All Questions
Tagged with sheaves or sheaf-theory
979 questions
6
votes
0
answers
212
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G-sheaves on spaces with a free G-action
Let $X$ be a topological space
equipped with an action of a group $G$. In the Tohoku paper, Grothendieck defined
"$G$-sheaves" on $X$ as sheaves equipped with $G$-action on the etale space,
...
- 9,237
17
votes
4
answers
1k
views
Can one glue De Rham cohomology classes on a differential manifolds?
Let $M$ be a differential manifold and $\mathcal H^k$ the presheaf of real vector spaces associating to the open subset $U\subset M$ the $k$-th de Rham cohomology vector space: $\mathcal H^k(U)=H^k_{...
- 47.5k
3
votes
2
answers
2k
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Relation between sheaf and group cohomology
Let $E=\mathbb{C}/L$ be an elliptic curve. Then $\mathbb{C}$ is contractible, and $L$ is the fundamental group of $E$. What's interesting is that we can find the cohomology of $E$, which is the same ...
- 63.9k
0
votes
1
answer
175
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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 $...
- 148k
1
vote
0
answers
215
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Artin-Winters proof of semi-stable reduction theorem: details
I've been reading through Artin-Winters proof of the semi-stable reduction theorem (Degenerate fibers and stable reduction of curves) and found myself confused about the following detail—
Let $\...
4
votes
1
answer
199
views
Conformal groupoid
I asked this over on Math.SE but it remained completely silent for over a week so I've deleted it and am reposting it here (I'm not really sure which site it fits better). The question itself is ...
- 439
3
votes
0
answers
530
views
Flasque sheaves on a site
This is a cross-post from MathStackexchange.
We define a flasque sheaf on a site as one whose first Čech cohomology vanishes for every covering of every object of the site. I know this definition is ...
- 15.9k
2
votes
0
answers
136
views
Homotopy fixed points vs coalgebras
Referring to the last part of this answer https://mathoverflow.net/a/225403/170683, I would like to understand how in the case of a Galois cover $f\colon X\to Y=X/G$ with Galois group $G$ (I guess ...
- 351
9
votes
2
answers
377
views
Cartesian-closed full subcategory of locally ringed spaces containing smooth manifolds
This coming fall, I will be teaching a course on differential topology to a small group of strong students. In preparation for it, I'm trying to find a category $\mathrm{GDiff}$ with the following ...
- 4,519
6
votes
0
answers
179
views
Is the category of diffeological spaces a full subcategory of locally ringed spaces?
It is known that the natural functor of smooth functions from the category of smooth manifolds into the category of locally ringed spaces is a full embedding (see, for example, here).
Is a similar ...
- 6,962
10
votes
1
answer
504
views
Is there a way to "puncture" a topos?
Let $E$ be a (Grothendieck) topos, e.g. $E = \text{Sh}(X)$ for a topological space $X$. And let $p = (p^*, p_*):\text{Set}\to E$ be a point of $E$, is there a way to "puncture" $E$ in some sense? By "...
CommunityBot
- 1
3
votes
0
answers
194
views
Hypercovers with sieves
Consider a covering family $\{Y_i \to X\}$ and the induced sieve $R \subseteq X$, the subpresheaf of all maps to $X$ that factor through some $Y_i$. The family gives me an induced Cech nerve $C_\...
- 1,094
2
votes
1
answer
130
views
Sheaves on families of genus 2 curves in Hassett's paper
Sorry for a maybe stupid long question but I'm reading the paper "Classical and minimal models of the moduli space of curves of genus two" by Brendan Hassett and I'm not able to unravel a ...
- 148k
25
votes
2
answers
3k
views
Are there (enough) injectives in condensed abelian groups?
The question is very simple : does $Cond(\mathbf{Ab})$, the category of condensed abelian groups (as defined in Scholze's Lectures in Condensed Mathematics), have enough injectives ?
Does it, in fact, ...
- 4,492
1
vote
0
answers
355
views
Global section of pullback of an ideal sheaf
For a local ring $R$ with maximal ideal $\mathfrak{m}\subset R$ and residue field $\kappa$, and a flat morphism $f\colon X\rightarrow \mathrm{Spec} R$ of schemes, we consider the short exact sequence ...
- 119
18
votes
4
answers
6k
views
Derived categories of coherent sheaves: suggested references?
I am interested in learning about the derived categories of coherent sheaves, the work of Bondal/Orlov and T. Bridgeland. Can someone suggest a reference for this, very introductory one with least ...
- 4,713
1
vote
0
answers
67
views
Is there an inverse image functor for sheaves on stacks?
I'm interested specifically in an inverse image functor between differentiable stacks, ie. stacks coming from Lie groupoids. Specifically, if I have a morphism of Lie groupoids $H\to G$ and I have a ...
- 1,198
7
votes
1
answer
465
views
When is a basis of a topological space a Grothendieck pretopology?
Bases of a topological space in point set topology will in general form a coverage on its category of inclusion on open subsets and on its category of inclusion on basic opens, but it takes a bit more ...
- 37.8k
3
votes
1
answer
249
views
Sheafifcation for the étale site
Let $X$ be a scheme and $\mathcal{F}$ a presheaf on $X_{ét}$.
For each $x_{i}\in X$, pick a geometric point $\bar{x}_{i}$ over $x$ and denote by $i_{\bar{x}_{i}}:\text{Spec}(k_{i})_{\text{ét}}\...
- 148k
15
votes
1
answer
1k
views
Grothendieck - sheaves as meter sticks
I'm trying to read parts of McLarty's Grothendieck on Simplicity and Generality. In the article, I read Grothendieck thought of sheaves over some topological space as meter sticks measuring it.
...
CommunityBot
- 1
4
votes
1
answer
515
views
Sheaf cohomology in non-commutative setup
Let $X$ be a topological space and $A$ a sheaf of noncommutative associative algebras over a fixed field $k$. My questions are:
1) Does the category of modules over A have enough injective?
2) If we ...
CommunityBot
- 1
9
votes
0
answers
545
views
Topos with enough projectives
It is often observed that every presheaf topos has enough projectives, as a corollary of the result that representables are projective and every presheaf is a colimit of representables. We also have ...
- 519
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
7
votes
1
answer
291
views
Direct and inverse image terminology
Let $f\colon X\to Y$ be a continuous map. Then $f$ induces a geometric morphism $f^\ast\dashv f_\ast\colon \mathrm{Sh}(X)\leftrightarrows\mathrm{Sh}(Y)$, whose left adjoint is called inverse image and ...
- 15.9k
12
votes
1
answer
561
views
Are there simple conditions on a category C which guaranty that Ind(C) is a Grothendieck topos?
The category of finite sets is not a Grothendieck topos, but its Ind category
Ind(Finite-Sets) = Sets
is a Grothendieck topos. Similarly, given a pro-finite group G, the Grothendieck topos of discrete ...
- 10.7k
5
votes
3
answers
680
views
Deequivariantisation of indecomposable sheaves
Let G be a connected group acting on a space X. All spaces should be reasonable, so e.g. G is a complex affine algebraic group acting on an algebraic variety X, with everything done using the usual ...
- 563
1
vote
0
answers
125
views
Homotopy of sheaves
On a certain topological space $X$ I want to think about sheaves up to homotopy, i.e., homotopies in the space of sheaves over $X$, and then see what homotopy classes of sheaves I get. Is there a good ...
- 11
4
votes
0
answers
457
views
Quantum sheaves
Are the following definitions known?
Consider H a Hilbert space. A "quantum topology" on H is a set Sigma of closed subspaces satisfying the following conditions:
(a) {0} and H lie in Sigma
(...
- 1,368
1
vote
0
answers
252
views
Pushforward of sheaves along finite etale map
Suppose $\pi : Y \to X$ is a finite 'etale map of degree d. I want a formula for $\pi_* \mathcal O_Y$. I'm happy with a formula in $K$ theory.
There is a $S_d$-torsor $P \to X$ of local isomorphisms $...
- 1,094
2
votes
0
answers
372
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 ...
CommunityBot
- 1
1
vote
0
answers
123
views
Motivic homotopy categories closed under subobjects and quotients
It is well known that the category $\mathbf{HI}_{\rm Nis}^{\rm tr}(k)$ of $\mathbb{A}^1$-local Nisnevich sheaves with transfers is closed under subobjects and quotients, from the highly nontrivial ...
- 16.9k
4
votes
0
answers
278
views
Are manifolds "naturally" ringed or locally ringed spaces?
My question is about whether it's more natural to see manifolds as ringed spaces or as locally ringed spaces. I think I have arguments for both points of view.
On the one hand, it's reasonable to ...
- 711
11
votes
1
answer
1k
views
Are groups determined by their morphisms from solvable groups?
$\newcommand{\Grp}{\mathrm{Grp}}$Consider the category of groups $\Grp$, and within it we have the solvable groups $S$. Then any group $G$ determines the functor from solvable groups: $$h_G:=\text{hom}...
- 63.9k
2
votes
0
answers
167
views
Explicit construction of a presentation of a constructible sheaf of $\mathbb{Z}$-modules
This question was prompted by the two following:
Constructible étale sheaves on X are étale algebraic spaces over X
Naive question about constructing constructible sheaves
If I have a ...
- 617
1
vote
0
answers
106
views
Invariance of categories of sheaves (on simplicial presheaves) under (local) weak equivalence
Let $\mathcal{C}$ be a Grothendieck site (secretly in my head I am thinking of Hausdorff topological spaces with open covers; if I am daring I might be thinking of the big etale site on complex ...
- 563
1
vote
1
answer
305
views
Interesting examples of direct image bundles
Let $\pi : W \rightarrow Y$ be a holomorphic fibration of complex manifolds. Let $L\rightarrow W$ be a holomorphic line bundle on its total space and denote by
$$E^k_q := R^q \pi_*L^k$$
the direct ...
- 42.9k
5
votes
0
answers
250
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
2
votes
1
answer
423
views
Purity of perverse cohomology sheaves
Let $f\colon X\to Y$ be a morphism of projective varieties over a finite field. Let $K$ be a perverse pure sheaf on $X$.
Are the perverse cohomology sheaves of $f_*(K)$ pure?
I am just learning the ...
- 148k
4
votes
1
answer
301
views
What extra structure does the group of automorphisms of a torsor carry?
Let $Y$ be a space, and $G$ a group. For simplicity we can take $G$ to be finite, $Y$ to be a point, if this avoids technical issues. Then for $X/Y$ a $G$ torsor, so a sheaf of sets with $G$ action on ...
- 6,239
2
votes
1
answer
177
views
Are vector bundles acyclic for $\Gamma_c$?
Let $X$ be a paracompact topological space or a manifold (which is not a particular case since the structure sheaves are different). It is well-known that vector bundles (more generally, $\mathcal{O}...
- 40.2k
3
votes
0
answers
83
views
Do rationally contractible presheaves have rationally contractible injective resolution
Given a presheaf $\mathcal{F}: Sm/k\rightarrow Ab$ we define a new presheaf $C\mathcal{F}= \varinjlim\limits_{X\times \{0,1\}\subset U \subset X\times \mathbb{A}^1}\mathcal{F}(U)$. The presheaf $\...
- 63.9k
4
votes
2
answers
419
views
Why abelian sheaves instead of $\mathcal{O}_X$-modules in topology and étale stuff?
Most often than not, the sheaves appearing in algebraic geometry (with the Zariski topology) are $\mathcal{O}_X$-modules, instead of simple abelian sheaves.
Now, when dealing with topological spaces (...
- 156k
25
votes
5
answers
3k
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Applications of the "other" definition of sheaves
In most literature, when you try to look for the definition of sheaves you will see the usual definition for presheaves as a functor from a topological space (or from a Grothendieck topology) to some ...
- 63.9k
6
votes
1
answer
327
views
Topology on cohomology of a sheaf of topological groups
Let $X$ be a topological space and $\mathcal{F}$ be a sheaf of commutative topological groups on $X$. I am interested in the following question:
Is there a natural way to introduce topology on $H^i(X,...
- 37.8k
33
votes
4
answers
6k
views
What (if anything) happened to Intersection Homology?
In the early 1990's, Gil Kalai introduced me to a very interesting generalization of homology theory called intersection homology, which existed for like 10 years back then I believe. Defined ...
- 4,713
2
votes
1
answer
474
views
How to find the smallest flabby sheaf containing a given sheaf?
None of the spaces $C^k(\mathbb{R}^n)$, with $0 \leq k \leq \infty$, is a flabby sheaf. However, they are respectively contained in the smallest flabby sheaves $C^k_{nd} (\mathbb{R}^n)$ of functions $...
- 19.6k
16
votes
6
answers
14k
views
What books should I read before beginning Masaki Kashiwara and Pierre Schapira's "Sheaves on Manifolds"
I am a beginner trying to learn about sheaves. I am trying to read Masaki Kashiwara and Pierre Schapira's book "Sheaves on Manifolds", but I find it is not easy for me to understand.
What ...
- 5,534
9
votes
0
answers
308
views
Refinement of hypercovers by ordinary covers
I am asking for references and discussions of statements of the form
Every bounded hypercover can be refined by an ordinary cover
By "bounded" I mean "finite height". E.g., are ...
- 4,519
2
votes
0
answers
265
views
Why does nearby cycles of a local system on $\mathbb{G}_m$ have same monodromy as local system?
Apologies if this belongs on MSE, but none of the tags I wanted existed, so I took it as a hint to post on MO.
Edit: here is my definition of nearby cycles. Suppose $X$ is a complex analytic space ...
- 272
8
votes
0
answers
750
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