Questions tagged [sheaf-theory]

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An special ideal in an integral domain [closed]

I am looking for an integral domain $D $ and an ideal $I $ of $D $ such that $I $ has infinitely many minimal prime ideal( overy $I $).
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Surjectivity of multiplicative map

Let $S$ be a smooth complex algebraic surface, and $\mathcal{F}$ be a coherent sheaf on $S$. I want to consider $W = S \times S$ and the coherent sheaf $\mathcal{G} = \mathcal{F} \boxtimes \mathcal{F}...
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121 views

When localization is indecomposable

We know that if $R $ is a domain then any localization of $R $ at any multiplicative subset of $R $ is indecomposable, that is, has no non trivial idempotents. Now let $R $ be a commutative ring with ...
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Existence of a non-degenerate global section is an open property?

Setting: $X$:projective surface over algebraically closed field $k$. $T$:scheme over $k$. $E$: Coherent sheaf on $X \times_k T$ , flat over $T$ and $\forall t \in T$, $E_t$ is rank 2 torsion-free ...
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Descent and co/ends

The bicategorical analogue of a coend, namely a universal extrapseudonatural transformation, is called a bicodescent object. As noted in arXiv:1709.01332 [math.CT], this notion goes back to Ross ...
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1answer
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Action on group $\operatorname{Ext}^i(\mathcal{L}, \mathcal{M})$ by scalar multiplication

Let $X$ be a proper scheme over field $k$ and $\mathcal{L}, \mathcal{M}$ two invertible $\mathcal{O}_X$-modules. Then $Hom_{\mathcal{O}_X}(\mathcal{L}, \mathcal{M}) \cong Hom_{\mathcal{O}_X}(\mathcal{...
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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 ...
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1answer
445 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 ...
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400 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?
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50 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 ...
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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)...
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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 $...
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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 $\mathcal{O}(K_S)$ ...
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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 ...
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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 $\...
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302 views

The Serre duality theorem intuition

It is a well known fact that proper scheme $X$ over $k$ has a up to isomorphism unique dualizing sheaf (EGA I, Hartshorne). This dualizing sheaf $\omega_X$ comes with two striking properties: (i) ...
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1answer
89 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 ...
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211 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 ...
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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 ...
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$m$-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 $...
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179 views

Explicit description of exponentials of etale spaces

It is well known that the category $\mathit{Sh}(X)$ of sheaves of sets on a topological space $ X $ is a topos. On the other hand, there exists a natural equivalence of categories between $\mathit{Sh}(...
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266 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 (...
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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 ...
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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 ...
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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 ...
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1answer
183 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 $\mathcal{L}$ ...
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1answer
301 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 ...
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732 views

Are there (enough) injectives in condensed abelian groups?

The question is very simple : does $Cond(\mathbf{Ab})$, the category of condensed abelian groups (as defined in Scholze's Lectures in Condensed Mathematics), have enough injectives ? Does it, in fact,...
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1answer
341 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 ...
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1answer
159 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 ...
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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 ...
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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 ...
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1answer
185 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,...
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1answer
106 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 $\...
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1answer
128 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{...
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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 ...
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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 ...
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1answer
349 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 $(X,\mathcal F)\to(Y,\mathcal G)$...
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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 ...
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1answer
166 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 ...
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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 $X \times X$ ...
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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 ...
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305 views

Does Peetre's theorem hold in complex analysis?

Let $E, F$ be two smooth vector bundles over a smooth manifold $M$. Peetre's theorem states that any $\mathbb{R}$-linear morphism $D: \mathcal{E} \to \mathcal{F}$ of the sheaves of sections of $E$ and ...
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322 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, ...
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1answer
200 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 ...
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188 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 ...
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165 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 ...
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1answer
181 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 ...
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52 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 ...
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87 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 ...

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