Questions tagged [sheaf-theory]
For questions about sheaves on a topological space.
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Devissage lemma (Mumford's & Oda's AG II)
This question is part II of my proof reading of Lemma of devissage
from Mumford's & Oda's Algebraic Geometry II, findable on page 81; Theorem 6.12:
Theorem 6.12 (“Lemma of devissage”). Let $K$...
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Coherent cohomological dimension and affine morphisms
For simplicity, all varieties in this question are quasiprojective varieties over an algebraically closed field of characteristic $0$.
The coherent cohomological dimension $cd(X)$ of a variety $X$ is ...
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$G$-torsor for topological space compared to that for sheaf of groups
I just read about the definitions about torsor of sheaf of groups and get a bit confused.
How does the notion of $G$-torsor for a topological space compared to that of a sheaf of groups? Is there a ...
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Donaldson-Thomas Theory and "Quantum Foam" for Mathematicians
Let $X$ be a smooth, projective Calabi-Yau threefold. From an algebro-geometric perspective, the Donaldson-Thomas invariants $\text{DT}_{\beta, n}(X)$ are virtual counts of ideal sheaves on $X$ with ...
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Image of Obstruction Map for Relative Quot-scheme
Let $f: X \to S$ be a projective morphisms between finite-type projective schemes, and $\mathcal{O}_{X}(1)$ an $f$-ample line bundle. Given an $S$-flat coherent $\mathcal{O}_{X}$-module $\mathcal{H}$ ...
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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 $...
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Checking $\mathbb{K}_{U\times(a,b)}\ast\mathbb{K}_{[0,\infty)}\simeq \mathbb{K}_{U\times[a,\infty)}[-1]$ in derived category $D(X\times\mathbb{R})$
Let $D(X\times\mathbb{R})$ be the derived category of sheaves of $\mathbb{K}$-vector spaces on a smooth manifold $X\times\mathbb{R}$ where $\mathbb{K}$ is a ground field. Let $p_1:X\times\mathbb{R}\...
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Local question and descent category for a quasi-coherent sheaf on $\mathbb{G}_m$-gerbe
Update: I removed what I thought was unecessary and tried to be more straightforward in the hope to get an answer.
Context:
Suppose I have a $\mathbb{G}_m$-gerbe $\mathcal{G}$ over a scheme $X$ with ...
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What is the logical progression in algebraic tools for studying spaces (varieties -> schemes, sheaves, topos etc.)?
Some algebraists (Cartier, Weil, Atiyah, etc.) sometimes speak of geometry as a long history of essentially asking the same question—"what is space, and how would one describe a space uniquely". ...
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Computing injective resolution of some constant sheaves
I follow the notations on "Sheaves on manifolds" written by Kashiwara-Schapira. Let $\mathbb{K}$ be a ground field and $X$ be a smooth manifold. Let $D(X)$ be the derived category of sheaves of $\...
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Direct image functor commuting with infinite direct sum of sheaves
Normally I would think this kind of question doesn't belong on overflow, but I haven't been able to find an answer anywhere else, so perhaps it is not so trivial.
Let $f: X \rightarrow Y$ be a ...
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Direct sums of invertible sheaves commuting with global sections and the functor of points approach
I am looking at the Stacks Project's treatment of the functor of points for projective space.
Let's restrict to the case that $S$ is a graded ring, generated by $S_{1}$ as an $S_{0}$ algebra. The ...
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Does the 1-category construction of a topos of presheaves extend to the 2-Category of Groupoids?
In the case of 1-categories, we know there is a functor category
$PSh(C):=[C^{op},Set]$, where $C$ is a small category,
and this functor category is a topos. I am hoping this will extend to the case ...
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Purity and skyscraper sheaves
In "The Geometry of moduli spaces of sheaves" a coherent sheaf $\mathcal{F}$ is defined to be pure of dimension $d$ if dim$(\mathcal{E})=d$ for all non-trivial proper subsheaves $\mathcal{E} \subset \...
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When is derived category of ringed space perfectly generated?
Let $(X,\mathcal{O})$ be a ringed space. Also assume that $X$ is nice, e.g. locally compact, Hausdorff, some type of finite dimension, ...
We can then consider $\mathcal{D}(\mathcal{O}\text{-}Mod)$. ...
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Is it always possible to write a derived manifold (in the sense of Spivak) as a homotopy colimit of principal derived manifolds?
Is it always possible to write a derived manifold as a homotopy colimit of principal derived manifolds (i.e. zero sets of smooth functions)? This is true for schemes and derived schemes, so it seems ...
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Why mu-stratifications?
In the microlocal theory of sheaves developed by Kashiwara and Schapira, there is the notion of a $\mu$-stratification, which is a stratification satisfying a stronger property ("$\mu$") than Whitney'...
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Restriction of Ext sheaves on closed subschemes
Let $f:X\rightarrow C$ be a morphism, where $C$ is a smooth curve. For $t\in C$ let $i_t:X_t = f^{-1}(t)\rightarrow X$ be the inclusion of the fiber of $f$ over $t$, and let $\mathcal{F}$ a coherent ...
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Equivariant sheafs and $G$ actions on modules
I am reading Simpson's paper on The Hodge filtration on nonabelian cohomology. In particular Chapter 5 (p.24) and I am confused about the notion of a group acting on an equivariant sheaf.
The set up ...
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Cokernel of section of a general coherent sheaf
Given a scheme $X$ and an $\mathcal{O}_{X}$-module $\mathscr{E}$, we know that a section $s \in H^{0}(X, \mathscr{E})$ is equivalent to a morphism $s :\mathcal{O}_{X} \to \mathscr{E}$. It is the ...
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On a weak notion of sheaves on topological spaces
First of all, I give my definition of weak sheaves:
By a weak sheaf on a topological space $ X $, we mean a presheaf
$F$ such that for all open covering $\{ U_i\}_{i\in I} $ of $X$ sheaf
...
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Homotopy Colimit of Čech Complex
I am studying homotopical cosheaves, and I came up with the following "conjecture".
We can see an "additive" precosheaf in chain complexes (such that corestrictions do not commute on the nose) as a ...
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Dual Abelian scheme (relative Picard functor) vs Ext sheaf
Let $A$ be an abelian scheme over some base scheme $S$.
Let $A^\vee$ be the dual abelian scheme, defined as $\text{Pic}^0_{A/S}$ where $\text{Pic}_{A/S}(T)=\text{Pic}(A_T)/\text{Pic}(A)$. (maybe some ...
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Vanishing of higher direct image of finite morphisms relative to the fppf topology
Let $f:X \to Y$ be a finite morphism of schemes.
Let $\mathcal{F}$ be a sheaf of abelian groups on the the etale site of $X$ then we know that $R^{i}f_{*} \mathcal{F} = 0$. Is this statement also true ...
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Quasi-coherent sheaves on affinoid space
From Conrad's notes on rigid geometry:
More specifically, Gabber has given an example of a sheaf of modules $F$ on the closed unit disk $B^1$ such that $F$ is locally a direct limit of coherent ...
CommunityBot
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Six Functor Formalism for Closed Inclusion
Let $X$ be a (finite-dimensional) topological manifold and suppose $i: Z \to X$ is the inclusion of a closed subspace. Then there are several derived functors on derived categories of sheaves of ...
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Equivalence of ordered and unordered cech cohomology.
Given a topological space X and a finite cover X = $\cup X_i$, one can define Cech cohomology of a sheaf of abelian groups F with respect to the cover $\{X_i\}$ in two different ways:
(Ordered): ...
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Examples of non-hypercomplete sheaves on affine schemes
Let $A$ be a commutative ring and let $\mathcal{O}$ be a sheaf of $E_{\infty}$-ring spectra on $\mathrm{Spec} A$ such that $\pi_0\mathcal{O} = \mathcal{O}_{\mathrm{Spec} A}$. Lurie provides a ...
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Any vector bundle can be twisted to have sections
Let $X$ be an integral scheme proper over $\mathbb{C}$. Let $V$ be a locally free $\mathcal{O}_X$-module of positive finite rank. Does there necessarily exist a locally free $\mathcal{O}_X$-module of ...
CommunityBot
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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 $\mathrm{Hom}(F, −):Qco(X)\rightarrow Ab$ ...
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Sheaf-type property for Derived Categories?
Suppose $X$ is a finite dimensional complex space (I'm happy to restrict to $X$ being a scheme of finite type over $\mathbb C$ as well). I'm wondering if the following sheaf-like properties hold for ...
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Mistake in Hartshorne's Exercise II.1.1?
This is really an elementary question, but let me state it. Exercise 1.1 of the second Chapter of Hartshorne's Algebraic Geometry ask to prove that the sheaf associated to the presheaf sending every ...
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Mathematical background required to learn about sheaves
Due to my interest in type theory (and higher type theory), I have found that learning about sheaves might be useful (for, e.g., sheaf models of type theories). There is Kashiwara and Schapira's ...
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Coherent locally free sheaves on projective varieties
Let $k$ be a field, $X$ be a connected smooth projective $k$-scheme. Let $f:X\rightarrow X$ be a finite $k$-morphism that is surjective on the underlying topological spaces. Suppose $f$ has degree $n$....
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Ext sheaves as extension by zero of locally free sheaves
Let $X$ be a complex projective manifold and let $\phi \colon E \hookrightarrow F$ an injection of locally free sheaves. Then we have a sequence of coherent sheaves
$$
0 \to E \to F \to F/E \to 0
$$
...
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Obstructions to abelian sheaf being quasi-coherent
Let $X$ be a Noetherian spectral topological space. A necessary condition for an abelian sheaf on $X$ to be quasi-coherent with respect to some affine scheme structure on $X$ is that its higher ...
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Affine scheme on spec(A) of a ring A as the sheafification of a pre-sheave on spec(A)?
It is obvious that there is a parallel between the definition of structure sheaf of $\operatorname{Spec}(A)$
versus the sheafification of a pre-sheaf.
The definition of the sheaf $\mathscr F^+$ ...
CommunityBot
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The (co)tangent sheaf of a topological space
Let $X$ be a topological space (assume additional assumptions if needed) and denote by $\mathcal O _X$ its sheaf of $\Bbbk$-valued continuous functions where $\Bbbk$ is $\mathbb{R}$ or $\mathbb{C}$ ...
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Equivariant Sheaf: Explanation on Stalks
I have a question about the explanation of the data defining a so called equivariant sheaf $F$ on a scheme X from wiki: https://en.wikipedia.org/wiki/Equivariant_sheaf. Let denote by $\sigma: G \...
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Ordinal-valued sheaves as internal ordinals
Let $X$ be a topological space (feel free to add some separation axioms like “completely regular” if they help in answering the questions). Let $\alpha$ be an ordinal, identified as usual with $\{\...
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A coherent sheaf is a vector bundle over subvariety?
Over a complex manifold, can every coherent sheaf be seen as a holomorphic vector bundle over an analytic subset?
Thanks in advance.
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Simplicial Sheaves?
I recently was wondering if there was a name for sheaves which were locally constant on the open simplexes in a simplicial complex. After some googling I stumbled across simplicial sheaves. I am ...
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Cohomology of tangent sheaf of a hypersurface
Let $X\subset\mathbb{P}^n$ be an irreducible and reduced hypersurface of degree $d$. How can one explicitly compute the dimension of the vector spaces $H^0(X,T_X),H^1(X,T_X),H^2(X,T_X)$? Here $T_X$ is ...
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Sheaves on Rectifiable Sets
Basic question: are there (co)homological or sheaf-based tools which might be useful in geometric measure theory?
Background: The jumping off point here is a simple analogy - geometric measure ...
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On the degrees of the normal and tangent sheaves of Gorenstein curves in Pn
It is known that if an irreducible curve $C$ is a local complete intersection in $\mathbb{P}^n$, then
$\wedge^{n-1}\mathcal{N}\otimes\omega_{\mathbb{P}^n}$ is isomomorphic to the dualizing sheaf $\...
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What is an example of a cokernel $B/\phi(A)$ in group schemes which does not have $A=\mu_d$ and requires the fppf topology to be a sheaf?
Let $S$ be affine. A bit of background: Let us think of $S$-group schemes as abelian sheaves over a given site (etale, Zariski, fppf, etc). When we take a cokernel of a morphism $\phi$ this category: $...
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On computability of sheafification
The question will feature some imprecise words but I believe that an expert could parse it to a precise question more or less uniquely.
Assume I have a reasonable topological space (say the ...
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Sheafification map is surjective
This is not a research level problem for sure. But, similar question was asked by some one else $2$ years back on Stack exchange has not received any attention. So, I thought it does not suit there....
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Differential forms as a sheaf on the site of all manifolds vs. sheaf on an individual manifold
The de Rham complex can be viewed as a sheaf $\Omega$ on the entire site $\mathsf{Man}$ of smooth manifolds via the usual pullback of differential forms $(f: M \to N) \mapsto (f^*: \Omega(N) \to \...
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If presheaf is zero on a covering is the sheaf zero?
Let $C$ be a site and $F$ an abelian presheaf on $C$. Suppose that for each object $U$ in $C$ there is a covering $\{ U_i\to U \}$ such that $F(U_i)=0$. Is it true that $F^{sh}=0$?
This should be ...
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