Questions tagged [sheaf-theory]

For questions about sheaves on a topological space.

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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}...
Gabriel's user avatar
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4 votes
2 answers
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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 (...
Gabriel's user avatar
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6 votes
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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,...
 V. Rogov's user avatar
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2 votes
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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 ...
Mathmank's user avatar
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7 votes
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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 ...
EJAS's user avatar
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10 votes
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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) \...
Chris Kuo's user avatar
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3 votes
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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 $\...
user127776's user avatar
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2 votes
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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 ...
user267839's user avatar
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6 votes
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641 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 ...
Exit path's user avatar
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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 ...
Doyoung Choi's user avatar
2 votes
1 answer
412 views

Higher direct image with compact support of a constant sheaf

Let $f: X \to Y$ be a locally trivial fibration between locally compact spaces with fiber $F$. It is well known that for a constant sheaf $A_X$ on $X$, the higher direct images $R^n f_* A_X$ are ...
Eduardo de Lorenzo's user avatar
1 vote
0 answers
158 views

When does a $D$-module think it’s a pullback along a smooth morphism?

Let $X$ and $Y$ be two algebraic varieties, and let $f: X \to Y$ be a morphism. Suppose $A$ is a holonomic $D$-module on $Y$. In this situation we can pull $A$ back to $X$ using either the $!$ or $*$ ...
Exit path's user avatar
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Local extension of holomorphic vector fields

Let $X$ be an open complex manifold, e.g., the complement of a simple normal crossing divisor $D$ in a (smooth) projective manifold $M$. Let $T^{1,0}X$ be the holomorphic tangent bundle of $X$. Let $K ...
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18 votes
2 answers
1k views

What are the points (and generalized points) of the topos of condensed sets?

What are the topos points of $\mathrm{CondSet}$, i.e. the geometric morphisms $\mathrm{Set} \to \mathrm{CondSet}$? More generally, is there a concise description of the geometric morphisms $\mathcal{E}...
Matthias Hutzler's user avatar
2 votes
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146 views

Torsors for nonabelian groups and maps to contracted products

$\newcommand\op{\mathrm{op}}$My question concerns torsors for a sheaf of groups $G$ that is not commutative, and left/right are messing me up. A left $G$-torsor is equivalent to a right $G^{\op}$-...
Leo Herr's user avatar
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6 votes
2 answers
583 views

Geometric meaning of coherent sheaves $\mathcal{F} \otimes \mathcal{O}_{\mathbb{P}^n}(d)$ over $\mathbb{P}^n$

Maybe it sounds like a silly question but I'm not able to figure out in my head the geometric meaning of "twisting" vector bundles (or more generaly coherent sheaf) over $\mathbb{P}^n$. I ...
gigi's user avatar
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1 vote
1 answer
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Decomposing an analytic function into two functions which vanish at $0$ and $\infty?$

This question comes from exercise I-10 of The Geometry of Schemes by Joe Harris (although this question is not about schemes). It is translated to less abstract language below: Consider the Riemann ...
Ma Joad's user avatar
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10 votes
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How to model (affine) schemes with a large sketch?

Guitart states in "Toute theorie est algebrique et topologique" as Proposition 17 that the category $\mathbf{Sch}$ of schemes is the category of models of a large mixed sketch. Presumably, ...
Martin Brandenburg's user avatar
3 votes
1 answer
216 views

Sheaves on sites given by a (regular) cd-structure

Let $C$ be a category equipped with a Grothendieck topology generated by a cd-structure (see https://ncatlab.org/nlab/show/cd-structure or Voevodsky's paper Homotopy theory of simplicial presheaves in ...
Josefien K's user avatar
15 votes
1 answer
2k views

How to motivate constructible sheaves

I'm writing some notes for some students which just finished a first course in scheme theory. There I would like to talk about constructible sheaves, but I found it hard to give a compelling ...
Gabriel's user avatar
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2 votes
1 answer
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Example of an Algebraic Space ("false" affine line with different tangents at origin)

I have a question about the following example from the Algebraic spaces and quotients by equivalence relation of schemes by Roy Mikael Skjelnes (page 12) of a presheaf quotient, which has associated ...
user267839's user avatar
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9 votes
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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 ...
Konrad Waldorf's user avatar
4 votes
1 answer
289 views

Functorial isomorphisms

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Sh{Sh}\DeclareMathOperator\PSh{PSh}$We know that a presheaf $\mathcal{F}$ on category $ \mathcal{C} $ gives a colimit preasheaf $ \mathcal{F}^{+} $ ...
BouaichiAnas's user avatar
4 votes
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276 views

Is the restriction of an injective sheaf on a closed subscheme still injective?

Let $X$ be a Noetherian scheme, and let $i:Z\to X$ be the inclusion of a closed subscheme $Z$. Let $\mathcal{I}$ be an injective sheaf of modules on $X$. Question. Is $i^*\mathcal{I}$ still an ...
Stabilo's user avatar
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9 votes
2 answers
726 views

Does inclusion from n-stacks into (n+1)-stacks preserve the sheaf condition?

I'm going to describe two situations that seem to contradict each other, and I'm interested to know precisely what's wrong with this reasoning. Let $M$ be a manifold, and consider the presheaf $C^*(-,...
David Corwin's user avatar
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3 votes
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938 views

Saturation of sheaves

Let $(X, \mathcal{O}_X)$ be a complex manifold, which we can take to be projective. A coherent subsheaf $\mathscr{F}$ of some sheaf $\mathscr{G}$ is said to be saturated in $\mathscr{G}$ if the ...
GradStudent's user avatar
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144 views

Sheaf-like reconstruction of a continuous function

Let $X$ and $Y$ be topological manifolds and let $\{(\phi_x,U_x)\}_{x \in X}$ and $\{(\psi_y,Y_y)\}_{y \in Y}$ be respective atlases of $X$ and $Y$; with each $\phi_x:U_x\rightarrow \mathbb{R}^n,\...
ABIM's user avatar
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1 vote
0 answers
104 views

Interesting property of a divisor contained in special fiber

Let $(A, \mathfrak{m}, \kappa=A/\mathfrak{m})$ be a local ring and $f:X \to \operatorname{Spec} (A)$ a scheme. Let $D \subset X$ a divisor on $X$ contained in special fiber $D \subset f^{-1}(\sigma_{\...
user267839's user avatar
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1 vote
1 answer
179 views

Dual of stable vector bundle on a Fano threefold

Let $E$ be a rank $2$ stable vector bundle on a prime Fano threefold of genus $8$, with Chern numbers $c_1=1, c_2=6, c_3=0$. Question. Is it true that $E(-1)=E^*$? What I am able to show is that ...
Kim's user avatar
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1 answer
170 views

Support of a coherent sheaf over a fiber product scheme

I'm trying to prove the following fact which I don't know if it is true since I am not able to find a counterexample: Let $X,S$ be two $K$-scheme of finite type with $K$ an algebraically closed field....
John117's user avatar
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6 votes
1 answer
396 views

Is there a Grothendieck correspondence for sheaves/stacks?

Given a category $\mathcal{C}$, the category of elements functor sets up an equivalence of categories $$ \mathsf{DFib}(\mathcal{C}) \cong \mathsf{PSh}(\mathcal{C}), $$ whereas the Grothendieck ...
Emily's user avatar
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4 votes
1 answer
183 views

Checking that (hyper) sheafification is fibrant in local projective model structure on simplicial presheaves

Context and Notation Let $X$ be a manifold and $\mathcal{C} = Op(X)$ be the category of open subsets of $X$ with inclusions. I then consider the projective model structure on the (simplicial model) ...
cheyne's user avatar
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1 vote
1 answer
108 views

Disjoint union of clopen sets such that the fibers has constant cardinality [closed]

Let $Z$ a compact set and $X$ a locally compact set. Let $p:Z\to X$ a local homeomorphism. Show that there exists $n≥1$ and $U_1,…,U_n$ open and closed sets of $X$ such that : $X=\sqcup_{i=1}^{n}U_i$ ...
MacFly's user avatar
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1 vote
1 answer
589 views

Computing Ext sheaves over complex projective plane

Let $X:=\mathbb{P}^2_K$ with $K$ algebraically closed field. Take $p\in X$ a point and $\mathcal{I}_p$ its ideal sheaf. One can prove (using Serre Duality and the exact sequence defining $\mathcal{I}...
John117's user avatar
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1 vote
0 answers
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Schur's lemma for sheaves with different reduced Hilbert polynomials

Recall Schur's Lemma for Gieseker-semistable sheaves, in particular the injectivity statement: Let $\psi : F \to G$ be a morphism of Gieseker-semistable sheaves. If $p(F)=p(G)$ and $F$ is stable, ...
alg_et_geom's user avatar
3 votes
1 answer
242 views

Automorphisms of Frobenius liftings and degeneration of the Hodge-de Rham spectral sequence

I am still studying Deligne and Illusie's paper (https://eudml.org/doc/143480), and I am again stuck, this time on pages 262/263. Assume $X\longrightarrow S$ is a smooth morphism of $\mathbb{F}_{p}$-...
The Thin Whistler's user avatar
6 votes
2 answers
509 views

commutativity of restriction and Gysin morphisms in a cartesian square

Let $X, Y, Z$ be compact topological manifolds $f: Y \to X, g: Z \to X$ be embeddings of submanifolds meeting transversely and let $W = Y \times_X Z$: $$ \begin{array}{ccc} Y & \to^f & X \\...
Dima Sustretov's user avatar
6 votes
1 answer
800 views

Putting sheaves to work for algebraic topology?

This is cross-posted from math.se after receiving points and no answers. I apologise if this question is too basic for MathOverflow. I'm refreshing my memory of ...
Siddharth Bhat's user avatar
3 votes
1 answer
309 views

Degeneration of Hodge-de Rham spectral sequence, exactness of a pairing and the trace morphism

If $X\longrightarrow S$, where $S=\operatorname{Spec}(k)$, is a proper smooth morphism of dimension $p$ between $\mathbb{F}_{p}$ schemes with Frobenius $F$, then there is an exact pairing $\wedge:\...
The Thin Whistler's user avatar
1 vote
0 answers
140 views

Sheafification in an arbitrary category

Let $\mathcal{F}$ be a presheaf valued in an arbitrary category $\mathcal{C}$ on a topological space $X$, with $\mathcal{C} $ has limits(or, $\mathcal{C}$ has equalizers so that a sheaf valued in $\...
jy z's user avatar
  • 101
48 votes
4 answers
7k views

Sheaf-theoretic approach to forcing

Inspired by the question here, I have been trying to understand the sheaf-theoretic approach to forcing, as in MacLane–Moerdijk's book "Sheaves in geometry and logic", Chapter VI. A general ...
Peter Scholze's user avatar
3 votes
0 answers
430 views

Does isomorphism on local rings imply the global isomorphism for the sheaf of spectra?

Let's assume we have a sheaf of spectra on some scheme. As an example I will assume that we are working with the $K$-theory sheaf. There are certain local to global spectral sequences, like descent ...
user127776's user avatar
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2 votes
1 answer
426 views

On the definition of a principal ideal sheaf

In his book Algebraic Geometry and Arithmetic Curves Qing Liu claims in Exercise 3.4, page 56, the following for a scheme $X$ and a global function $f\in \mathcal O_X(X)$: "The map $U\mapsto f\...
lefuneste's user avatar
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0 votes
0 answers
97 views

Quotient of $\text{Proj}(A)$ by the action of a finite group

Let $X$ be $ \operatorname{Proj}(A)$ for some graded ring A, and let $G$ be a finite group acting on $A$ with morphisms of graded rings; consequently $G$ acts on $X$. I know I can write $X = \bigcup_{...
waphety's user avatar
1 vote
1 answer
319 views

Flat familiy of coherent sheaves over a scheme

I'm studying the moduli problem of locally free sheaves over a connected smooth projective curve on an algebraically closed field, from the Lecture Notes of Victoria Hoskins, and I cannot fully ...
John117's user avatar
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2 votes
1 answer
362 views

Locally free sheaves and vector bundles over smooth connected projective curve

Let $X$ be a connected smooth projective curve over an algebraically closed field $K$. Let $\mathcal{F}$ be a locally free sheaf on $X$ and $\mathcal{E}$ a subsheaf of $\mathcal{F}$, which is again ...
John117's user avatar
  • 395
4 votes
1 answer
384 views

Push-out in the category of coherent sheaves over the complex projective plane

I'm trying to deal with an example of a rank two vector bundle over the complex projective plane which is non slope-stable (because the associated sheaf of sections has a coherent subsheaf of equal ...
John117's user avatar
  • 395
17 votes
2 answers
560 views

In the internal language of the topos of sheaves on a topological space, can we define locally constant real-valued functions?

For the purposes of this question, in a Grothendieck topos, we will call “definable” the objects and relations obtained from the terminal object, the natural numbers object and the subobject ...
Gro-Tsen's user avatar
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2 votes
1 answer
190 views

How to compute cup product of derived limits / presheaf cohomology

I have a finite category $\mathcal{C}$, along with a functor $F \colon \mathcal{C}^{\mathrm{op}} \to \mathsf{GradedCommRings}$. If $F_j$ is $j$-th graded piece of $F$, then I write $H^i(\mathcal{C},...
Mr. Palomar's user avatar
5 votes
1 answer
197 views

Cohomology of doubly pinched torus via spectral sequences

Let $f:T^2\to Y$ be a resolution of singularities where $Y$ is a torus with two "pinched" points (or, if you prefer, two copies of $\mathbb{P}^1$ meeting at two points). I'm interested in ...
EJAS's user avatar
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