All Questions
Tagged with homological-algebra sheaf-theory
27 questions with no upvoted or accepted answers
27
votes
0
answers
469
views
Are these comparison morphisms between Čech and Grothendieck cohomology the same?
For better or for worse, there is more than one approach to comparing Čech cohomology $\smash{\check{H}}^\bullet(\mathfrak{U},X;\mathscr{F})$ of a sheaf $\mathscr{F}$ on a space $X$ w.r.t the cover $\...
- 1,021
20
votes
0
answers
3k
views
Idea of presheaf cohomology vs. sheaf cohomology
Let $X$ be a topological space and $U$ an open cover of $X$.
In this thread Angelo explained beautifully how presheaf cohomology (Cech cohomology) relates to sheaf cohomology:
The zeroth Cech ...
- 319
10
votes
0
answers
212
views
Does the category of $G$-equivariant sheaves have enough injectives?
The question is related to this one.
Let $k$ be a field and $X$ be a topological space. We consider Sh$(X)$, the category of sheaves of $k$-vector spaces on $X$.
Let $G$ be a topological group which ...
- 9,019
6
votes
0
answers
451
views
Yoneda product on Ext
Let $M$ be a closed manifold and consider the constant sheaf $\mathbb{R}_M$. I have heard that the Yoneda product on $\operatorname{Ext}(\mathbb{R}_M, \mathbb{R}_M)= H^*(M; \mathbb{R})$ coincides with ...
- 177
6
votes
0
answers
171
views
Could we characterize injective objects in the category of $G$-equivariant sheaves?
Let $k$ be a field and $X$ be a topological space. We consider Sh$(X)$, the category of sheaves of $k$-vector spaces on $X$.
Let $G$ be a topological group which act on $X$ continuously from the left....
- 9,019
5
votes
0
answers
290
views
About the left adjoint of $f^*$
In lots of different cases (Verdier duality, Grothendieck duality, étale cohomology, ...) the very existence of a (right) adjoint to the sheaf functor $f_!$ gives useful information. (I'm going to ...
- 711
5
votes
0
answers
113
views
How "commutative" are Cech cochains of a sheaf of commutative (dg) algebras?
Let $X$ be a topological space and $\mathcal{F}$ be a sheaf of commutative dg-algebras over $X$. Let $\mathfrak{U}$ be a fixed open covering of $X$ and $C^\bullet(\mathfrak{U},\mathcal{F})$ be the ...
5
votes
0
answers
271
views
K-flat, K-flabby resolution
Let $X$ be a topological space and $F$ a flat sheaf of abelian groups. It is well known that taking the Godement resolution of $F$ gives rise to a "flabbyflat" or "flasqueflat" resolution, in the ...
- 501
5
votes
0
answers
380
views
Does the de Rham complex induce a functorial soft resolution of the category of cochain complexes of sheaves of vector spaces on a smooth manifold?
I apologize in advance if this is pretty straightforward; I'm a differential geometer and physicist by training so my homological algebra and homotopy theory are a bit weak.
Question: Let $M$ be a ...
- 6,025
4
votes
0
answers
195
views
Example of pseudocoherent complex which is not locally quasi-isomorphic to a strict pseudocoherent one
I ask this question here even if I posted it also on math.stackexchange (recieving no answer so far) because I have read some analogous question but for perfect complexes on this site, even though ...
- 351
4
votes
0
answers
205
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 ...
- 1,761
4
votes
0
answers
432
views
Reference request: sheaf-theoretic operations in the classical topology?
Like many graduate students before trying to learn something about étale cohomology and Deligne's proof(s) of the Riemann hypothesis part of the Weil conjectures, I am hunting for references detailing ...
- 273
4
votes
0
answers
367
views
Cohomological dimension of the category of sheaves
Let $X$ be an $n$-dimensional manifold. Then for any sheaf $\mathcal{F}$ on $X$, the cohomology $H^i(X; \mathcal{F})$ vanishes for $i > n$.
Let $k$ be a field, and let $\mathrm{Shv}_k(X)$ be the ...
- 25.6k
3
votes
0
answers
81
views
Image of Obstruction Map for Relative Quot-scheme
Let $f: X \to S$ be a projective morphisms between finite-type projective schemes, and $\mathcal{O}_{X}(1)$ an $f$-ample line bundle. Given an $S$-flat coherent $\mathcal{O}_{X}$-module $\mathcal{H}$ ...
- 1,701
3
votes
0
answers
430
views
Computing injective resolution of some constant sheaves
I follow the notations on "Sheaves on manifolds" written by Kashiwara-Schapira. Let $\mathbb{K}$ be a ground field and $X$ be a smooth manifold. Let $D(X)$ be the derived category of sheaves of $\...
- 213
3
votes
0
answers
422
views
What kind of ringed space $X$ has the property that a locally free sheaf is projective in Qcoh$(X)$?
It is well known that for an affine scheme $X$, every finitely generated locally free sheaf $\mathcal{E}$ is projective in the category Qcoh$(X)$. i.e. the functor $\text{Hom}_{\text{Qcoh}(X)}(\...
- 9,019
3
votes
0
answers
115
views
Characterization of global sections (which are not products) of a sheaf which is locally a product
In order to compute certain group cohomology sets I have come upon a construction which seems rather general concerning sheaves which are locally products. So I will state the problem here in a ...
- 397
2
votes
0
answers
137
views
details of a dévissage argument for constructible sheaves
I am working on the following Künneth-type isomorphism from [SGA5, exposé III, 2,3]:
$\mathrm{Settings}.$ Let $X_1, X_2$ be separated finite type schemes over the spectrum of a field $S=\mathrm{Spec}...
- 375
2
votes
0
answers
239
views
Do we have $\underline{\operatorname{Ext}}^i_\text{fppf}(\widehat{\mathbb{G}}_a,\mathbb{G}_m)=0$ for $i>0$?
Let $k$ be a characteristic zero field and consider the category $(\mathsf{Sch}/k)_\text{fppf}$ of schemes over $k$ with the fppf topology. I know that $\underline{\operatorname{Hom}}(\mathbb{G}_a,\...
- 711
2
votes
0
answers
128
views
On the generalization of a Cech-to-sheaf type spectral sequence
Let $(X,O_{X})$ be a ringed space and $\mathcal{F^{\bullet}}$ be a bounded below complex of $O_{X}$-modules on $(X,O_{X})$. On [Stacks Project 20.25.1][1] it is shown that there is a weakly convergent ...
- 541
2
votes
0
answers
371
views
How to deduce Künneth from its relative version (in cohomology of sheaves)
Let $p:X\to S$ and $q:Y\to S$ be morphisms of "spaces" over $S$. We have an isomorphism
$$f_!(M\boxtimes N)=p_! M\otimes q_!N$$
in the derived category of "sheaves" over $S$, where ...
- 711
2
votes
0
answers
143
views
Computing derived functor of a complex with non-acyclic terms
Let $A^\bullet =(\dots\to A^i\to A^{i+1}\to\dots)$ be a bounded below complex in an abelian category $\mathcal{A}$ with sufficiently many injectives. Let $F\colon \mathcal{A}\to \mathcal{B}$ be an ...
- 21.8k
1
vote
0
answers
58
views
Which sheaves are good for calculating extraordinary restriction?
Let $X$ be a sufficiently nice locally compact Hausdorff space and let $i:Y\subset X$ be the inclusion map of a sufficiently nice closed subspace. For example, one could take $X$ to be a locally ...
- 23.5k
1
vote
0
answers
213
views
Zero in colimit of sheaves category
This question is motivated by showing that the category $\mathbf{Sheaves} (X)$ from the open subset excluding the empty set category to the category of abelian group $\mathbf{Ab}$ has enough injective ...
- 1,168
1
vote
0
answers
42
views
Checking $\mathbb{K}_{U\times(a,b)}\ast\mathbb{K}_{[0,\infty)}\simeq \mathbb{K}_{U\times[a,\infty)}[-1]$ in derived category $D(X\times\mathbb{R})$
Let $D(X\times\mathbb{R})$ be the derived category of sheaves of $\mathbb{K}$-vector spaces on a smooth manifold $X\times\mathbb{R}$ where $\mathbb{K}$ is a ground field. Let $p_1:X\times\mathbb{R}\...
- 213
1
vote
0
answers
202
views
Soft sheaves on indiscrete paracompact spaces
Let $X$ be some space, I have basically 2 questions:
1 - Are sheaves on paracompact but not Hausdorff spaces acyclic? I've been doing some reading and some authors say that soft sheaves on ...
- 417
0
votes
0
answers
421
views
Split and pure exact sequence of sheaves
Let $X$ be a topological space and
$$\varepsilon \ :\ 0 \to A \to B \to C \to 0$$
be an exact sequence of sheaves of ${\cal O}_X$-modules. $\varepsilon$ is said to be pure if
for each point $x\in X$,...
- 63