All Questions
Tagged with derived-categories sheaf-theory
44
questions
6
votes
0
answers
91
views
Is the derived category of sheaves localised at pointwise homotopy equivalences locally small?
In order to define the cup and cross products in sheaf cohomology, Iversen makes computations in an intermediate derived category. If $K(X;k)$ is the triangulated category of cochain complexes of ...
- 871
4
votes
1
answer
322
views
Gluing objects of derived category of sheaves
Let $X$ be a locally compact topological space (may be assumed to be a stratified space with finite stratification).
Let $\{U_i\}$ be an open finite covering. Assume that over each $U_i$ we are given ...
- 21.3k
2
votes
1
answer
229
views
Derived category of local systems of finite type on a $K(\pi,1)$ space: an explicit counterexample
Let $X$ be a nice enough topological space. I am mostly interested in smooth complex algebraic varieties. One may ask whether the bounded derived category of the category $\mathrm{Loc}(X)$ of local ...
- 1,061
2
votes
1
answer
398
views
Hypersheaves vs derived category of sheaves
This question arose from Peter Scholze's notes on six functor formalisms, specifically lecture VII in the proof of proposition 7.1.
We fix a LCH space $X$ and consider the functor $D(\mathrm{Ab}(X)) \...
- 121
-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
2
votes
0
answers
138
views
Push-forward of a locally constant sheaf using two homotopic maps
Let $X,Y$ be compact smooth manifolds. Let $f,g\colon X\to Y$ be smooth submersions
(in particular, locally trivial bundles) which are homotopic to each other (in the class of smooth maps, not ...
- 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
2
votes
1
answer
219
views
Dualizing complex of the cone over a manifold
Let $M$ be a smooth (or just topological) closed manifold. Let $C(M)$ denote the cone over $M$, i.e.
$C(M)$ equals to $M\times [0,\infty)$ with $M\times \{0\}$ contracted to a point. The image of $M\...
- 21.3k
0
votes
1
answer
167
views
Fourier transform for constructible sheaves on spheres
Let $S_1 = S_2 = S^d$ be two copies of the $d$-dimensional sphere. Let $p_i : S_1 \times S_2 \to S_i$ be the projection, $j : U \to S_1 \times S_2$ the inclusion of the complement of the diagonal and $...
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
8
votes
0
answers
680
views
What's the definition of a microlocal sheaf?
I'm slowly becoming familiar with what microsupport of a sheaf is, but none of the references I've seen give a definition of what a microlocal sheaf should be in general.
In this paper of ...
- 191
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
4
votes
0
answers
120
views
Understanding a step in proof of sheaf version Verdier duality
Warning: This question is likely low-level for MathOverflow. My apology that there is almost surely something basic I miss.
So all proofs I can find factors through a particular statement, which goes ...
- 1,764
3
votes
0
answers
69
views
Characterization of degeneracy of spectral sequence of a fiber bundle at the second term
Let $f\colon E\to B$ be a fiber bundle of compact manifolds with fiber $F$. Assume that the push-forward $Rf_*(\underline{\mathbb{F}})$ in the derived category of the constant sheaf with coefficients ...
- 21.3k