# Questions tagged [sheaf-theory]

For questions about sheaves on a topological space.

894
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### Quotient sheaf obtained from a quotient of $\mathrm{SL}_2$ having a section

$\DeclareMathOperator\SL{SL}$Let $K$ be a field and let $G$ be a $K$-defined, closed, algebraic subgroup of $\SL_2$. Denote by $\mathcal{C}$ the site whose objects are $K$-schemes, with your favorite $...

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### Sheaf theory in TDA

I was wondering wether anyone had any examples as to why it more useful to consider a sheaf theory approach to TDA problems.
I am familiar with some of the benefits of using cellular cosheaves to ...

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### Who introduced the notion of ringed spaces?

My question is very concise, please forgive it.
Who introduced the concept of ringed space?
My first try would be that they were introduced by Cartan in his study of analytic functions with sheaves. ...

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### Cellular (co)Sheaves and applications

In the last few years there have been efforts made to generalise the theory of peristence homology and cohomology to deal with sheaves for example Russold - Persistent sheaf cohomology. This as i ...

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### Question of deforming a sheaf

Consider over $\mathbb C$. Let $Artin/\mathbb C$ denote the category whose objects are Artinian local $\mathbb C$-algebras with residue field $\mathbb C$, and morphisms are local ring maps preserving ...

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### not quite the sheaf condition

Let $Sets$ be the category of finite sets and all maps. I have come accross several example of functors $F : Sets^{op} \to Sets$ which satisfy the condition below:
-- There exists an integer $k$ such ...

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### Do we have $\underline{\operatorname{Ext}}^i_\text{fppf}(\widehat{\mathbb{G}}_a,\mathbb{G}_m)=0$ for $i>0$?

Let $k$ be a characteristic zero field and consider the category $(\mathsf{Sch}/k)_\text{fppf}$ of schemes over $k$ with the fppf topology. I know that $\underline{\operatorname{Hom}}(\mathbb{G}_a,\...

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### Do we have $\underline{\operatorname{Ext}}^i_\text{fppf}(\mathbb{G}_a,\mathbb{G}_m)=0$ for $i>0$?

Let $k$ be a characteristic zero field and consider the category $(\mathsf{Sch}/k)_\text{fppf}$ of schemes over $k$ with the fppf topology. I know that $\underline{\operatorname{Hom}}(\mathbb{G}_a,\...

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### Unifying categorical equivalences and dualities for the functors : Gelfand spectrum and Zariski spectrum, Structural sheaf and Continuous sections

I would be very grateful for any references I might be led to, from a categorical point of view for the functors:
$\textsf{Spec}_{\mathscr{Z}\textrm{arisky}}(-)$, related to $\mathcal{O}(-)$, which ...

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### Canonical product in sheaf cohomology

EDIT: Let $\mathcal{F},\mathcal{G}$ be sheaves of abelian groups on a topological space $X$. Then there exists a canonical cup product
$$H^i(X,\mathcal{F})\otimes_\mathbb{Z}H^j(X,\mathcal{G})\to H^{i+...

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### Global sections of relative characteristic of log-smooth curves

$\DeclareMathOperator\Spec{Spec}$I am currently learning about log-geometry and try to understand the theory in the example of curves with basic log-structure over a general base $S$. Especially, I am ...

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### When morphism of complexes is homotopic to 0?

Let $f\colon A^\bullet\to I^\bullet$ be a morphism of bounded below complexes in an abelian category. Assume all $I^i$ are injective objects. Assume also that $f$ induces the zero map on cohomology.
...

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### Geometric interpretation of sheaf cohomology

Please forgive me for the informal and naïve nature of my question, as I am a beginner in algebraic geometry.
In the famous book by Hartshorne, sheaf cohomology is defined as a certain derived functor....

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### References on principal $\mathbf{C}$-bundles, where $\mathbf{C}$ is a category?

Is there any treatment on principal "categorical" bundles - principal $\mathbf{C}$-bundles, where $\mathbf{C}$ is some (topological) category?
I know that one can define "categorical ...

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### flatness of restriction of structure sheaf over ring of global sections

Let $X$ be an affine scheme. $U \subseteq X$ open. Then I want to show that $\mathcal{O}_X(U)$ is flat over $\mathcal{O}_X(X)$.
But I want to prove it only by knowing the definition of structure sheaf ...

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### Is this double quotient of $\operatorname{SL}_2$ representable by an algebraic space or a scheme?

$\DeclareMathOperator\SL{SL}$Let $B$ be a Borel subgroup (upper triangular matrices), and let $\Gamma := \langle \sigma\rangle$ be the group generated by a (hyperbolic) element of $\SL_2/\mathbb{Q}_p$ ...

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### Bijective map between two definition for extension for sheaf on base

Let $F$ be sheaf of sets on base $\mathscr B =${$B_i,i \in I$}, there are two equivalent definitions for the extention for $F$, denoted by $\mathscr F $:
Def1:$\mathscr F(U) $ := {$(f_p \in F_p)_{p \...

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### Recommendations for getting into sheaves with emphasis on differential geometry and algebraic topology

I want to study the theory of sheaves from a categorical point of view with an emphasis on applications in algebraic topology and differential geometry and I'm looking for a good introductory book to ...

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### Unit of the inverse/direct image sheaf adjunction in terms of étale spaces

$\def\sF{\mathcal{F}}
\def\sect{\operatorname{Sect}}$[I am reposting here this question from MSE, since thus far I received no answers there and maybe here I attract the attention of other people that ...

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### Subobject classifier for sheaves on large sites with WISC

Let $\mathsf{C}$ be a possibly large category with a Grothendieck topology satisfying the Weakly Initial Set of Covers condition: there is for each $X$ a set (not a proper class) of covering families ...

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### What is $\text{Hom}(\mathcal{F}\times\mathcal{G}, \mathbb{A}^1)$?

Let $S$ be an affine scheme and let $\text{Aff}(S)$ be the site of affine $S$-schemes. Let $\mathcal{F}$ and $\mathcal{G}$ be a pair of sheaves on $\text{Aff}(S)$, and let $\mathbb{A}^1_S$ be the ...

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### 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))\...

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### Why does the associated sheaf vanish?

I am learning local cohomology from Hartshorne’s book Local Cohomology.
My question is about understanding a line in the proof of proposition 1.11 in this book.
The set-up for proposition 1.11 is that ...

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### Push-forward of a locally constant sheaf using two homotopic maps

Let $X,Y$ be compact smooth manifolds. Let $f,g\colon X\to Y$ be smooth submersions
(in particular, locally trivial bundles) which are homotopic to each other (in the class of smooth maps, not ...

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### The stack $\operatorname{GL}_2/B$

Let $F$ be the functor from the category of affinoid Tate algebras over $\mathbb{Q}_p$ to the category $\mathrm{Sets}$, which maps an affinoid $\operatorname{Spm} R$ to the set of orbits $\...

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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 $H$ ...

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### Using higher topos theory to study Cech cohomology

It seems to me that many classical statements about Cech cohomology should without much effort follow from Lurie's HTT, but I am struggling to fill out the details. I want to show that, given an ...

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### Understanding the Lagrangian Lift of a given isotopy

I was looking at the paper "Sheaf quantization of Hamiltonian isotopies
and applications to non-displaceability
problems" by Guillermou-Kashiwara-Schapira(GKS). So firstly, GKS doesn't ...

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### About the support of a holonomic D-module

Let $X$ be a smooth algebraic variety over $\mathbb{C}$ and let $M^\bullet\in\mathsf{D}^b_\text{h}(\mathcal{D}_X)$ be a complex of D-modules with holonomic cohomologies. We define the support of $M^\...

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### "Quasi-coherent" vector spaces in Sch/S

$\DeclareMathOperator\Vec{Vec}\newcommand\Sch{\mathrm{Sch}}\DeclareMathOperator\Hom{Hom}$Let $S$ be a base scheme. Let me write $\Vec(S)$ to denote the category of $\mathbb A_S$-vector space objects ...

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### Singular cohomology to cohomology of quasi-coherent sheaf

Let $X$ be a projective nonsingular variety (integral Noetherian scheme of finite type that is proper over a field $k=\overline{k}$ such that $\Omega^1_X$ is locally free). Suppose one knows the ...

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### The site and the space

There is a (seemingly simple) statement in the literature on sheaf theory, namely,
If $E$ is the site of opens of a topological space $X$, the notion of sheaf over $X$ coincides with that of sheaf of ...

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### Isbell duality between algebras and sheaves

nLab says on Isbell duality, the following:
A general abstract adjunction
$(\mathcal{O} \dashv \operatorname{Spec}) : \mathrm{CoPresheaves} \leftrightarrows \mathrm{Presheaves}$
relates (higher) ...

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### Do the covariant maps of a sheaf with transfer automatically satisfy a dual gluing axiom?

I'm interested in describing a notion of equivariant sheaves that uses Mackey functors on topoi. We can describe a genuine G-spectrum as a spectral Mackey functor on the topos of finite G-sets. If we ...

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### What is the functor of points of the moduli scheme of stable sheaves?

Let $\Bbbk$ be an algebraic closed field of characteristic zero. Let $(\mathrm{Sch}/\Bbbk$ denote the category of locally Noetherian schemes. Let $B$ be a projective scheme over $\Bbbk$. Let $L$ be an ...

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### Subsheaves of Spec K, K a field

$\DeclareMathOperator\Spec{Spec}\newcommand\Ring{\mathrm{Ring}}\newcommand\op{^\text{op}}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Sh{Sh}$In the category of schemes the objects of the form $\...

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### 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}...

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### Basic question on projective bundles

Let $\mathcal{E}$ be a coherent sheaf on an irreducible scheme $S$ ($S$ can be pretty nice, say noetherian of finite type), and let $\mathbf{P}(\mathcal{E}):=\mathrm{Proj}(\mathrm{Sym}(\mathcal{E}))$ ...

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### Canonicity in split sequence in cotangent spaces

Let $X$ be a locally ringed space. We have for a point $p$ the exact sequence
$$0\to \mathfrak{m}_p^2\to \mathfrak{m}_p\to \mathfrak{m}_p/\mathfrak{m}_p^2 \to 0$$
where $\mathfrak{m}_p$ is the maximal ...

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### Resolutions of unbounded complexes: Condition ($\ast$) in Spaltenstein's paper

In the paper "Resolutions of unbounded complexes" (Compositio Math., vol. 65, no. 2, pp. 121-154) N. Spaltenstein generalizes the 6 functor formalism to unbounded complexes of sheaves over ...

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### When the sheaf of principal parts is reflexive?

Following EGA IV, we can construct the sheaf of principal parts of a sheaf $\mathcal{E}$ over a scheme $X$ by $\mathcal{P}_X^n(\mathcal{E}) = \mathcal{P}_X^n \otimes \mathcal{E}$, where $\otimes$ is ...

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### Does the (Vistoli-)sheafification induce isomorphism?

Given a presheaf, in Angelo Vistoli's 2007 Notes on Grothendieck topologies,
fibered categories
and descent theory
there is a construction of the sheafification (Proof for theorem 2.64).
Note: In ...

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### On the generalization of a Cech-to-sheaf type spectral sequence

Let $(X,O_{X})$ be a ringed space and $\mathcal{F^{\bullet}}$ be a bounded below complex of $O_{X}$-modules on $(X,O_{X})$. On [Stacks Project 20.25.1][1] it is shown that there is a weakly convergent ...

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### Norm of sections on $T$-invariant subspace of a homogenous space over $\mathbb{Q}_p$

Let $K=\mathbb{Q}_p$ and $G$ a split reductive group over $K$ with split maximal torus $T$. Furthermore, $X$ is a smooth projective variety over $K$ with a free $G$-action. Let $U \subset X$ be a $T$-...

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### Is the Grothendieck topology equivalent to its Singleton Grothendieck topology?

I'm using the definition of a Grothendieck topology in Angelo Vistoli’s 2007 Notes on Grothendieck topologies,
fibered categories
and descent theory and I found on nLab about superextensive site, that ...

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### Is there a Čech-like way of computing $H^\bullet(X,M^\bullet)$ or even $\mathsf{R}f_* M^\bullet$?

Let $X$ be a topological space (or a site) and let $M$ be a sheaf on $X$. If $X$ is paracompact, or if $X$ is a noetherian separated scheme and $M$ is quasi-coherent, or if $X$ is quasi-projective ...

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### Some question about (semi-)stable sheaves

Let $X$ be a projective normal variety over $\mathbb C$, I have several questions about semi-stable sheaves:
Question 1. Suppose that $E$ is a pure sheaf such that $HN_*(E)$ is the Harder-Narasimhan ...

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### Is the determinant line bundle of a coherent sheaf functorial (between sheaves of the same rank)?

The determinant line bundle of a coherent sheaf $\mathcal{F}$ on an $n$-dimensional (smooth) analytic space is defined as
\begin{equation}
\det \mathcal{F} := \bigotimes_i^n (\det \mathcal{E}_i)^{⊗...

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### Joins of (closed) subschemes and Zariski-local Z-functors

$\newcommand\Aff{\mathrm{Aff}}\newcommand\cRing{\mathrm{cRing}}\newcommand\Sch{\mathrm{Sch}}$Equip $\Aff = \cRing^\text{op}$ with the Zariski Grothendieck-topology. There are nested categories:
$$\Aff\...

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### Is $\mathbb{C}^*$ not irreducible, or is every locally constant sheaf on $\mathbb{C}^*$ constant?

I am running into contradiction from the following set of definitions, propositions, and assumptions. Can anyone spot where I'm off?
Definition A sheaf $\mathcal{F}$ on a topological space $X$ is ...