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3
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0answers
57 views

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
votes
0answers
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 ...
4
votes
0answers
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 ...
1
vote
0answers
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, $...
2
votes
0answers
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}...
6
votes
1answer
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 $T\to p_*p^*T$ ...
0
votes
0answers
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 (...
0
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0answers
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_{...
5
votes
1answer
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 ...
1
vote
0answers
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 $n\leqslant m$ ...
1
vote
1answer
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 $...
0
votes
0answers
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 $...
3
votes
0answers
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 ... \...
1
vote
0answers
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,...
1
vote
0answers
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 ...
0
votes
1answer
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 $X$, ...
2
votes
0answers
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} \...
1
vote
1answer
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 :...
3
votes
1answer
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 $...
1
vote
1answer
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 ...
1
vote
0answers
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 ...
1
vote
0answers
95 views

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 ...
2
votes
1answer
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 ...
5
votes
1answer
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....
0
votes
2answers
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 $X$,...
5
votes
1answer
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 ...
-4
votes
1answer
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(...
2
votes
0answers
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 : ...
3
votes
3answers
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 ...
11
votes
1answer
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 ...
5
votes
1answer
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 ...
6
votes
1answer
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 ...
2
votes
0answers
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 ...
1
vote
0answers
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 $\...
5
votes
1answer
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 ...
2
votes
0answers
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 $\mathcal{H}=(R^1\pi_*\mathbb{Q}_\ell)^\vee$...
1
vote
0answers
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 ...
3
votes
0answers
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 ...
5
votes
1answer
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 ...
2
votes
1answer
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; ...
2
votes
0answers
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 \...
2
votes
0answers
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 (...
2
votes
0answers
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. ...
0
votes
1answer
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 ...
2
votes
1answer
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 $\...
3
votes
0answers
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 ...
15
votes
1answer
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 ...
2
votes
0answers
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?
2
votes
0answers
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 ...
5
votes
1answer
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 ...