Tagged Questions
2
votes
2answers
78 views
Equivalence of categories and derived functors.
Don't know if this kind of a dumb question but let $A$ and $B$ be abelian categories and suppose they're equivalent: there are two functors $P: A \rightarrow B$ and $Q: B \rightarrow A$ satisfying the ...
6
votes
1answer
110 views
How to compute Hom in derived category?
Let $X$ be a smooth variety, $D^{b}(X)$ be the derived category of bounded coherent sheaves.Then there is a definition of $Hom(F^{\cdot},G^{\cdot})$ which is the derived functor of $Hom(F^{\cdot},-)$. ...
5
votes
0answers
58 views
Example where Čech and derived functor cohomologies don't agree.
I'm studying sheaf cohomology, and I've seen that Čech and derived functor cohomologies agree, at least on paracompact Hausdorff topological spaces.
Is there a simple example of a topological space ...
4
votes
0answers
73 views
Cartan-Eilenberg resolutions, adapted classes and acyclic resolutions
I may get grilled for this but here I go: Let $\mathcal{A}$ be an abelian category with enough injectives. What I want to know is VERY VERY specific. Let's say I have a complex in $\mathcal{A}$
$0 ...
2
votes
0answers
71 views
Computing Derived Pullback on the Complement
Let $X$ be a scheme and $\iota: Z\hookrightarrow X$ the embedding of a closed subscheme $Z$; let $j: U\hookrightarrow X$ be the open complement. Suppose $\mathcal{F}$ is a coherent sheaf on $X$.
...
4
votes
0answers
142 views
Composition of derived functors and comparison between hypercohomology and sheaf cohomology
I had a few questions about compositions of derived functors, the comparison between hypercohomology, and sheaf cohomology and the following theorem from the Gelfand, Manin homological algebra book:
...
9
votes
0answers
209 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 ...
11
votes
2answers
485 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 ...
