The sheaf-theory tag has no usage guidance.

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### Cohomology in the sheaf of functions locally constant along leaves of foliation

According to this paper:
https://www.jstor.org/stable/pdf/2043718.pdf?refreqid=excelsior%3A25a90472dc9886aedb1ce2bc20ab7951
(posted in: Cohomology of a sheaf of functions locally constant along a ...

**3**

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38 views

### Relation between the orientation sheaves of the interior and the boundary of a topological manifold

Let $(M, \partial M)$ be an $n$-dimensional topological manifold with boundary. Let $\mathcal{O}_{M \setminus \partial M}$ and $\mathcal{O}_{\partial M}$ denote the orientation sheaves of $M \setminus ...

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122 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 ...

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70 views

### example of rank 2 torsion free sheaf with no global sections that is not stable

Ìn the book "Vector bundles on complex projective spaces" the authors prove in Chapter 2 Lemma 1.2.5 that, if $E$ is a rank $2$ reflexive sheaf on $\mathbb{P}^n$, then $E$ is stable if, and only if, $...

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118 views

### Defineing a Sheaf of rings over a topological space

Let $X$ be a topological space and let $R$ be a commutative ring with $1$ such that for each $x\in X$ there exists a multiplicatively closed subset $S_x$ of $R$ such that for each $a\in R$ if $\frac{a}...

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**1**answer

241 views

### Cokernel of map of étale sheaves

Let $p:\mathbb{G}_m\to \operatorname{Spec} k$ be the structure map, and let $T$ be an algebraic $k$-torus viewed as an étale sheaf over $k$. Why is the cokernel of the canonical map $T\to p_*p^*T$ ...

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76 views

### Examples of degree zero, rank one reflexive sheaves without r-th roots

Let $X$ be a normal, projective surface (or more generally a variety) over $\mathbb{C}$ (i.e., $X$ is irreducible). Fix a polarisation on $X$. I am looking for examples of rank one, degree zero (...

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58 views

### A relation between an radical ideal and its $J$-radical

Let $R$ be a commutative ring with $1$. For an ideal $I$ of $R$ the $J$-radical of $I$, denoted by $J-rad(I)$, is the intersection of all maximal ideals of $R$ containing $I$, that is, $J-rad(I)=\cap_{...

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**1**answer

131 views

### Products, coproducts and equalizers in category of lattices

Background: Let $\mathbf{Lat}$ be the 2-category of lattices which can be viewed as a subcategory of the 2-cateogry of posets $\mathbf{Pos}$, that is, objects in $\mathbf{Pos}$ that have all finite ...

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77 views

### A special family of prime ideals

I am looking for a commutative ring $R$ with identity which has a family $F:=\{p_i\}_{i\in A}$ of prime ideals such that for each $n\in \mathbb{N}$ there exists $m\in \mathbb{N}$ with $n\leqslant m$ ...

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**1**answer

88 views

### An ideal and its J-radical

Let $R$ be a commutative ring with $1$ and $I$ be an ideal of $R$. Now let $J=\cap_{I\subseteq m\in Max(R)}m$. Set $A:=\{p\in Spec(R): I\subseteq p\}$ and $B:=\{p\in Spec(R): J\subseteq p\}$, where $...

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142 views

### Sheafification map is surjective

This is not a research level problem for sure. But, similar question by some one else $2$ years back on Stack exchange has not received any attention. So, I thought it does not suit there.
Suppose $...

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108 views

### What is the filtration in Leray's spectral sequence?

Leray's theory of spectral sequences considers a continuous map $f : X \to Y$ between topological spaces. The statement is that there exists a filtration
$$H^n(X,k) = F^0H^n(X,k) \supseteq ... \...

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117 views

### A family of prime ideals and a family of maximal ideals

Let $R$ be a commutative ring with $1$ and let $\{\frak{p}_\alpha\}_{\alpha\in A}$ and $\{\frak{m}_\beta\}_{\beta\in B}$ be a family of prime ideals and a family of maximal ideals of $R$, respectively,...

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48 views

### Continuous maps vs filtrations construction of the Leray spectral sequence

The Leray spectral sequence of a continuous map $f : X \to Y$ between two topological spaces can be constructed, as far as I understand, in two ways, one very general, and the other a bit more ...

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**1**answer

159 views

### Sheaf cohomology relative to a closed subspace

Let $i : A \hookrightarrow X$ be a closed subspace of a topological space $X$, and $j : Z := X \setminus A \hookrightarrow X$ denote its open complement. Given a sheaf $F$ of abelian groups on $X$, ...

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42 views

### On the limit of a directed system of sheaves

I find myself in the following situation:
Let $(\mathcal{F}_i, f_{i,j})$ be a directed system of sheaves, defined on an exhausting family of closed topological subspaces $$X_i \subset X_{i+1} \...

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**1**answer

131 views

### Commutativity between functors on sheaves of abelian groups

I am trying to understand certain properties of sheaf theory, but I'm having trouble finding the notions to answer my questions. I'd be really glad if someone could help me with the following. Let $f :...

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**1**answer

85 views

### Interesting (non) examples of singular support

I'm trying to better understand singular support of sheaves on smooth manifolds---to this end: What are examples of conical subsets of $T^*X$ that cannot arise as the singular support of a sheaf on $...

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**1**answer

163 views

### Direct image of reflexive sheaf via finite, flat map

Suppose $f: X \rightarrow Y$ is a finite, flat (hence locally free) morphism of curves (i.e. schemes of dimension 1, not smooth or even reduced). Suppose $L$ is a reflexive sheaf on $X$, locally free ...

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52 views

### maps between two Leray spectral sequences based on maps on cochain complexes

Let $f_1:X_1 \to Y_1$ and $f_2:X_2 \to Y_2$ be two continuous maps between topological spaces. I am trying to understand under what condition there could be a map relating the Leray spectral sequences ...

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### Maps between Leray spectral sequences

Let $f_1 : X_1 \to Y_1$ and $f_2 : X_2 \to Y_2$ be two continuous maps between topological spaces. Suppose that there exists a commutative diagram of singular cohomology groups (say with coefficients ...

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**1**answer

107 views

### Leray spectral sequence for continuous functions on pairs of topological spaces

Let $A \subset X$ and $B \subset Y$ be topological spaces, with $A$ and $B$ closed, and $f: X \to Y$ be a continuous functions such that $f(A) \subset B$.
The Leray spectral sequence (with complex ...

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**1**answer

153 views

### What does an ideal correspond to in the internal language of sheaves?

Suppose I have a sheaf $\mathcal F$ in some topos $\mathrm{Sh}(\mathcal C)$. Then this becomes the sheaf of rings from algebraic geometry when described as a ring in the internal language of the topos....

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**2**answers

248 views

### Can we recover a sheaf from its original presheaf on a basis

This question might be completely totological (I apologize in advance if it is the case):
suppose that we are given two sheaves $\mathcal{F}, \mathcal{G}$ of Abelian groups on a topological space $X$,...

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**1**answer

217 views

### On the Leray spectral sequence and sheaf cohomology

I'm having trouble understanding the construction of the Leray spectral sequence for continuous maps (not necessarily fibrations). More precisely, given a continuous map $f : X \to Y$ between two ...

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**1**answer

773 views

### Why sheaves are important and why do we care about them? [closed]

Presheaves are contravariant functors from a category $C$ to the category $Set$, that is functors $P$:
$$P:C^{op}\to Set.$$
For every topology $J$ on $C$ we can generate a reflexive subcategory
$$Sh(...

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142 views

### Weaker version of smooth base change for étale sheaves

Consider the cartesian square of schemes
$$ \require{AMScd}
\begin{CD}
X' @>{g'}>> X \\
@V{f'}VV @VV{f}V \\
S' @>>{g}> S
\end{CD}
$$
and the base change map
$$ \eta : ...

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**3**answers

363 views

### A few general questions about pre-sheaves and sheaves

I am no specialist in sheaf theory, so I would be glad to get some help regarding the following:
I have a pre-sheaf $F$ of abelian groups above a topological space $X$, and I have found an open ...

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**1**answer

364 views

### Sheaf-theoretically characterize a Riemannian structure?

A smooth structure on a topological manifold can be characterized as a sheaf of local rings, see for example the discussion here.
Q: Is there a way to characterize a Riemannian structure on a smooth ...

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**1**answer

211 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 ...

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233 views

### Kozsul resolution of $\mathcal{O}_X$

Let $i: X \hookrightarrow Y$ be a closed embedding of smooth algebraic varieties. In the book D-modules, perverse sheaves and representation theory the authors say that there exists a locally free ...

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102 views

### Quotient of a sheaf by group action and representabillity

Let $X$ be a scheme and $S$ be a sheaf of sets over the fppf topology of $X$. Let $G$ be a group scheme over $X$ and there is an action of $G$ on $S$. Now, I want to look at the quotient $G \setminus ...

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82 views

### Global sections of twisted dualizing sheaf of Hirzebruch surface

Let consider a Hirzebruch surface $S= \mathbb{P}(\mathcal{E})$ over $\mathbb{P}^1$ with invariant $e \ge 0$ where $\mathcal{E}= \mathcal{O}_{\mathbb{P}^1}(e) \oplus \mathcal{O}_{\mathbb{P}^1}$.
Let $\...

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**1**answer

217 views

### A topos where every object is internally projective but not every object is projective

An object $P$ in a topos $\mathcal{E}$ is said to be projective if $Hom_{\mathcal{E}}(p,-)$ preserves epis, internally projective if $(-)^P$ preserves epis.
Can anyone give an example of a topos ...

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145 views

### Usage of Leray spectral sequence

Maybe this is an elementary stuff for experts, I could not figure it out by myself. Let $\pi:G\to S$ be an elliptic curve with zero section $e:S\to G$. Take $\mathcal{H}=(R^1\pi_*\mathbb{Q}_\ell)^\vee$...

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113 views

### A question on Hilbert polynomial and flat morphisms

Note: this question was previously asked at math.stackexchange (under the same title) to no avail.
I have been recently reading Dr. Kaledin's notes on algebraic geometry. There is a statement in ...

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55 views

### Canonical map in the direct image of $\mathscr{D}_X$

Let $f : X \to Y$ be a proper holomorphic map between holomorphic manifolds. We work with $\mathscr{D}$-modules. Consider the transfer bi-modules $\mathscr{D}_{Y\leftarrow X}.$ Can one find a ...

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**1**answer

220 views

### Sheaf cohomology with support vanishes

I am trying to solve the exercise 2.4 chapter III in Hartshorne's "Algebraic Geometry". For this I would like to prove for a sheaf $F$ of Abelian groups on a topological space $X$ and $U$ open subset ...

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**1**answer

155 views

### Why are sheaves of a coverage the same as those on its generated Grothendieck coverage?

Preliminaries: There are lots of variations on the settings in which we define sheaves. I am concerned with the details linking the general definition below to the Grothendieck topology it generates; ...

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104 views

### Exterior tensor product of $D$ Modules

The exterior tensor product of sheaves of modules is defined as:
$M \boxtimes N = p_1^{*}M \otimes_{\mathcal{O}_{X \times Y}} p_2^{*}N \cong \mathcal{O}_{X \times Y} \otimes_{p_1^{-1}\mathcal{O}_X \...

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158 views

### Singularities of reflexive sheaves

I am studying reflexive sheaves (on $\mathbb{P}^3$) by the Hartshorne's paper ''Stable reflexive sheaves''. As far I understood, reflexive sheaves fail to be locally free at a finite number of points (...

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105 views

### epimorphism of fppf sheaves is an fppf morphism

I asked this question on math.stackexchange (https://math.stackexchange.com/questions/2693471/epimorphism-of-fppf-sheaves-is-an-fppf-morphism) but didn't get an answer. Maybe someone here can help.
...

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**1**answer

131 views

### The cohomology of meromorphic functions

Let $A$ be a sheaf such that $$A(U) = \{ f \in \mathbb M(U): f \in \mathbb{O}(U \backslash\{p_1,\ldots, p_n\}) \ \mbox{with at worst a simple pole at}\ p_i \} $$ where $\mathbb M(U)$ means the set of ...

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**1**answer

84 views

### Defining a sheaf from its values on a prebase (plus little more structure)

Let $X$ be a topological space and $\mathcal{B}$ a base of the topology (i.e. it is closed under intersection and every open set is the union of elements from $\mathcal{B}$). Any functor from $\...

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155 views

### Locality in Grothendieck Topologies

Let $\mathcal{C}$ be a category and $\mathcal{J}$ be a Grothendieck topology on it (i.e., $(\mathcal{C},\mathcal{J})$ is a site). Then what is a good notion of locality in it?
I came up with the ...

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**1**answer

569 views

### A sheaf is a presheaf that preserves small limits

There is a common misconception that a sheaf is simply a presheaf that preserves limits. This has been discussed here before many times and I believe I understand it well enough.
However when reading ...

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89 views

### Pullback of a constant sheaf over arbitrary sites

Given a geometric morphism between arbitrary Grothendieck topoi, $f:\mathcal{Sh(D)}\to\mathcal{Sh(C)}$, does the pullback $f^{-1}$ (i.e, the left adjoint) take constant sheafs to constant sheafs?

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44 views

### Support of étale sheaves

Let $X$ be a scheme, $i: Z\to X$ a closed subscheme, $j: U\to X$ its complement in $X$. Assume the codimension of $Z$ in $X$ is large (at least $2$).
Let $A$ be an étale sheaf on $U$, $B$ an étale ...

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**1**answer

206 views

### Homotopy cosheaf?

Let $C$ be a site, $\mathbf{S}$ some ($\infty$-? homotopy?) category of spaces.
Question. What do you call a (covariant!) functor $F:C\to \mathbf{S}$ enjoying the following property: for every ...