All questions with tag [math: sheaf-theory]


Composing covers with epis

I am beginner of sheaf-theory and beg your pardon for this maybe silly question. Let $\mathcal{C}$ be a Grothendieck site and $T$ the category of sheaves on $\mathcal{C}$ and let $f:X\rightarrow Y$ be an epic morphism in $T$ into a representable sheaf $Y$. I have a general lack of understanding how such epic morphisms look like and this leads to...
2022-07-25 17:47:06

About the etale space

When one defines the etale space of a presheaf $\mathscr F$ on a topological space $X$, would be assumed that $X$ is a $T0$-space (i.e. for every $x$, $y$ in $X$ exists an open set containg one of them but not the other point)? If $X$ is not $T0$, I'm not sure that the stalks of $\mathscr F_x$ are disjoint each other for all $x\in X$.
2022-07-25 17:45:56

Pullback of a locally constant sheaf by a function whose domain is simply connected

Let $\mathcal{A}$ be a locally constant sheaf on a topological space $X$ and let $\sigma:\Delta_p\to X$ denote a singular $p$-simplex. Writing the pullback of $\mathcal A$ by $\sigma$ as $\sigma^\ast(\mathcal A)$, Bredon's book on sheaf theory (page 26 in the second edition) says: Since $\mathcal{A}$ is locally constant and $\Delta_p$ is simply...
2022-07-25 17:18:35

Locally closed subspaces and sheaves.

(Bredon page 11.) Let $A$ be a locally closed subspace of $X$ and let $\mathcal{B}$ be a sheaf on $A$. It is easily seen (since $A$ is locally closed) that there is a unique topology on the point set $\mathcal{B}\cup(X\times\{0\})$ such that $\mathcal{B}$ is a subspace and the projection onto $X$ is a local homeomorphism (we identify $A\times\{0...
2022-07-25 17:18:31

When does a sheaf exist with prescribed stalks?

I think the intended question is clear, but let me attempt to formulate it precisely: If $X$ is a topological space, and $f$ is a function on $X$ such that $f(x)$ is an abelian group for every $x \in X$(I'll leave the codomain a mystery, it's a bit neater than phrasing this in terms of a functor), is there a sheaf $\mathcal{F}$ such that $\mathc...
2022-07-25 17:17:36

Confused about Hypercohomology terminology and meaning

check this: Given a sheaf complex $F^\bullet$, let's say I want to compute the hypercohomology of this complex, if we consider the bicomplex of sheaves $C^\bullet(F^\bullet) = (C^p(F^q))\quad (p,q\in\mathbb{Z})$, where $C^\bullet(F^q)$ is the Godement resolution of the sheaf $F^q$. The hypercohomology of $F^\bullet$ is the cohomology of the comp...
2022-07-25 17:06:49

Twisting a sheaf by a Divisor

My question is how to define the twisting of a sheaf $\mathcal{L}$ by a divisor $D$. In specific I'm interested in the twisting of the canonical bundle $\omega$ of a non-compact Riemann surface $X$ by a divisor of points. (The points are the missing points of the compactification, in my case they are a finite number.) My guess on the definition...
2022-07-25 13:31:48

How much is a topological space $X$ determined by the category of sheaves of abelian groups on $X$?

Well, the title pretty much says it all. We have a functor $$\mathsf{Sch}_{Ab} : \mathsf{Top} \to \mathsf{Cat}$$ which takes a topological space $X$ to the category $\mathsf{Sch}_{Ab}(X)$ of sheaves of abelian groups on $X$. Every continuous map $f:X \to Y$ gives rise to a functor $f_*:\mathsf{Sch}_{Ab}(X) \to \mathsf{Sch}_{Ab}(Y)$, which takes ...
2022-07-25 13:11:18

Questions about effective Cartier divisors

Let $f:X\rightarrow S$ be a morphism of schemes. The definition of an effective Cartier divisor in $X/S$ given in Katz-Mazur (what is called relative effective Cartier divisor in the Stacks Project) is: a closed subscheme $i:D\hookrightarrow X$ such that the ideal sheaf $I(D)$ is invertible and $f\circ i:D\rightarrow S$ is flat. I have two (pos...
2022-07-25 12:47:52

is the pushforward of a flat sheaf flat?

Let $f:X \to Y$ be a morphism of schemes and let $F$ be an $\mathcal{O}_X$-module flat over $Y$. Is $f_*F$ flat over $Y$? What's wrong with this argument? [EDIT: as Parsa points out, the (underived) projection formula does not hold for arbitrary modules on Y] Let $N' \to N$ be an injection of modules over $Y$, we need to show that $f_*F \otimes ...
2022-07-20 14:36:31

The precise definition of a "sheaf of rings"

Let $X$ be a topological space. Let $Op(X)$ be the category of open sets. A sheaf of abelian groups is a (contravariant) functor $F:\mathcal{C} \to Ab$ satisfying the sheaf condition: the following sequence is exact $$ 0 \to F(U)\to \prod_i F(U_i) \to \prod_{i,j} F(U_{ij})$$ in the category of abelian groups. Now, one can replace "category ...
2022-07-17 14:40:11

Canonical example of a cosheaf

Sheaves can, like all modern-day mathematical building and constructions and also abstractions, be counterproductive monsters yet, like all such building and constructions, a couple of instances can permit one to imagine them merely as a generalisation of an all-natural object (ie. the collections of neighborhood features on a topological room)....
2022-07-16 11:43:00

Do the locally integrable functions on the real line form a sheaf, and can they be defined in this fashion?

In his notes: (Wayback Machine, Pete Clark outlines an axiomatic approach to the Riemann Integral. He doesn't use the language of sheafs, but it seems implicit in his definition before Theorem 1. He goes on to show that the fundamental theorem of calculus follows, and then that such an integral exists...
2022-07-16 11:38:36

A problem on Čech cohomology

Let $X$ be a topological room and also $$0\to\mathcal{F}^{\prime\prime}\to\mathcal{F}\to\mathcal{F}^\prime\to 0$$ be a specific series of sheaves on $X$. Just how can I show that $$H^1(X,\mathcal{F}^{\prime\prime})\to H^1(X,\mathcal{F})\to H^1(X,\mathcal{F}^{\prime})$$ is specific?
2022-07-15 00:41:27

Exactness of direct limit of other than modules

Taking straight restrictions is a specific functor in the group of components. It has actually been reviewed thoroughly below. What I ask is: I recognize that taking straight restrictions is not a specific functor in various other groups. Our teacher stated it when reviewing Cech cohomology of sheaves, defining the n - th Cech cohomology team a...
2022-07-14 02:25:10

Why isn't the associated sheaf transformation onto?

What is an instance of a presheaf $F$ such that the common morphism $F\to \tilde F$ to the linked sheaf is not onto on all areas, i.e. there exists an $X$ and also $F(X)\to \tilde F(X)$ is not onto?
2022-07-14 02:17:18

Does the restriction functor (big to small) commute with the inverse functor?

I would like to know whether the restriction functor commutes with the inverse image functor. (I basically follow the terminology in "Topological and Smooth Stacks".) Let $X$ be a topological space. The small and big sites of $X$ are denoted $\mathrm{Op}(X)$ and $\mathbf{T}/X$ respectively. A small sheaf over $X$ is a sheaf over the s...
2022-07-13 23:00:02

Is the presheaf "represented" by an ind-group scheme a sheaf?

Let $S$ be a scheme and $(G_i,\varphi_{ij})_{i\in I}$ an inductive system of $S$-group schemes. By a sheaf on $S$ I mean an fppf sheaf. Each $G_i$ represents a sheaf on $S$, and the presheaf colimit of the $G_i$ is the functor $T\mapsto\varinjlim G_i(T)$ on locally finitely presented (or locally finite type or whatever you prefer) $S$-schemes $T...
2022-07-13 22:43:16

Pullback and pushforward

Say I have a scheme $X$, irreducible and of finite type over a field $k$, and a closed subscheme $Y$ of $X$ with associated closed immersion $i: Y \to X$. Consider a sheaf $F$ on $X$ (for the étale topology). How should I think about the sheaf $i_* i^* F$ and the kernel of $F \to i_* i^* F$ in a more or less "concrete" way? Is it poss...
2022-07-12 11:51:40

An example of a Grothendieck topology

A Grothendieck topology on a category $\mathcal{C}$ with finite limits consists of, for each object $U$ in $\mathcal{C}$ a collection $\text{Cov}(U)$ of sets $\{ U_i \to U \}$ such that Isomorphisms are covers, e.g if $V \to U$ is an isomorphism then $\{ V \to U \} \in \text{Cov}(U)$ Transitivity: If $\{V_i \to U \}$ and $\{V_{ij} \to V_i \}$ a...
2022-07-12 11:50:28