Questions tagged [sheaf-theory]

For questions about sheaves on a topological space.

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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+...
asv's user avatar
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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 ...
Matthias's user avatar
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-1 votes
1 answer
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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. ...
asv's user avatar
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9 votes
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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....
George's user avatar
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5 votes
2 answers
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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 ...
Tanny Sieben's user avatar
1 vote
1 answer
179 views

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 ...
Hamed Khalilian's user avatar
2 votes
0 answers
168 views

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 $\SL_2/\mathbb{Q}_p$ ...
kindasorta's user avatar
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11 votes
3 answers
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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 ...
Ofek Aman's user avatar
  • 111
6 votes
1 answer
242 views

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 ...
Rylee Lyman's user avatar
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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 ...
kindasorta's user avatar
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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))\...
kindasorta's user avatar
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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 ...
Boris's user avatar
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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 ...
asv's user avatar
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133 views

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 $\...
kindasorta's user avatar
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3 votes
1 answer
288 views

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 $H$ ...
Puzzled's user avatar
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9 votes
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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 ...
Markus Zetto's user avatar
2 votes
1 answer
201 views

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^\...
Gabriel's user avatar
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4 votes
3 answers
477 views

"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 ...
Nico's user avatar
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0 answers
191 views

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 ...
locally trivial's user avatar
3 votes
0 answers
178 views

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 ...
user234212323's user avatar
10 votes
0 answers
494 views

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) ...
Ilk's user avatar
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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 ...
Doron Grossman-Naples's user avatar
1 vote
0 answers
73 views

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 ...
Display Name's user avatar
6 votes
1 answer
440 views

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 $\...
Nico's user avatar
  • 775
3 votes
1 answer
132 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}...
Gabriel's user avatar
  • 943
4 votes
2 answers
596 views

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 $\mathbf{P}(\mathcal{E}):=\mathrm{Proj}(\mathrm{Sym}(\mathcal{E}))$ ...
rfauffar's user avatar
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109 views

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 ...
Arturo's user avatar
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11 votes
1 answer
374 views

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 ...
algori's user avatar
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3 votes
0 answers
140 views

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 ...
gabriel fazoli's user avatar
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1 answer
169 views

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 ...
Muster Maxfrau's user avatar
2 votes
0 answers
128 views

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 ...
Fernando Peña Vázquez's user avatar
2 votes
0 answers
115 views

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 $T$-...
KKD's user avatar
  • 463
3 votes
0 answers
157 views

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 ...
Muster Maxfrau's user avatar
4 votes
2 answers
220 views

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 ...
Gabriel's user avatar
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1 vote
1 answer
202 views

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 ...
Li Yutong's user avatar
  • 3,362
4 votes
1 answer
407 views

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)^{⊗...
Carlos Esparza's user avatar
1 vote
0 answers
103 views

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\...
Nico's user avatar
  • 775
0 votes
0 answers
199 views

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 ...
locally trivial's user avatar
2 votes
1 answer
214 views

Dualizing complex of the cone over a manifold

Let $M$ be a smooth (or just topological) closed manifold. Let $C(M)$ denote the cone over $M$, i.e. $C(M)$ equals to $M\times [0,\infty)$ with $M\times \{0\}$ contracted to a point. The image of $M\...
asv's user avatar
  • 21.1k
1 vote
1 answer
186 views

Concrete sheaves

On the nLab, given a local $S$-topos $E$, a concrete sheaf is defined as an object that is separated with respect to the local isomorphisms (the morphisms that are inverted by the global sections ...
NDewolf's user avatar
  • 193
1 vote
1 answer
186 views

Chern class of torsion sheaf support on a point

Let $X$ be a smooth projective surface. Let $p$ be a closed point of $X$. Let $k(p)$ be the corresponding skyscraper sheaf, then actually we could use Grothendieck-Riemann-Roch to calculate the Chern ...
Mike's user avatar
  • 165
3 votes
0 answers
137 views

Johnstone's Elephant - Lemma C2.1.7 confusion

I don't understand the proof of (ii) in the Johnstone's Elephant: Lemma 2.1.6 is: Now consider $\bigcup_{f \in R} f \circ f^*(S)$. This is my notation for the sieve Johnstone references in the proof ...
Emilio Minichiello's user avatar
1 vote
0 answers
91 views

NSR superstring as a map of supermanifolds

On one hand, I know that the NSR superstring is described by a map $\Phi: \Sigma \to X$, where $\Sigma$ is a supermanifold with local coordinates $(\sigma,\theta)=(\sigma^0,\sigma^1 | \bar{\theta},\...
Alec's user avatar
  • 11
16 votes
1 answer
431 views

Zorn's lemma for Grothendieck sites

In every treatment of Grothendieck sites I can find, flasque sheaves are not defined in the way one would naïvely expect from ordinary sheaf cohomology; namely instead of saying that "restriction ...
cat man's user avatar
  • 163
19 votes
2 answers
369 views

Why is $1$ not a dense sub-site in a group with the trivial Grothendieck topology?

A friend of mine had the following question while reading the section "C2.2 The topos of sheaves" in "Sketches of an Elephant". Let $G$ be a group (considered as a category with ...
Arshak Aivazian's user avatar
3 votes
1 answer
427 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 ...
Muster Maxfrau's user avatar
1 vote
0 answers
84 views

Tensoring by a soft flat sheaf

Let $X$ be a paracompact Hausdorff space and $R$ a commutative ring. All sheaves below will be sheaves of $R$-modules on $X$. A sheaf $S$ is soft if every section of $S$ over a closed subset can be ...
algori's user avatar
  • 23.2k
6 votes
0 answers
208 views

Is the right adjoint to presheaf direct image interesting?

Let $X\overset{f}{\to}Y$ be a continuous map. It induces on presheaves a classical adjunction inverse image ⊣ direct image. However, the direct image functor has a further right adjoint, defined by ...
Arrow's user avatar
  • 10.3k
6 votes
1 answer
428 views

Where can I find a definition of $\underline{H}^p(X, \mathscr{F})$?

Let $X$ be a topological space and $\mathscr{F}$ a sheaf on $X$. In the paper Tropical cycle classes for non-archimedean spaces and weight decomposition of de Rham cohomology sheaves by Yifeng Liu, ...
Jakob Werner's user avatar
  • 1,093
3 votes
1 answer
219 views

Compatibility of pullbacks with an equivalence relation

This question was originally posted last week in Math Stack Exchange (see here). I'm currently working on the proof of the existence of the sheafification in Angelo Vistoli’s 2007 Notes on ...
Muster Maxfrau's user avatar

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