Questions tagged [sheaf-theory]

For questions about sheaves on a topological space.

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1 vote
2 answers
599 views

Methods of sheaf theory for solving Diophantine equations

What are some examples of sheaf theory used to either provide solutions to Diophantine equations, or to state that no such solutions exist?
4 votes
2 answers
834 views

Singular support of an irreducible perverse sheaf

I was studying Sheaves on Manifolds by Kashiwara and Schapira, and while the singular support seems like a complicated invariant I cannot seem to find a counterexample to the following: Let $X$ be a ...
13 votes
1 answer
602 views

How strong a set theory is necessary for practical purposes in sheaf theory?

Is it known how much of ZFC is actually necessary for the basic, familiar constructions and theorems in sheaf theory, along the lines of section II.1 (and its exercises) in Hartshorne's "Algebraic ...
3 votes
0 answers
80 views

Embedding abelian categories into abelian sheaves

The Yoneda functor from an abelian category into sheaves of abelian groups is shown to be exact in The Stacks Project, Lemma 19.9.2. I like this proof because it is constructive and it doesn't use ...
12 votes
0 answers
494 views

Is the morphism of sheaves $(R \mapsto GL(R((h)))) \rightarrow (R \mapsto PGL(R((h))))$ surjective in Zariski topology?

Consider two functors given by $R \mapsto GL(R((h)))$ and $R \mapsto PGL(R((h)))$ for a ring $R$. It is easy to see that these functors are sheaves in Zariski topology (in fact for any affine variety $...
2 votes
2 answers
473 views

What is the geometric description of the set of isomorphism class of $G$-torsors over a site $C$?

Let $X$ be a topological space and $G$ be a topological group. Let $\tilde{G}$ be the sheaf of groups defined by the sheaf of sections of the product $G$ bundle $ \pi_1:X\times G \rightarrow X $. (1)...
5 votes
1 answer
452 views

Residue of the canonical sheaf along subvariety

Let $S$ be a smooth projective surface over an algebraically closed field $k$ and $C \subset S$ a singular curve. Let us denote by $K_S$ the class of canonical divisor of $S$ and $\mathcal{O}(K_S)$ ...
18 votes
0 answers
440 views

Is there a model category describing shape theory?

Is there a nice model structure on some category of topological spaces compatible with shape theory? In particular, weak equivalences should induce isomorphisms on sheaf cohomology. As an example, ...
4 votes
0 answers
119 views

Understanding a step in proof of sheaf version Verdier duality

Warning: This question is likely low-level for MathOverflow. My apology that there is almost surely something basic I miss. So all proofs I can find factors through a particular statement, which goes ...
2 votes
0 answers
136 views

Degree of a divisor along a subscheme

I'm curious about a computation of Prop2.3 in The gonality conjecture on syzygies of algebraic curves of large degree by Ein and Lazarsfeld. Let $C$ be a smooth projective curve carrying a pencil $\...
3 votes
1 answer
198 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 ...
1 vote
1 answer
1k views

Kashiwara-Schapira Trilogy

I’m going to start reading Kashiwara-Shapira’s trilogy Categories and Sheaves, Sheaves on Manifolds, and Perverse Sheaves with someone soon. Flipping through the table of contents for Sheaves on ...
0 votes
0 answers
361 views

When are the cotangent and tangent sheaves isomorphic?

Let $X$ be an $S$-scheme. Under what conditions, if any, is the cotangent sheaf $\Omega_{X/S}$ isomorphic to the tangent sheaf $\Theta_{X/S}$ as $\mathcal{O}_X$- modules? For example, given a ...
2 votes
0 answers
110 views

$m$-regularity of sheaves

This is Lemma 1.4 on Green and Lazarsfeld's Some results on the syzygies of finite sets and algebraic curves. Let $X$ be a closed subscheme of $\mathbb{P}^r$. Suppose the ideal sheaf $\mathcal{I}$ of $...
3 votes
2 answers
376 views

Is any constant Zariski sheaf already a Nisnevich sheaf?

Lat $A$ be a set and $\underline{A}$ the associated constant Zariski sheaf on the category $Sm/S$ of schemes which are smooth over $S$ for a fixed base scheme $S$. Is $\underline{A}$ already a (...
16 votes
2 answers
2k views

Understanding the definition of stacks

First of all I should apologies if this question does not count as a research level one. I asked the same question on MathUnderflow and didn't receive any answer. Let me cross post (copy and paste) it ...
8 votes
0 answers
335 views

What is a morphism of ∞-sites?

Recall that a morphism of sites is a covering-flat functor that preserves covering families. Morphisms of sites can be identified with those geometric morphisms of induced toposes for which the ...
3 votes
1 answer
370 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 ...
1 vote
1 answer
1k views

Pullback map on global sections surjective

Let $X,Y$ be two irreducible, projective $k$-schemes. $k$ is assumed to be algeraically closed. Consider a dominant morphism $f: X \to Y$ between them which is not an isomorphism! Let $\mathcal{L}$ ...
6 votes
1 answer
675 views

The Yoneda pairing, hypercohomology, and cup product

Let $\mathcal{F}$ and $\mathcal{G}$ be coherent analytic sheaves on $\mathbb{P}^n$. Let $\mathcal{F}_\bullet$ be a locally free resolution of $\mathcal{F}$. In Principles of Algebraic Geometry by ...
6 votes
1 answer
465 views

Prove category of constructible sheaves is abelian

Let $X$ be a nice enough topological space, perhaps a complex algebraic variety with its analytic topology. I'm hoping someone could help me prove that the category $\text{Constr}(X)$ of ...
3 votes
1 answer
544 views

Jordan–Hölder sequence for $\mu$-semi stable sheaves

Let $X$ be a smooth variety over $\mathbb{C}$, and let $\omega \in \operatorname{Pic}(X)_\mathbb{R}$ be an ample class. I would like to know if any $\mu_\omega$-semistable sheaf $E \in \operatorname{...
3 votes
1 answer
190 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 ...
3 votes
0 answers
69 views

Characterization of degeneracy of spectral sequence of a fiber bundle at the second term

Let $f\colon E\to B$ be a fiber bundle of compact manifolds with fiber $F$. Assume that the push-forward $Rf_*(\underline{\mathbb{F}})$ in the derived category of the constant sheaf with coefficients ...
2 votes
0 answers
194 views

Only discrete topology gives trivial topos?

Given a Grothendieck site $\mathsf{(C,\tau)}$, if the associated Grothendieck topos $\mathsf{Shv(C,\tau)}$ is trivial, i.e. $\mathsf{Shv(C,\tau)}$ consists of the terminal sheaf $*$ only, can we ...
4 votes
1 answer
223 views

Morphisms of flat families of sheaves

$X$: projective scheme over a scheme $S$. $E, F$: $\mathscr{O}_X$-modules, flat/$S$ $\phi$: $E \rightarrow F$ : morphism s.t. $\phi_t$: $E_t \rightarrow F_t$ is zero morphism for all $t \in S$ Then,...
1 vote
1 answer
120 views

Relative version of the cohomology product

Let $f\colon X\to Y$ be a continuous map of 'nice' topological spaces (e.g. $f$ is a smooth map of smooth manifolds; $f$ might be assumed to be proper although I am not sure it is relevant). Let $\...
2 votes
0 answers
94 views

What does a partial map classifier look like as a sheaf?

[Cross-posted from M.SE, where it didn't get an answer] In constructive logic, it's possible for a set $X$ to satisfy $$\forall x,y \in X.\, x = y$$ while being non-trivial. Such a set is called a ...
3 votes
0 answers
195 views

What's wrong with higher dimensional nearby cycles?

Suppose we have a complex algebraic variety $X$ with a map $f: X \to \mathbb{C}$ with $Y=f^{-1}(0)$. Let $\overset{\sim}{\mathbb{C}}$ be the universal cover of $\mathbb{C}-\{0\}$ and consider the ...
6 votes
1 answer
463 views

Category of spaces/sheaves

Consider the following category $\mathcal C$: An object of $\mathcal C$ is a pair $(X,\mathcal F)$ where $X$ is a space and $\mathcal F$ is a sheaf on $X$. A morphism $(X,\mathcal F)\to(Y,\mathcal G)$...
29 votes
3 answers
3k views

Sheaf description of $G$-bundles

Now, among algebraic geometers, at least, it is well known that there is an equivalence between locally free $\mathcal{O}_X$-modules of rank $n$ and vector bundles of rank $n$. So, equivalently, ...
4 votes
0 answers
324 views

Properties of microlocalization

Let $i: M\hookrightarrow X$ be the inclusion of a closed submanifold in a smooth manifold $X$. I denote by $T_MX$ the normal bundle to $M$ in $X$, by $T^{\ast}_MX$ its dual bundle, and by $D^b(X)$ the ...
2 votes
1 answer
533 views

Pushforward in Compactly Supported Cohomology

Suppose $X,Y$ are locally compact Hausdorff spaces and $f:X\to Y$ is a topological submersion of relative dimension $n$. By this we mean that for all points $x\in X$, there exists an open neighborhood ...
0 votes
0 answers
113 views

cohomology of curves

Let $X$ be a smooth projective complex curve. Consider the diagonal $\Delta$ in $X \times X$, and $\mathcal{O}(\Delta)$ the associated line bundle. If $j$ is the inclusion of $\Delta$ in $X \times X$ ...
4 votes
0 answers
183 views

Infinity-categorical exceptional push-forward

Classically, if $f:X\to Y$ is a map of locally compact Hausdorff topological spaces, one can define the exceptional push-forward functor $f_!:Sh(X;k)\to Sh(Y;k)$ among $k$-valued sheaves for, say, a ...
5 votes
1 answer
482 views

Big etale topos vs small etale topos

Are they equivalent? That is, given a sheaf of sets $\mathscr{F}$ defined on the small etale site on $X$, is there an essentially unique way to extend it to a sheaf on the big etale site on $X$? If ...
1 vote
0 answers
211 views

Zero in colimit of sheaves category

This question is motivated by showing that the category $\mathbf{Sheaves} (X)$ from the open subset excluding the empty set category to the category of abelian group $\mathbf{Ab}$ has enough injective ...
3 votes
1 answer
284 views

What to call a morphism of sites inducing an equivalence on categories of sheaves?

Is there a standard term for a morphism of sites $(C,J)\to(C',J')$ which induces an equivalence on sheaf categories $\operatorname{Sh}(C',J')\xrightarrow\sim\operatorname{Sh}(C,J)$? The nLab entry ...
0 votes
0 answers
179 views

Theorem on Formal Schemes

I have few questions about the proof of Thm 7.5 from P. Wagreich's paper "Elliptic Singularities of Surfaces" (page 452): Denote by $\mathcal{N}$ the kernel of $A \to H^0(O_Z)$ (sorry, I haven't ...
12 votes
2 answers
1k views

How to visualize the Microsupport of a Sheaf?

I am looking through Persistent homology and microlocal sheaf theory to learn a bit on barcodes. They are require the notion of a microsupport of a sheaf, looks like it could be a rather concrete ...
2 votes
0 answers
62 views

For which locally ringed spaces is the structure sheaf given by LRS morphisms to the real line?

Let $\mathsf{LRS}_{\mathbb R}$ denote the category of locally $\mathbb R$-ringed spaces. Given a locally ringed space $(X,\mathcal O_X)$, write $C_{(X,\mathcal O_X)}^p$ for the hom-sheaf on $X$ of ...
3 votes
0 answers
120 views

Topological Brauer group and sheaf of $C^{\infty}$-functions

Given a smooth manifold $X$; by definition the topological Brauer group $B(X)$ is the group of classes of Azumaya algebras over the sheaf $\mathcal{O}_X$ of $\mathbb{C}$-valued functions on $X$. If ...
3 votes
1 answer
235 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 ...
7 votes
3 answers
778 views

Encounters with partitions of unity

Not sure how this would be received here. This question is about smooth partitions of unity. Let $M$ be a manifold. Consider an open cover $\{U_\alpha\}_{\alpha\in \Lambda}$ of $M$. A collection of ...
1 vote
0 answers
258 views

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 $K$...
5 votes
0 answers
245 views

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 ...
4 votes
1 answer
211 views

$G$-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 ...
18 votes
0 answers
529 views

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 ...
3 votes
0 answers
79 views

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 $\mathcal{H}$ ...
3 votes
1 answer
295 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 $...

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