All Questions
Tagged with sheaves or sheaf-theory
979 questions
4
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0
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252
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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 \...
- 1,325
4
votes
0
answers
369
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.
...
- 587
4
votes
0
answers
121
views
Reference to an explixit construction of a locale from a measurable space
In A sheaf theoretic approach to measure theory shows that measures on a measurable space are equivalent to measures on some locale whose open sets are the $\sigma$-ideals of the $\sigma$-algebra. The ...
- 391
4
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0
answers
195
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Question on de Rham complex with distributional coefficients
Let $X$ be a smooth manifold (usually assumed to be paracompact). Let us denote by $\underline{\Omega}^{p,-\infty}_X$ the sheaf of real valued $p$-forms with distributional coefficients in the ...
- 21.8k
4
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0
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432
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Reference request: sheaf-theoretic operations in the classical topology?
Like many graduate students before trying to learn something about étale cohomology and Deligne's proof(s) of the Riemann hypothesis part of the Weil conjectures, I am hunting for references detailing ...
- 273
4
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0
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536
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When is a coherent subsheaf determined by its global sections
I am reading an article in which a proof is based on defining a subsheaf by only giving its global sections.
The exact setting is that, one has a surjective finite morphism $f:Y\to X$ between ...
- 333
4
votes
0
answers
262
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Applications of cosheaf homology?
What are some applications of cosheaf homology within mathematics?
Some ones I've heard of Sheaves (not cosheaves) are computing global sections and the Picard Group with a sheaf on projective space.
- 913
4
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0
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367
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Is a locally finitely generated sheaf of modules finitely generated on sections?
"Let $(X, \mathcal{O})$ be a ringed space. A sheaf of modules $\mathcal{F}$ on $X$ is finitely generated if for all $a \in X$ there exists a neighbourhood $U$ of $a$, an integer $n$ and a surjective ...
- 191
4
votes
0
answers
367
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Cohomological dimension of the category of sheaves
Let $X$ be an $n$-dimensional manifold. Then for any sheaf $\mathcal{F}$ on $X$, the cohomology $H^i(X; \mathcal{F})$ vanishes for $i > n$.
Let $k$ be a field, and let $\mathrm{Shv}_k(X)$ be the ...
- 25.6k
4
votes
0
answers
110
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Convolution of DQ-Modules
On page 92 of Deformation Quantization Modules Kashiwara and Schapira define two different convolution products for DQ-modules that differ by whether one uses $Rp_{13*}$ or $Rp_{13!}$ to push forward....
- 41
4
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0
answers
261
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Can one construct Freyd-Mitchell's embeddings that respect sheafifications?
For a certain presheaf $P$ with values in an abelian category $A$ satisfying AB5 and its sheafification $S$ (with respect to a small Grothendieck site) I would like to prove: $S(f):S(X)\to S(Y)$ is ...
- 16.9k
4
votes
0
answers
166
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Homotopy-theoretic measure of operations on sheaves failing to be sheaves
Here's something I've been wondering about for a few weeks:
Consider a topological space $X$ and a sheaf of rings $\mathscr O_X$ on $X$. Suppose $\mathscr{F}$ and $\mathscr{G}$ are $\mathscr O_X$ ...
user1437
4
votes
0
answers
457
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Quantum sheaves
Are the following definitions known?
Consider H a Hilbert space. A "quantum topology" on H is a set Sigma of closed subspaces satisfying the following conditions:
(a) {0} and H lie in Sigma
(...
- 1,368
4
votes
0
answers
382
views
Exercise concerning locally constant presheaves [closed]
Let $\mathscr{F}$ be a presheaf of abelian groups on some topological space $X$. We say that $\mathscr{F}$ is locally constant if there exists an open cover $\mathcal{U}$ of $X$ (i.e. $X=\bigcup_{U\in\...
- 4,902
3
votes
1
answer
363
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Why isn't the localization $C[W^{-1}]$ (locally) small when $C$ is small and $W$ admits a calculus of (right) fractions?
In the presence of a calculus of (right) fractions, one may prove that every equivalence class of the general localization---the quotient of $F(UC +_{obj W} W^{op})$, the free category on the ...
- 352
3
votes
1
answer
345
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Normal bundle of a linear subspace
Let $X\subset\mathbb{P}^N$ be a smooth scheme theoretical complete intersection, and $H\subset X$ a linear subspace. Denote by $N_{H,X}$ the normal bundle of $H$ in $X$.
If $\dim(H) = 1$, that is $H$ ...
- 8,998
3
votes
1
answer
3k
views
Cohomology of tangent bundles
Let $X$ be a smooth scheme and $Z\subset X$ a smooth subscheme. Consider the blow-up
$$\pi:\widetilde{X}:=Bl_{Z}X\rightarrow X$$
of $X$ along $Z$.
What is the relation between the cohomology of the ...
- 8,998
3
votes
1
answer
893
views
system of local coefficients on X, locally constant sheaves and orientation sheaves
Hi,
I try to understand the orientation sheaves. When searching it in the google, i meet new areas such as local coefficient system and locally constant sheaves. I realize that any system of local ...
- 225
3
votes
2
answers
1k
views
Equivalence of Definitions of Twisted Sheaf $ \mathcal {O}(1)$
Let $\mathcal {O}(-1)$ be the tautological line bundle $X$ of $ \Bbb CP^1$, where $X=\{(z,l) \in \Bbb C^2 \times \Bbb CP^1 : z \in l \}$ together with canonical projection $X \to \Bbb CP^1$ (line ...
- 6,038
3
votes
1
answer
418
views
Showing morphism of sheaves is zero
I work in derived category $D^b(X)$ of constructible sheaves on a reasonable space $X$. Let $j\colon U\to X$ be an open inclusion and $i\colon Y\to X$ the closed complement. Let $M,N\in D^b(X)$ and ...
- 1,595
3
votes
1
answer
583
views
Simplifying the definition of a geometric context using sieves?
On Pages 1-3 of Cours 2 of Toën's Master Course on Stacks, he defines the notion of a Geometric context with a rather extensive list of axioms (they take up about two pages over and above the ...
- 19.6k
3
votes
1
answer
265
views
Most general context where a "disjoint sum" definition of a direct limit is applicable and always exist
I am a bit out of my element here so I'm hopefully not saying something stupid.
Anyways, wikipedia gives two ways to define direct limits, one for "algebraic structures" and one for general ...
- 3,307
3
votes
2
answers
488
views
Application of sheaves theory in ring theory
Is there any text that gives some applications of sheaves theory in commutative ring theory? In the other word, is any results in commutative ring theory that be verified by sheaves method?
- 49
3
votes
2
answers
566
views
Vanishing of sheaf cohomology with compact support
Let $X$ be a smooth manifold. Let $F$ be a sheaf of $\mathbb{R}$-vector spaces on $X$. I have three closely related questions.
1) Under what sufficient conditions on $F$ for any compact subset $K\...
- 21.8k
3
votes
3
answers
687
views
Nature of Invertible Sheaves in which there are no global sections.
EDIT: Let me try to make the question clearer.
Consider the invertible sheaves $\mathcal{O}(d)$ over the projective space $\mathbb{P}^n$ where $d\in \mathbb{Z}$. Now, if $d>0$, among many ...
- 2,132
3
votes
1
answer
308
views
The size of sheafification
Let $X$ be a small site. Let $\aleph$ be an infinite cardinal, such that $|Ob(X)|\leq \aleph$ and $|Mor(X)|\leq \aleph$, where $Mor(X)$ is the set of all morphisms.
We define the size of a presheaf $...
- 501
3
votes
1
answer
782
views
Is the cokernel of a map of sheaves a seperated presheaf?
The cokernel of a map of sheaves is not necessarily a sheaf until you sheafify. In every example I have seen of the cokernel failing to be a sheaf it is the glueability axiom that fails while the ...
- 31
3
votes
1
answer
284
views
Is this functor $\mathcal{F}: \text{Sch}/\mathbb{Q}\longrightarrow \text{Sets}$ a sheaf?
Consider the functor $\mathcal{F}: \text{Sch}/\mathbb{Q}\longrightarrow \text{Sets}$, defined by sending a scheme $X$ with coordinate ring $\mathcal{O}(X)$ to the set of orbits $B(\mathcal{O}(X))\...
- 2,907
3
votes
1
answer
747
views
Are cohomology functors sheaves?
Question is the following:
Is the functor $H^n_{dR}:\text{Man}\rightarrow \text{Set}$ a sheaf with respect to open cover topology on $\text{Man}$?
More generally, are cohomology functors sheaves in ...
- 6,023
3
votes
1
answer
195
views
Finitely generated sheaf of algebras over geometric points
I would like to ask if the following is true or not: Let $S$ a scheme and $X$ a $S$-scheme which is proper and flat. Let $\mathcal{F}$ a sheaf of $\mathcal{O}_{X}$-algebras over $X$. Let's suppose ...
- 439
3
votes
1
answer
638
views
Can not tell colimits from limits
Proposition 71 here reads:
Let $X$ be a concentrated scheme and $F$ a quasi-coherent sheaf of modules. The
following are equivalent:
(a) The functor $\mathrm{Hom}(F, −):Qco(X)\rightarrow Ab$ ...
user139957
3
votes
1
answer
459
views
Help understand a calculation involving RHom of sheaves on manifolds
I am reading a paper and there is some computation of RHom of sheaves that I don't understand. I hope this is the right place to ask.
It is this paper, example 3.10 , page 25
arxiv.org/pdf/1005.1517v4....
- 1,893
3
votes
2
answers
552
views
Projectivity of free O_X modules with respect to the sheafy hom?
I've heard that given a ringed topos $(X,\mathcal{O}_X)$, the functor $Hom_{\mathcal{O}_X-\operatorname{Mod}}(\mathcal{O}_X, -)$ often fails to be exact. Is this only the case for the unenriched hom (...
- 19.6k
3
votes
1
answer
466
views
Finding global sections of a sheaf of sets using (some kind of) sheaf cohomology?
Let $X$ be a compact manifold, say, and $G$ a Lie group, and $H$ a closed Lie subgroup such that $M \cong G/H$ is a homogeneous space. (For my purposes, $X$ and $M$ would be a smooth projective ...
- 1,763
3
votes
1
answer
513
views
Proof without sieves: Equivalent grothendieck topologies have the same sheaves
I'm currently learning about sheaf theory with Angelo Vistoli’s 2007 Notes on Grothendieck topologies,
fibered categories and descent theory. And in page 35, there is the following definition of a ...
- 119
3
votes
1
answer
225
views
Subspace inclusion with non-vanishing higher direct images
I'm looking for concrete topological intuition for the derived pushforward.
Let $f:X\to Y$ be a continuous map. The derived pushforward $\mathbf Rf_\ast$ takes a sheaf $F$ to the sheafification of ...
- 10.5k
3
votes
3
answers
348
views
Constants sheaves on an open subset
Let $X$ be a topological space and $U\subset X$ an open subset. Let's work in the category of sheaves of abelian groups on $X$. Consider the constant sheaf on $U$, $\mathbb{Z}_U$, given by $\mathbb{Z}...
- 15.2k
3
votes
1
answer
260
views
Etale cohomology of relative elliptic curve
Let $E_a: y^2 = x(x-1)(x-a)$ be a smooth proper relative elliptic curve over $\text{Spec}(A)$, with $a\in A$, and assume $\text{Spec}(A)$ is a $\text{Spec}(\mathbb{Q}_p)$-scheme.
Let $R^1f_*\mathbb{Q}...
- 2,907
3
votes
1
answer
212
views
Reference for original Leray definition of a sheaf
Leray originally defined sheaves over closed sets. Is there any easily readable (i.e. obtainable through the Internet and written in English) reference that explicitly states the definition using ...
3
votes
1
answer
147
views
What do we know about a sheaf $M$ if we know its derived fibers $\mathsf{L}x^* M$, for $x\in X(k)$?
Let $X$ be a scheme over a field $k$. (Feel free to assume that $X$ is an algebraic variety, if needed.) Also, let $M^\bullet$ be a complex in the derived category of quasi-coherent sheaves $\mathsf{D}...
- 711
3
votes
1
answer
201
views
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 ...
- 1,802
3
votes
1
answer
249
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}}\...
- 481
3
votes
1
answer
225
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 ...
- 33
3
votes
1
answer
331
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:\...
- 1,805
3
votes
1
answer
549
views
Question about adjoint of forgetful functor from condensed abelian groups to condensed sets
There is a forgetful functor from condensed abelian groups to condensed sets. According to Scholze's notes, this has an adjoint $T \mapsto \mathbb{Z}[T]$ (which is the sheafification of the functor ...
3
votes
1
answer
428
views
What sort of object represents skyscaper sheaves on the etale site of $\mathbb{Z}_p$?
By SGA 4 IX Proposition 2.7, any constructible sheaf $\mathcal{F}$ on a qcqs scheme $X$ can be represented as an equalizer of two etale maps between representable (by schemes) sheaves. This would ...
- 15.4k
3
votes
1
answer
901
views
Cohomological Brauer group vs classical
Let $X$ be a smooth scheme over $\mathbb{C}$.
A $O_X$-algebra $A$ is called Azumaya algebra on $X$
if locally it's ismorphic to matrix algebra: ie for
every $p \in X$ there exist open $U \subset X$ ...
- 6,038
3
votes
1
answer
382
views
Is the perfection (perfect closure) presheaf a sheaf?
The perfection of a ring $A$ of prime characteristic $p$ is the perfect ring $A_\rm{pf}=$ lim{$A\to A\to ...$} where all maps are Frobenius. It does not commute with products, as was shown by YCor in ...
- 533
3
votes
1
answer
515
views
For what kind of sheaves can we always extend a sheaf map from a closed subset to the whole space?
Let $X$ be a topological space. We know that a sheaf on $X$ is call soft if for any closed subset $Z$ of $X$, a section on $Z$ can be always extend to a section on $X$.
Now we consider a similar ...
- 9,019
3
votes
3
answers
600
views
A question on flasque sheaf
Let $0\to \mathscr{F}'\to\mathscr{F}\to\mathscr{F}''\to 0$ be an exact sequene of sheaves. It is well known that $\mathscr{F}$ flasque iff $\mathscr{F}''$ flasque provided $\mathscr{F}'$ is flasque. ...
- 153