All Questions
Tagged with derived-categories cohomology
19
questions
4
votes
0
answers
196
views
Cohomological methods in intersection theory and derived categories
Are there any enumerative questions akin to: “What is the number of planes containing a given line tangent to a given cubic surface in $\mathbb{P}^3$” that we can answer using derived categories? I've ...
- 373
0
votes
0
answers
87
views
$\operatorname{Ext}$-group in the category of modules versus in the subcategory of finitely generated ones
I am trying to refine my understanding of derived categories.
Let $\text{Mod}_R$ and $\text{Mod}^f_R$ be respectively the categories of modules and finitely generated modules over a Notherian ring $R$ ...
- 1,479
5
votes
0
answers
316
views
Appropriate notion of derived category over condensed set
If we have a compact Hausdorff space $S$, then my understanding is that the appropriate notion of the derived category of sheaves of condensed abelian groups is to consider the derived category $D_{\...
- 163
2
votes
0
answers
206
views
Hochschild cohomology of a sheaf of associative algebras
Assume that $X$ is a complex manifold. Let $\delta: X\to X\times X$ be the diagonal map. Assume that $\mathcal{A}_X$ is a $\mathbb C_X$-algebra and $\mathcal{M}_X$ is a left $\mathcal{A}_X\otimes_{\...
- 599
4
votes
1
answer
438
views
Perverse sheaves on the complex affine line
Show that a perverse sheaf on $\mathbb{A}^1(\mathbb{C})$ (the complex plane with the analytic topology) is a bounded complex $A$ of sheaves of $\mathbb{Q}$-vector spaces with constructible cohomology ...
- 1,479
5
votes
0
answers
156
views
Extension groups in quotient categories
Let $\mathcal{A}$ be an abelian category and let $\mathcal{B}$ be a Serre subcategory of $\mathcal{A}$. We can form the quotient category $\mathcal{A}/\mathcal{B}$, and the canonical functor $Q:\...
- 1,479
3
votes
0
answers
206
views
2 K3s and cubic fourfolds containing a plane
Two K3 surfaces show up when talking about cubic fourfolds containing a plane. Let $P\subset X\subset \mathbb{P}^5$ be the plane inside the cubic. Since $P$ is cut out by 3 linear equations then $X$ ...
- 3,737
4
votes
0
answers
160
views
Reference for equivariant derived Künneth formula
I'm looking for a reference for the following statement in as much generality as possible, assuming it is correct.
Let's $X$ and $Y$ be "spaces" with a $G$-action. We can take the $G$-product defined ...
- 41
3
votes
0
answers
393
views
Stalks of perverse cohomology sheaves?
For a complex of sheaves $\cal{F}^{\bullet}$ on a variety $X$, a useful fact is that the stalks of the cohomology sheaves of $\mathcal{F}^{\bullet}$ agree with the cohomology groups of the complex of ...
- 1,701
5
votes
1
answer
362
views
When is $\mathcal{D}(\mathcal{F}):\mathcal{D}(\mathcal{A})\to \mathcal{D}(\mathcal{B})$ fully faithful?
Let $\mathcal{A}$ and $\mathcal{B}$ be two abelian categories and let $\mathcal{F}:\mathcal{A}\to \mathcal{B}$ be an additive functor. Assume that $\mathcal{F}$ is exact and let $\mathcal{D}(\mathcal{...
- 1,479
2
votes
0
answers
76
views
Cohomology of sheaves on $X \cup_{Z} Y$
I am in the following situation, I have two schemes $X$, $Y$ and two closed immersions $Z \rightarrow Y$, $Z \rightarrow X$. Everything is smooth. I am interested in calculating morphisms in the ...
- 1,315
6
votes
1
answer
889
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 ...
- 2,739
5
votes
1
answer
349
views
Base change and the octahedron axiom
I am trying to understand "de Cataldo, Migliorini. The perverse filtration and the Lefschetz hyperplane theorem. Annals of Mathematics, 171(2010), 2089-2113." My question is about one detail in the ...
- 143
8
votes
1
answer
284
views
Injective model structure on sheaves of bounded complexes of $A$-modules
The following might be very well known for people who works with model categories, but I do not find the answer.
Let $A$-be a ring. Denote $\mathbf{Ch}_+(A)$ the category of positive degree cochain ...
- 2,811
9
votes
4
answers
3k
views
Is there a (satisfying) proof that cellular cohomology is isomorphic to simplicial cohomology that doesn't use relative cohomlogy?
That singular and de Rham cohomologies of a smooth manifold are isomorphic has two proofs that I know of. The classical one uses Stokes' theorem to give the isomorphism explicitly. The second proof ...
- 5,869