All Questions
12 questions
42
votes
4
answers
8k
views
Serre intersection formula and derived algebraic geometry?
Let $X$ be a regular scheme (all local rings are regular). Let $Y,Z$ be two closed subschemes defined by ideals sheaves $\mathcal I,\mathcal J$. Serre gave a beautiful formula to count the ...
- 30.5k
17
votes
2
answers
1k
views
Grothendieck spectral sequence when one of the functors is contravariant
Let $f \colon X \rightarrow S$ be a morphism of schemes. I am interested in computing the cohomology groups of
$$
\mathbf{R}\mathscr{H}om(\mathbf{R}f_* \mathcal{O}_X, \mathcal{O}_S)
$$
in terms of $\...
- 2,663
6
votes
1
answer
929
views
Different definitions of derived functors
In principle one uses the notion of derived category, and the other doesn't.
Suppose $F: \mathcal A \to \mathcal B$ is a left exact (additive) functor between abelian categories, and suppose the ...
- 2,789
6
votes
0
answers
656
views
The derived version of the Grothendieck spectral sequence
Consider the (very well known) Grothendieck spectral sequence for composition of functors $\mathcal F: \mathcal A \to \mathcal B$ and $\mathcal G: \mathcal B \to \mathcal C$ between abelian categories ...
- 61
5
votes
0
answers
396
views
Description of connecting maps of Derived functors
Let $C$ be an abelian category with enough injectives and $F$ be a left exact additive functor. Consider the short exact sequence $0 \to A' \to A \to A ''\to 0$. Therefore, we have connecting maps $\...
- 493
4
votes
2
answers
811
views
Motivation/intuition behind the definition of delta-functors and related concepts
I originally posted this on Maths SE, but then realised that the question probably fits MO better, as my objective was to gain different perspectives regarding the matter.
Why are $\delta$-functors ...
- 1,516
4
votes
2
answers
809
views
Two basic questions on derived categories
Let $\mathcal{A}, \mathcal{B}$ be two abelian categories with sufficiently many injective objects (in my case these are categories of sheaves of vector spaces on a manifold).
Let $f_*\colon \mathcal{A}...
- 21.8k
2
votes
1
answer
239
views
How to complete $i^*i_*\mathcal{F}\to \mathcal{F}$ into an exact triangle for a smooth divisor $i: X\hookrightarrow Y$?
Let $Y$ be a smooth algebraic variety and $i: X\hookrightarrow Y$ be a smooth divisor. We consider the derived functors $i^*: D^b_{coh}(Y)\to D^b_{coh}(X)$ and $i_*: D^b_{coh}(X)\to D^b_{coh}(Y)$. By ...
- 9,019
2
votes
1
answer
342
views
A question on some lemmas in Orlov's "Triangulated Categories of Singularities and D-Branes in Landau-Ginzburg Models" (Exts vanishing)
I'll write the two lemmas I have questions about, and then ask my questions. For reference, I'm using the following definition of Gorenstein:
$\mathbf{Definition\ 1.15}$ A local noetherian ring $A$ ...
- 841
2
votes
0
answers
244
views
Do we have a left adjoint of $i^*$ for a closed immersion $i$?
Let $i: X\hookrightarrow Y$ be a closed immersion of varieties. We have the derived pullback functor $i^*: D^b_{coh}(Y)\to D^b_{coh}(X)$.
My questions is: can we construct a left adjoint of $i^*$ in ...
- 9,019
1
vote
2
answers
431
views
Action on group $\operatorname{Ext}^i(\mathcal{L}, \mathcal{M})$ by scalar multiplication
Let $X$ be a proper scheme over field $k$ and $\mathcal{L}, \mathcal{M}$ two invertible $\mathcal{O}_X$-modules. Then $Hom_{\mathcal{O}_X}(\mathcal{L}, \mathcal{M}) \cong Hom_{\mathcal{O}_X}(\mathcal{...
- 6,018
0
votes
1
answer
519
views
A functorial isomorphism in derived category
This question is a direct continuation of Question 1 in this post: Two basic questions on derived categories
Let $f\colon \mathcal{A}\to\mathcal{B}$ be a left exact functor between two abelian ...
- 21.8k