Questions tagged [sheaf-theory]

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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 ...
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  • 1,658
6 votes
0 answers
85 views

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, ...
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9 votes
2 answers
773 views

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 $\...
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1 vote
0 answers
186 views

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 $G$ ...
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  • 311
15 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_{...
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0 votes
1 answer
115 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 $...
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1 vote
0 answers
116 views

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 $\...
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2 votes
1 answer
107 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 ...
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3 votes
0 answers
266 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 ...
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  • 153
7 votes
0 answers
192 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 ...
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1 vote
0 answers
95 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 ...
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  • 101
8 votes
2 answers
234 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 ...
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4 votes
0 answers
109 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 ...
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3 votes
0 answers
122 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_\...
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  • 924
2 votes
1 answer
102 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 ...
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1 vote
0 answers
55 views

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....
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  • 317
1 vote
0 answers
140 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 ...
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1 vote
0 answers
49 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 ...
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7 votes
1 answer
245 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 ...
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  • 1,015
3 votes
1 answer
213 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}}\...
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  • 409
4 votes
1 answer
198 views

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 $S$ ...
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8 votes
1 answer
494 views

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 ...
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  • 4,005
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 ...
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  • 1,317
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 ...
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  • 1,317
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 ...
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4 votes
3 answers
592 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 ...
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1 vote
0 answers
112 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 ...
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  • 11
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 $...
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  • 924
4 votes
0 answers
243 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 ...
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  • 1,317
6 votes
0 answers
101 views

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 $X$,...
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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}...
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  • 1,537
2 votes
0 answers
42 views

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 $\mathcal{C}^\mathrm{op}$-...
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1 vote
0 answers
73 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 ...
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  • 503
2 votes
0 answers
141 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 ...
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1 vote
1 answer
218 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 ...
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1 vote
0 answers
116 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 ...
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4 votes
0 answers
183 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 $D_{\mathcal{S}}(X)$ ...
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2 votes
1 answer
286 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 ...
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  • 19k
4 votes
1 answer
184 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 ...
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  • 1,537
2 votes
0 answers
254 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 ...
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  • 1,317
2 votes
1 answer
130 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}...
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  • 1,317
4 votes
2 answers
307 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 (...
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  • 1,317
6 votes
1 answer
233 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,...
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2 votes
0 answers
118 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 ...
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  • 273
7 votes
0 answers
289 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 ...
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  • 151
10 votes
1 answer
302 views

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) \...
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  • 475
3 votes
0 answers
78 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 $\...
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  • 4,926
2 votes
1 answer
198 views

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 ...
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  • 1,015
6 votes
1 answer
432 views

What do nearby/vanishing cycles have to do with Fourier transforms?

Let $E$ be a vector bundle on some smooth algebraic variety and $E^*$ its dual. Suppose $A$ is a sheaf (constructible or a $D$-module) on $E$. Given a linear function $f$ on $E$, we may compute the ...
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  • 2,640
2 votes
0 answers
95 views

About condition for structure sheaf of a scheme being compact object in a category of sheaf of module over X [duplicate]

I found the condition for one direction : Categorical interpretation of quasi-compact quasi-separated schemes This article said that if $X$ is quasi compact and quasi separated, $\mathscr{O}_X$ is a ...
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