I have some questions regarding the definition of these minimal resolutions. Precisely, I want to define them with the least conditions and I have some problems with definitions in some books.
1) In Eisenbud's Commutative Algebra, he defines the minimal free resolution over a LOCAL RING as a free resolution satisfying $\partial(F_i)\subseteq mF_{i-1}$ and he uses NAK to prove the the lemma in the picture. I believe the proof is wrong since NAK only applies to finitely generated module.
2) Intuitively, like Eisenbud mentioned in the introduction of Chapter 19, $F_i$ is constructed as a free module whose rank is equal to the minimal number of generators of $Im(\partial_i)$. Does it mean we count the infinite case (i.e. $F_0$ could be $\oplus_I R$)?
3) No book I found states the definition of a minimal injective resolution. I assume $E_i$ is the injective envelop of $\partial_{i-1}$. However, the definition would require nothing about $R$ although in many articles and books, they always treat minimal injective resolution over a Noetherian local ring. Do I miss any details?