7
votes
2answers
338 views

Recovering an abelian category out of its derived category

I'm trying to learn more about derived category stuff and my curiosity has made me to ask these questions. Sorry if I'm being sloppy, I'm a new learner. In Wikipedia it has been stated that since ...
1
vote
0answers
250 views

Additive functors preserving quasi-isomorphism

Let $F: \mathcal{A} \rightarrow \mathcal{B}$ be an additive functor between abelian categories (with enough injectives and projectives) and $K^\cdot, L^\cdot$ objects of $\textrm{Ch}(\mathcal{A})$. ...
5
votes
1answer
461 views

Is the bounded derived category of coherent sheaves of a variety a small category?

The question is in the title. I am trying to apply the Mitchell (Freyd-Mitchell?) embedding theorem, which states that for every small abelian category $A$, there exists a ring $R$ such that A ...
0
votes
1answer
604 views

Terminology - subcategories of Abelian categories

Hello, I have terminological question. Consider the following properties of a full subcategory $B \subset A$, where $A$ is an abelian category, and we assume $B$ to be closed under finite direct ...
7
votes
2answers
718 views

The composition of derived functors - commutation fails hazardly?

Hello, When we have left exact functors $F: A \to B , G: B \to C$ (between abelian categories), we would like sometimes to state that $D(GF)=D(G)D(F)$ (functors between bounded below derived ...