# Questions tagged [sheaf-theory]

A Sheaf $\mathcal F$ on a topological space $X$ captures local data $\mathcal F(U)$ given on open sets $U\subseteq X$ and how such data can be restricted to smaller open sets or glued together. In typical cases, $\mathcal F(U)$ is a set of functions defined on $U$ and an element of $\mathcal F(V)$, $V\subseteq U$ is obtained by restricting the domain and not all elements of $\mathcal F(U)$ can be obtained by restricting a global section $\in\mathcal F(X)$.

1,827 questions
Filter by
Sorted by
Tagged with
66 views

### Weak Nullstellensatz in locally ringed spaced?

Is there a version of the weak Nullstellensatz valid in general locally ringed spaces? The statement I imagine is something like this: Let $(X, \mathcal{O}_X)$ be a locally ringed space (possibly ...
32 views

### looking for a proof or a reference of classical result

Let $X$ be a complex manifold of dimension $n$, $\Omega_{X}$ the sheaf of top degree forms, $\mathcal{D}_{X}$ the sheaf of holomorphic differential operators of infinite order and $q$ the natural ...
119 views

### Global section of tensor product of sheaves

I want to know if there is an easy proof for the following statement: Let $X$ be a non-singular, projective variety ( maybe it is easier Surfaces=X). Let $E$ be a locally free sheaf; then there ...
95 views

### Categories for which every contiuous sheaf is representable

I'm interested in locally small, cocomplete categories $\mathbf{C}$ such that every limit preserving functor $$\mathbf{C}^\mathrm{op}\to\mathbf{Set}$$ is representable. Is there a name for such ...
46 views

### proving that a functor is an equivalence of categories

I would like to prove that a functor is an equivalence of categories. I tried to use that equivalence of categories is the same as fully faithful plus "surjective". I found a difficulty verfying this ...
61 views

47 views

### Extension Classes $Ext^1 _X (\mathcal{F},\mathcal{G})$ of Sheaves

Let $k$ be a field and $X$ be a proper $k$ scheme. Futhermore let $\mathcal{H}$ be a coherent $\mathcal{O}_X$-module. Grothendieck's Finiteness Theorem says that for all $i \ge 0$ the cohomology ...
141 views

### Why is the bounded functions not a sheaf?

I know that the presheaf of bounded functions is not a sheaf but I don't see why. I checked in wikipedia they say that this presheaf does not verify the axiom of "Glue". For me it verifies this axiom. ...
29 views

109 views

### Is there a classifying topos for locales?

Is there a Grothendieck topos $F$ such that, for any Grothendieck topos $E$, the category of geometric morphisms $$E\rightarrow F$$ is equivalent to the category of locales internal to $F$? I suspect ...
129 views

### Flat sections of flat vector bundles

Let $E\to X$ be a vector bundle with flat connection $\nabla$. Is there a canonical way to construct a vector subbundle of $E$ from the flat sections ($\nabla \sigma=0$)? More specifically, I would ...
38 views

54 views

### Weakening the gluing axiom for sheaves

The gluing axiom states that given an open cover $\{U_i\}$ of some open set $U$, if I have a section $s_i$ on each $U_i$ such that the restrictions agree on intersections, then there exists a unique ...
101 views

104 views

104 views

### Further examples of stalks

I am currently learning about stalks for the first time. In my exploration online about the topic, I routinely run into the same three examples: In a constant sheaf associated with an abelian ...
38 views

63 views

### Is the definition of a sheaf in Serre's paper “Coherent Algebraic Sheaves” equivalent to the standard definition?

In Serre's paper "Coherent Algebraic Sheaves", the definition of a sheaf is so different from other book's I have ever seen. I want to know whether it is equivalent to the standard definition.
44 views

### Show that $\text{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ is an algebraic variety

I'd like to show that $\text{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$ is a variety. Is it even true that $\mathbb{H} = \{ x + iy : y > 0 \}$ is an algebraic variety? There's no metric, so it's ...
124 views

### Restriction of sheaf on a closed subscheme is in general not quasi-coherent?

In Hartshorne 5.2.4, we have If $Y$ is a closed subscheme of a scheme $X$, then the sheaf $\mathcal{O}_{X|Y}$ is not in general quasi-coherent in $Y$. I'm having a hard time believing this, ...
81 views

52 views

### Construction of “etale associated-sheaf” in Rotman's

On pg 282 Prop 5.68 Rotman makes the following construction given a presheaf $P$ of abelian groups over a space $X$. The construction is as follows: For $P^{et}:= (E^{et}, p^{et},X)$. I am ...
134 views

### Showing $d\pi$ is non-zero in module of Kähler differentials
While reading about Kähler differentials I came across the seemingly innocent statement that $d\pi \in \Omega_{\mathbb{R/Q}}$ in non-zero. This is kind of "obvious" as if $\alpha$ is algebraic then ...