4
votes
1answer
252 views

Fourier-Mukai transform for abelian varieties

Let $A$ be an abelian variety over $\mathbb{C}$, $L$ be a very ample line bundle on $A$, then the dual abelian variety is $\hat{A} \cong A/K(L)$ with $K(L)$ the kernel of surjective morphism $A \to ...
-2
votes
1answer
283 views

Derived Category.

Question 1: Let $X$ be a scheme. Then generally for the complex $C^{\bullet}$ in $D^b(X)$, we define $R\Gamma(C^{\bullet})\colon$ = Complex obtained by applying $\Gamma$ to the injective resolution ...
1
vote
1answer
195 views

Balanced dualizing complexes according to A. Yekutieli

I am reading A. Yekutieli's original article on dualizing complexes for noncommutative algebras and I found a problem I cannot solve. First, some background. We start with a field $k$ and a ...
1
vote
1answer
71 views

is the orthogonal complement of a saturated sequence saturated?

Suppose I have a smooth projective variety $X$, and a semi-orthogonal decomposition of its bounded derived category: $$D^b(X)= < A, E_1, E_2, ... , E_n >$$ where the $E_i$ are fully faithful, ...
3
votes
1answer
176 views

what are mutations of sheaves all about?

Suppose I have a smooth projective variety $X$ and a semi-orthogonal decomposition of its bounded derived category of coherent sheaves $D^b(X)$. Then I can apply right or left mutations to the full ...
1
vote
2answers
426 views

Theorem on composition of derived functors, question about proof

I got a question about a proof I found in Gelfand-Manin's "Methods of homological algebra" (Page 200): Theorem 1. Let $\mathcal{A}, \mathcal{B}, \mathcal{C}$ be three abelian categories, $F: ...
3
votes
1answer
339 views

Unbounded complexes, resolutions and computation of derived functors

Hey guys, let $F: \mathcal{A} \rightarrow \mathcal{B}$ be a left exact functor between abelian categories with enough injectives, let $K \in Kom(\mathcal{A})$ be an unbounded complex, I've heard that ...
2
votes
2answers
272 views

Obtaining derived functors from derived functors of similar complexes or “bluntly truncated” unbounded complexes (without adding 0's to the left)

I don't know if I'm actually using the right terminology here, to be clear I'm going to state explicitly what I'm trying to figure out to see if I can be pointed in the right direction: Let $F: ...
5
votes
0answers
456 views

functor before cat?

As i read the literature, derived functors were there several years before derive categories - right?
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 ...
1
vote
1answer
544 views

Adjunctions between derived functors

Given an adjunction $F\dashv G$ between functors between Abelian categories, we know that $F$ is right exact and $G$ is left exact so there are derived functors $LF$ and $RG$ between (bounded above, ...
3
votes
1answer
477 views

Perfect complexes and RGamma(X,F) without mentioning derived categories

Let $A$ be a commutative noetherian ring. Let $K_{parf}(A)$ be the full subcategory of the homotopy category $K(A)$ of $A$-modules whose objects are bounded complexes of finitely generated ...
12
votes
1answer
861 views

Why does the naive definition of compactly supported étale cohomology give the wrong answer?

Illusie's article about ├ętale cohomology available here (in French) mentions that the standard definition of compactly supported cohomology (and higher direct images with compact support) does not ...