4
votes
0answers
48 views

The functor $\underline{\mathbf{R}}^if_*$

Let $f: X \to Y$ be a proper morhpism of varieties, and $\mathcal{F}$ be a sheaf on $X$. Then we have $f_* \mathcal{F}$ as a sheaf on Y and we also have a higher derived functor $\mathbf{R}^i ...
1
vote
0answers
73 views

Derived push-forward of projective sheaf

Let S,X be schemes and $s \in S$ be a closed point. Let $D(X)$ be the derived category of complexes of sheaves. Let $$i_s: X \cong {s} \times X \hookrightarrow S \times X$$ be the natural embedding. ...
15
votes
1answer
420 views

Homological methods in algebraic geometry

This question will probably seem quite silly to those well-versed in algebraic geometry (about which I admittedly hardly know anything); in the preface of Atiyah-Macdonald's book on commutative ...
3
votes
2answers
79 views

Reference request: localisation of categories

I'm trying to track down a result mentioned in Verdier's Categories Derivees apearing in SGA 4.5. In chapter 2, 3.1 (page 280 in SGA 4.5) Verdier mentions that categorical localisations always exist, ...
11
votes
1answer
220 views

What is a concrete example of why one wants to have a *derived category* in algebraic geometry?

My question asks for a concrete (and hopefully easy) example, why one wants to derive things in algebraic geometry. I heard, that a resolution of an object by free ones behaves much better than the ...
5
votes
3answers
198 views

What does it mean to have exact derived functors?

Let $F:\mathcal A\to \mathcal B$ be a functor between abelian categories. Suppose $F$ is, say, left exact (plus additive and covariant). We have built its right derived functors $R^iF$. I see no ...
9
votes
1answer
169 views

Ext between two coherent sheaves

Let $X$ be a smooth projective variety over a field $k = \overline k$. From Hartshorne we know, that $\textrm{dim} \, H^i (X,F)<\infty$ for any coherent sheaf $F$. How to show, that all $Ext^i ...
6
votes
1answer
190 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
1answer
136 views

Sheaves on $\mathbb{P}^n \times \mathbb{P}^m$, and a commutation relation for derived functors of global sections and tensor products on it.

I'll state my questions first and then provide some background. Question 3 is by far my most important one. We work over $k=\mathbb{C}$ whenever necessary. Is it true that $\text{Pic}(\mathbb{P}^n ...
9
votes
2answers
152 views

What does $Tor_{R}^n(M,N)$ represent?

Let $R$ be a commutative ring and $M$ and $N$ be $R$-modules (I am not sure if one really needs commutativity in the following). It is well-known that $Ext_{R}^n(M,N)$ for $n>1$ parametrizes ...
2
votes
0answers
130 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 $X$. ...
0
votes
1answer
89 views

Right derived functor of diagonal morphism equals direct image on line bundles?

Let $X$ be a smooth projective variety. The map $i:X\to X\times_k X$ induced by the identity is a closed immersion. Denote its image by $\bigtriangleup$. We have ...
4
votes
5answers
314 views

Derived category and so on

I am looking for an introductive reference to the theory of derived categories. Especially I need to start from the very beginning and I need to know how to use this in examples which comes from ...