# Questions tagged [sheaf-theory]

The sheaf-theory tag has no usage guidance.

822
questions

2
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### Line bundle whose pushforward is a complex of vector bundles

If $E\to X$ is a holomorphic vector bundle, it is well known that the tautological line bundle $\mathcal{O}_E(1)$ over the projectivization $\pi:\mathbb{P}(E^*)\to X$ satisfies
$$\pi_*\mathcal{O}_E(1)=...

0
votes

0
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73
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### Criteria for a sheaf to be locally free over subvariety

Let $X$ be a compact complex manifold and $\mathcal{F}\to X$ a sheaf.
Is there a regularity criteria (or a condition) for $\mathcal{F}$ that determines whether we there exists a closed subvariety $i:...

3
votes

1
answer

158
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### Is this a true weakening of the quasi-coherence property?

Let $R$ be a commutative Noetherian ring, and $X=$Spec$(R)$ the associated affine scheme. Let $F$ be a sheaf of $O_X$-modules. Consider the following condition
(#) For all containments $V \subseteq ...

6
votes

0
answers

86
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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
votes

2
answers

784
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### Is the ideal product presheaf a sheaf? Do we have any reasons to believe it will be / it won't?

My question is about one of those several concepts in algebraic geometry who everybody uses but nobody defines or introduces properly.
Given a ringed space $(X,\mathcal{O}_X)$ and ideal sheaves $\...

1
vote

0
answers

190
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### Question regarding affine fibre bundles

Let $f:X\to Y$ be a morphism of affine varieties such that it is a fibre bundle with fibre $F$. Let $\pi_1(Y)=\Gamma$ be a free group (non abelian) of finite rank and $\pi_1(F)$ is a finite group $G$ ...

15
votes

4
answers

1k
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### 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_{...

0
votes

1
answer

117
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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 $...

1
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0
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116
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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 $\...

2
votes

1
answer

108
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### 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 ...

3
votes

0
answers

267
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### 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 ...

7
votes

0
answers

195
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### 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 ...

1
vote

0
answers

95
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### 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 ...

8
votes

2
answers

234
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### 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
votes

0
answers

109
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### 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 ...

3
votes

0
answers

122
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### 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_\...

2
votes

1
answer

102
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### 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 ...

1
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0
answers

55
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### Dimension of support of quotient of reflexive sheaves

Let $X$ be a smooth projective variety and $M,N$ be two reflexive sheaves on $X$ such that $N\subset M$ and $M/N\neq 0$ is a torsion sheaf. Then I think $\mathrm{codim} Supp(M/N)\geq 2$ can not happen....

1
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0
answers

140
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### 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 ...

1
vote

0
answers

49
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### 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 ...

7
votes

1
answer

247
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### 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 ...

3
votes

1
answer

213
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### 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}}\...

4
votes

1
answer

198
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### Sheafification of presheaf of trivial vector bundles is the stack of vector bundles

This is a deliberately vague question, possibly obvious to experts. Let $F$ be a field. Over the (say, fpqc) site of $F$-schemes, we may define a presheaf $T^{\textrm{pre}}$ that takes a scheme $S$ ...

8
votes

1
answer

495
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### What's the point of a point-free locale?

In [1, example C.1.2.8], a locale $Y$ (dense in another locale
$X$) without any point is given. I fail to understand the point
of such point-less locale - Why can't we identify those as the
trivial ...

7
votes

2
answers

427
views

### Is there a relation between Gelfand duality and the spectrum of a ring (with its Zariski topology)?

Compare the following two results:
Thm A) Let $A$ be a commutative $C^*$-algebra and let $X$ be its Gelfand spectrum. Gelfand duality says that there's a natural isometric $*$-isomorphism from $A$ to ...

5
votes

0
answers

254
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 ...

7
votes

1
answer

191
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 ...

4
votes

3
answers

593
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### 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 ...

1
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0
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112
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### 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 ...

1
vote

0
answers

68
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 $...

4
votes

0
answers

243
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### 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 ...

6
votes

0
answers

101
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### Cech cohomology on product covers & Fréchet sheaves

My question is about the paper [Ka67].
Let $S, T$ be sheaves of nuclear Fréchet spaces over paracompact topological spaces $X, Y$, respectively; in particular, if $V \subset U$ are open subsets in $X$,...

11
votes

1
answer

1k
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### 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}...

2
votes

0
answers

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### Canonical coresolutions of cosheaves

Let $M$ be a sufficiently nice topological space (e.g. smooth manifold).
Recall that a (pre-)cosheaf in a category $\mathcal{C}$ over a topological space $M$ is essentially a $\mathcal{C}^\mathrm{op}$-...

1
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0
answers

73
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### 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 ...

2
votes

0
answers

141
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### 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 ...

1
vote

1
answer

218
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### 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 ...

1
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0
answers

116
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### 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 ...

4
votes

0
answers

183
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### 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 $D_{\mathcal{S}}(X)$ ...

2
votes

1
answer

287
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### 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 ...

4
votes

1
answer

184
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### 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 ...

2
votes

0
answers

255
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### 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 ...

2
votes

1
answer

130
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### 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}...

4
votes

2
answers

307
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### 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 (...

6
votes

1
answer

233
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### 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,...

2
votes

0
answers

118
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### 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 ...

7
votes

0
answers

289
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### 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 ...

10
votes

1
answer

302
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### Taking the category of sheaves is symmetric monoidal

Let $M$ and $N$ be topological spaces.
Let $\operatorname{Sh}(M)$ denote the presentable $\infty$-category of space-valued sheaves on $M$.
It seems to me that the equivalence
$$\operatorname{Sh}(M) \...

3
votes

0
answers

78
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### 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 $\...

2
votes

1
answer

198
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### Some facts about sheafification functor on étale site

I'm studying the book Etale cohomology and the Weil conjecture by Freitag, Kiehl and I have some questions on the subchapter introducing the machinery associating to an étale presheaf
a sheaf (that is ...