All Questions
17
questions
-1
votes
1
answer
171
views
When morphism of complexes is homotopic to 0?
Let $f\colon A^\bullet\to I^\bullet$ be a morphism of bounded below complexes in an abelian category. Assume all $I^i$ are injective objects. Assume also that $f$ induces the zero map on cohomology.
...
- 21.3k
3
votes
1
answer
137
views
What do we know about a sheaf $M$ if we know its derived fibers $\mathsf{L}x^* M$, for $x\in X(k)$?
Let $X$ be a scheme over a field $k$. (Feel free to assume that $X$ is an algebraic variety, if needed.) Also, let $M^\bullet$ be a complex in the derived category of quasi-coherent sheaves $\mathsf{D}...
- 1,139
5
votes
0
answers
233
views
Formality of a category of constructible sheaves
Let $X= S^1 \wedge S^1$ be a wedge of circles. Then $X$ admits a natural stratification $\mathcal{S}$ as a union of two disjoint open intervals $I_1, I_2$ and a point $\{*\}$.
Let $D_{\mathcal{S}}(X)$ ...
- 373
1
vote
0
answers
96
views
$L^r_M = i_* \circ \hat{L}^{r-1}_M \circ i^*$ by the projection formula and the Poincare duality
This is a question arising when I am reading
M. A. A. de Cataldo, L. Migliorini - The Hard Lefschetz Theorem and the topology of semismall maps, Ann. sci. École Norm. Sup., Serie 4 35 (2002) 759-772.
...
- 1,094
2
votes
0
answers
90
views
Cone of morphism in families
I am working in derived category $D^b(X)$ of coherent sheaves on a smooth projective varitey.
Let $E,F$ be two sheaves on $X$, with $\mathrm{R}Hom(E,F)=k\oplus k[-1]$, I consider the following ...
- 1,962
3
votes
0
answers
198
views
What's wrong with higher dimensional nearby cycles?
Suppose we have a complex algebraic variety $X$ with a map $f: X \to \mathbb{C}$ with $Y=f^{-1}(0)$. Let $\overset{\sim}{\mathbb{C}}$ be the universal cover of $\mathbb{C}-\{0\}$ and consider the ...
- 2,969
6
votes
1
answer
1k
views
How to understand the Fourier-Sato transform and microlocalization functors?
Given a smooth real vector bundle $\pi: E \to M$ I can look at the (bounded from below) derived category of sheaves on $E$. Since $E$ admits a very natural action of $\mathbb{R}^{\geq 0}$ by scaling, ...
- 272
4
votes
0
answers
198
views
Sheaf-type property for Derived Categories?
Suppose $X$ is a finite dimensional complex space (I'm happy to restrict to $X$ being a scheme of finite type over $\mathbb C$ as well). I'm wondering if the following sheaf-like properties hold for ...
- 1,731
1
vote
0
answers
165
views
Espace étalé for derived category
It is known that for a sheaf $\mathcal{F}$ on $X$, we can associate $X_\mathcal{F}$, the étalé space of $\mathcal{F}$ over $X$ such that section of $X_\mathcal{F}$ coincides with section of $\mathcal{...
- 677
2
votes
0
answers
166
views
Restriction of scalars from an Azumaya algebra and preservation of perfect/compact objects of the derived categories
An Azumaya variety over a field is by definition a pair $(X,\mathcal A_X)$, where $X$ is an algebraic variety of finite type over that field and $\mathcal A_X$ is a sheaf of Azumaya algebras, namely ...
- 2,079
3
votes
1
answer
448
views
K-injective (also known as hoinjective) complexes of sheaves of modules
Let $(X,\mathcal O_X)$ be a ringed space (if necessary, assume that it is a scheme with suitable hypotheses). Given two complexes of sheaves $\mathcal F$ and $\mathcal G$ of $\mathcal O_X$-modules, ...
- 2,079
1
vote
1
answer
472
views
Relation between local cohomology and open immersions
Let $X$ be a noetherian scheme $U \subset X$ an open subset with complement $Z = X- U$. Assume $Z$ is cut out by the ideal sheaf $\mathcal{I} \subset \mathcal{O}_X$. We have exact sequences:
$$0 \to \...
- 7,689
0
votes
1
answer
500
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 ...
- 21.3k
4
votes
2
answers
805
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}...
- 21.3k
4
votes
1
answer
366
views
flat descent for perverse sheaves
Let $E \in D^{b}_{c}(X,\overline{\mathbb{Q}}_{l})$ where $X$ is a $k$ scheme of finite type for a field $k$.
Let $Y\rightarrow X$ a finite flat surjective morphism such that $f^{*}E$ is perverse and ...
- 3,452