All Questions
Tagged with homological-algebra sheaf-theory
77 questions
93
votes
3
answers
11k
views
What is homology anyway?
Disclaimer: I don't feel qualified to ask this question and yet it's been troubling me for some time now and I lost my patience and decided to ask to get some kind of answer. If there are any stupid ...
- 7,789
62
votes
8
answers
14k
views
Sheaf cohomology and injective resolutions
In defining sheaf cohomology (say in Hartshorne), a common approach seems to be defining the cohomology functors as derived functors. Is there any conceptual reason for injective resolution to come ...
user709
48
votes
8
answers
8k
views
When are there enough projective sheaves on a space X?
This question is being asked on behalf of a colleague of mine.
Let $X$ be a topological space. It is well known that the abelian category of sheaves on $X$ has enough injectives: that is, every ...
- 65.4k
35
votes
5
answers
4k
views
Heuristic explanation of why we lose projectives in sheaves.
We know that presheaves of any category have enough projectives and that sheaves do not, why is this, and how does it effect our thinking?
This question was asked(and I found it very helpful) but I ...
- 4,842
28
votes
1
answer
3k
views
Two points of view about Borel-moore homology
They are several ways to define the Borel-Moore homology on a locally compact space $X$.
The first one is by analogy with the singular homology but instead of using finite chains, we use locally ...
- 1,017
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
5
answers
2k
views
Equivalence of ordered and unordered cech cohomology.
Given a topological space X and a finite cover X = $\cup X_i$, one can define Cech cohomology of a sheaf of abelian groups F with respect to the cover $\{X_i\}$ in two different ways:
(Ordered): ...
- 10.5k
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
18
votes
2
answers
1k
views
Measuring a presheaf's failure to be a sheaf?
Apologies for the vagueness of question.
Background
this thread has some nice examples of presheaves failing to be sheaves.
Question
Is there a generic way to measure "how badly" a presheaf fails ...
- 928
14
votes
3
answers
1k
views
Counterexamples to gluing complexes of sheaves
Note: I asked the question below last week on MathSE but received no answer.
Background:
I have read the claim that perverse sheaves behave more like sheaves than like complexes of sheaves. This ...
- 937
12
votes
2
answers
1k
views
How to visualize the Microsupport of a Sheaf?
I am looking through Persistent homology and microlocal sheaf theory to learn a bit on barcodes. They are require the notion of a microsupport of a sheaf, looks like it could be a rather concrete ...
- 22.8k
11
votes
1
answer
406
views
Resolutions of unbounded complexes: Condition ($\ast$) in Spaltenstein's paper
In the paper "Resolutions of unbounded complexes" (Compositio Math., vol. 65, no. 2, pp. 121-154) N. Spaltenstein generalizes the 6 functor formalism to unbounded complexes of sheaves over ...
- 23.5k
11
votes
1
answer
812
views
Understanding the purely formal part of the sheaf theoretic (cohomological) framework for representation theory
By now I have the impression that many statements in representation theory can be phrased a lot more elegantly using cohomological language. Therefore I'm trying to understand a bit better the sheaf ...
- 7,789
11
votes
1
answer
866
views
Serre spectral sequence for de Rham cohomology
Suppose we a given a fibration of manifolds $p\colon E\to M$ with a path connected fiber $F$ and simply connected $M$, then we have the Serre spectral sequence with
$$
E_2^{p,q} = H^p(M,\underline{H^...
- 2,305
10
votes
3
answers
2k
views
Where can I find a proof of the de Rham-Weil theorem?
Where can I find a proof of the de Rham-Weil theorem?
Does anyone know?
- 360
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
9
votes
1
answer
1k
views
Splitting of exact triangles in derived category
Let $A\to B\to C\to A[1]$ be a distinguished triangle in a (bounded below) derived category of an abelian category.
Is there a necessary and sufficient condition that it splits, namely $B\simeq A\...
- 21.8k
8
votes
1
answer
586
views
Injective objects in Mor(Ab)
Consider the abelian (Grothendieck) category $\mathcal{C} := \mathrm{Fun}(\{0<1\},\mathrm{Ab}) = \mathrm{Mor}(\mathrm{Ab})$. Objects are morphisms $(A \to B)$ of abelian groups, morphisms are ...
- 63.1k
7
votes
2
answers
611
views
Are subfunctors of left exact functors also left exact?
Consider the direct image functor $f_*: Sh(X) \rightarrow Sh(Y)$, let $X$ and $Y$ be topological spaces, let $f: X \rightarrow Y$ be a continuous map, let $G \in Sh(X)$ be a sheaf. I was reading this ...
- 360
7
votes
1
answer
310
views
On the ordered set of real numbers, does sheaf+cosheaf imply constant?
I have a technical question about unbounded chain complexes. I couldn't think of a descriptive title for it.
Let $P$ be a chain complex of contravariant functors on $\mathbf{R}$ (the real numbers ...
- 4,949
7
votes
1
answer
353
views
Does the category of cosheaves have enough projectives?
Given a general topological space $X$ does the category $\mathbf{coShv}(X,\mathbf{Mod}_R)$ have enough projectives ? I know that under some conditions this is true, for example if $X$ is a cell ...
- 173
6
votes
1
answer
441
views
Where can I find a definition of $\underline{H}^p(X, \mathscr{F})$?
Let $X$ be a topological space and $\mathscr{F}$ a sheaf on $X$. In the paper Tropical cycle classes for non-archimedean spaces and weight decomposition of de Rham cohomology sheaves by Yifeng Liu, ...
- 1,153
6
votes
1
answer
2k
views
Information and intuition packed in the Chern character for coherent sheaves
even after quite some time learning it, I still get somehow puzzled by the Chern character. Let me recall some stuff to get notation and setting.
Let us consider a smooth projective algebraic variety ...
- 155
6
votes
1
answer
326
views
Spectral sequence generalizing Čech cohomology
Let $X$ be a 'nice' topological space. Let $\left(U_i\right)_{i\in I}$ be a finite open covering of $X$. Let $\mathcal{F}$ be a sheaf of abelian groups.
For a subset $A\subset I$ denote $$U_A:=\cap_{...
- 21.8k
6
votes
0
answers
452
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
2
answers
3k
views
Are presheaves of constant functions sheaves?
I was reading 'An introduction to homological algebra' by Rotman, and on page 279 in the section about sheaves, example 5.64, Rotman gives an example of a constant presheaf $\mathcal{P}$ that's not ...
- 456
5
votes
2
answers
830
views
Closed monoidal structure on the derived category of sheaves
Given a topological space X, i'd like to find Der X - the derived category of sheaves of abelian groups on X - to be a closed monoidal category. Hom should be cohomological and the internal-hom should ...
- 3,249
5
votes
1
answer
3k
views
Question about hypercohomology / spectral sequence of a complex of "almost-acyclic" sheaves
I have a very particular situation involving a (non-exact) complex $K$ of coherent sheaves on a nonsingular projective variety $X$, and I need to compute the hypercohomology of the complex. The ...
- 545
5
votes
1
answer
512
views
Do we have $\underline{\operatorname{Ext}}^i_\text{fppf}(\mathbb{G}_a,\mathbb{G}_m)=0$ 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
5
votes
1
answer
331
views
Is the sheaf associated to a differential structure of a specific type?
On a set $X$, let us define a set $\mathcal{D}$ of functions from $X$ to $\mathbb{R}$. Consider first the initial topology $\tau_\mathcal{D}$ on $X$ with respect to $\mathcal{D}$, i.e. the coarsest ...
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
2
answers
228
views
Is there a Čech-like way of computing $H^\bullet(X,M^\bullet)$ or even $\mathsf{R}f_* M^\bullet$?
Let $X$ be a topological space (or a site) and let $M$ be a sheaf on $X$. If $X$ is paracompact, or if $X$ is a noetherian separated scheme and $M$ is quasi-coherent, or if $X$ is quasi-projective ...
- 711
4
votes
1
answer
335
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 ...
- 21.8k
4
votes
4
answers
696
views
Canonical product in sheaf cohomology
EDIT: Let $\mathcal{F},\mathcal{G}$ be sheaves of abelian groups on a topological space $X$. Then there exists a canonical cup product
$$H^i(X,\mathcal{F})\otimes_\mathbb{Z}H^j(X,\mathcal{G})\to H^{i+...
- 21.8k
4
votes
1
answer
895
views
When does derived pullback commute with infinite products?
Let $f:X \to Y$ be a morphism of reasonable schemes (qcqs). Let $f^*: D(Y) \to D(X)$ be the pullback defined on the derived unbounded categories of quasi-coherent sheaves.
Question: When does $f^*$...
- 7,789
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
4
votes
1
answer
291
views
Exactness of $j_!$ in abelian category recollement
Consider a recollement situation, with notation the same as on the nLab page. That is, we have adjunctions $i^* \dashv i_* \dashv i^!$ and $j_! \dashv j^* \dashv j_*$ between the abelian categories $\...
- 6,025
4
votes
1
answer
781
views
Restricting a Soft Sheaf to an Open is again Soft?
Hi everyone! Answered to my satisfaction in the comments - thanks nosr and Jacob Bell! :)
Let $X$ be Hausdorff, locally compact, paracompact. Consider $\mathcal{F}$ a soft sheaf on $X$: as there are ...
- 329
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
1
answer
363
views
Why isn't the localization $C[W^{-1}]$ (locally) small when $C$ is small and $W$ admits a calculus of (right) fractions?
In the presence of a calculus of (right) fractions, one may prove that every equivalence class of the general localization---the quotient of $F(UC +_{obj W} W^{op})$, the free category on the ...
- 352
3
votes
1
answer
638
views
Can not tell colimits from limits
Proposition 71 here reads:
Let $X$ be a concentrated scheme and $F$ a quasi-coherent sheaf of modules. The
following are equivalent:
(a) The functor $\mathrm{Hom}(F, −):Qco(X)\rightarrow Ab$ ...
user139957
3
votes
1
answer
459
views
Help understand a calculation involving RHom of sheaves on manifolds
I am reading a paper and there is some computation of RHom of sheaves that I don't understand. I hope this is the right place to ask.
It is this paper, example 3.10 , page 25
arxiv.org/pdf/1005.1517v4....
- 1,893
3
votes
2
answers
552
views
Projectivity of free O_X modules with respect to the sheafy hom?
I've heard that given a ringed topos $(X,\mathcal{O}_X)$, the functor $Hom_{\mathcal{O}_X-\operatorname{Mod}}(\mathcal{O}_X, -)$ often fails to be exact. Is this only the case for the unenriched hom (...
- 19.6k