3
votes
2answers
91 views

Derived functors definition

I´m searching for a reference that defines $n^{th}$derived functors in an analogous way to the definition given in Mitchell´s "Theory of Categories" for the $0^{th}$ derived functor of $T$ covariant ...
0
votes
0answers
70 views

What is the cohomological dimension of a functor?

Let $F:\mathcal{C}\rightarrow \mathcal{D}$ be a functor between abelian categories. Could anyone explain what the cohomological dimension the functor $F$ is? We may need some additional condition to ...
11
votes
0answers
289 views

Why do universal $\delta$-functors annihilate injectives?

Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Suppose $\mathcal{A}$ has enough injectives, and consider a universal (cohomological) $\delta$-functor $T^\bullet$ from $\mathcal{A}$ to ...
9
votes
2answers
463 views

Derived functors of torsion functor

Let $A$ be a domain. For every $A$-module $M$ consider its torsion submodule $M^{tor}$ made up of elements of $M$ which are annihilated by a non zero-element of $A$. If $f \colon M \to N$ is a ...
0
votes
1answer
240 views

derived functors and acyclics

I'm not sure how I can show the following: If F is a left exact functor from an abelian category A to an abelian category B, whose derived functor RF in the sense of derived categories exists, then ...
15
votes
2answers
687 views

Meaning of “efface” in “effaceable functor” and “injective effacement”

I'm reading Grothendieck's Tōhoku paper, and I was curious about the reasoning behind the terms "effaceable functor" and "injective effacement". I know that in English, to efface something means ...