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37 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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0answers
127 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
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0answers
101 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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0answers
114 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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0answers
45 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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1answer
89 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
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0answers
41 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
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1answer
123 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
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1answer
76 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
132 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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0answers
50 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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0answers
92 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
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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 ...
2
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0answers
83 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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2answers
225 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 $X$,...
4
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1answer
210 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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1answer
536 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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0answers
135 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
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3answers
351 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
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1answer
346 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
210 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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1answer
224 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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0answers
96 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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0answers
80 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
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1answer
211 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
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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 $\mathcal{H}=(R^1\pi_*\mathbb{Q}_\ell)^\vee$...
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0answers
110 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
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0answers
54 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
212 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
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1answer
152 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
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0answers
101 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
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0answers
152 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
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0answers
102 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
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1answer
128 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
83 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
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0answers
152 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
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1answer
555 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
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0answers
82 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
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0answers
42 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
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1answer
201 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 ...
6
votes
2answers
261 views

Do pseudo-differential operators form a sheaf of algebras?

Let $M$ be a smooth manifold. I have been trying to figure out from the literature I know whether (any flavor of) pseudo-differential operators form a sheaf of algebras (w.r.t. the usual topology on ...
10
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1answer
259 views

Is every endomorphism of the sheaf of holomorphic functions on a disk a differential operator?

Let $D= \{z\in \mathbb{C}:|z| < 1\}$ be the unit disk. And consider the sheaf of holomorphic functions $\mathcal{O}_{D}$. Question (?) : Is there a sheaf endomorphisms $\phi : \mathcal{O}_D \to \...
2
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1answer
91 views

On sections into Banach bundles over a compact manifold

Let $M$ be a smooth, compact manifold and $\xi: \mathcal B \to M$ a smooth complex Banach bundle over $M$. Here, smooth is understood to be in the Fréchet-sense. Further, let $p: V \to M$ be an ...
4
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1answer
210 views

Why are Regular Categories assumed to be finitely complete?

Regular categories may equivalently defined as those with: finite limits coequalizers of kernel pairs pulback stable regular epis or finite limits pullback stable regular epi/mono factorization ...
5
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0answers
82 views

How “commutative” are Cech cochains of a sheaf of commutative (dg) algebras?

Let $X$ be a topological space and $\mathcal{F}$ be a sheaf of commutative dg-algebras over $X$. Let $\mathfrak{U}$ be a fixed open covering of $X$ and $C^\bullet(\mathfrak{U},\mathcal{F})$ be the ...
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0answers
173 views

Can a presheaf fail to have a nontrivial sheafification?

I'm studying Vakil's Foundations of Algebraic Geometry, and working through the exercises in 2.4 about compatible germs as a method for constructing a sheafification of a given presheaf. Can this ...
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0answers
81 views

How to “intersect” or “refine” a pair of abstract simplicial complexes

Let $S,T$ be abstract simplicial complexes. Is there a (unique) abstract simplicial complex that gives me the most of what is in common with $S$ and $T$? I'm thinking of this as an "intersection," ...
7
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1answer
178 views

How are the left and the right group of a bitorsor related?

This question arose from my answer to To what extent does a torsor determine a group: it turns out that I do not know one thing about it. Let $G$, $G'$ be groups in some nice enough category (you may ...
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0answers
109 views

Induced Morphism on Fibre Product

Let $X$ be a proper $k$-scheme and $k \subset k'$ a field extension. Consider the fibre product \ base change $X' = X \otimes _k k'$. Let $\mathcal{F} \in Coh(X)$ and $p: X' \to X$ the canonical ...
4
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0answers
139 views

Weierstrass model of an elliptic curve: a line bundle over the base

Let $S$ be a Weierstrass model of an elliptic surface (for me it works better to understand it as an elliptic fibration), that is a map $\pi : S \to C$ where $C$ is a compact Riemann surface. ...