G=\hom_G(\mathbb Z,A)$, so the derived functors of $(-)^G$ are the derived functors of $\hom_G(\mathbb Z,-)$, which are exactly $\operatorname{Ext}_G(\mathbb Z,-)$.

• Second, to compute $\operatorname{Ext}_G(\mathbb Z,A)$ he knows he can either use an injective resolution of $A$ and apply to it the functor $\hom_G(\mathbb Z,-)$, or take a projective resolution of $\mathbb Z$ and apply to it the functor $\hom_G(-,A)$.

Well: he picks the second option.