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 $\mathcal F(V)$, ...

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29 views

Torsion in cotangent sheaf of the cusp

Let $X$ be the cuspidal curve $y^2 = x^3$. This is a singular curve with a unique singularity in the origin. One can easily see this using the Jacobi criterion. How does one show explicitly that the ...
2
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1answer
48 views

Injectivity of associate map of affine scheme homomorphism

Let $R$ be a ring and the corresponding $(\text{Spec } R, \mathcal{O}_{\text{Spec } R})$ be the affine scheme where $\mathcal{O}_{\text{Spec } R}$ is the structured sheaf of rings. By definition, the ...
3
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1answer
54 views

$\text{Hom}_{\mathcal{O}_X}(\cdot, \mathcal{G})$ is an exact functor

Suppose that $(X, \mathcal{O}_X)$ is a ring space. If $\mathcal{F}, \mathcal{G}$ are sheaves of $\mathcal{O}_X$-modules then the assignment $$U \mapsto \text{Hom}_{\mathcal{O}_X|U}(\mathcal{F}|U, ...
3
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1answer
42 views

Local properties of morphisms of schemes

In Hartshorne Proposition II.5.8, he shows, given a morphism $f \colon X \to Y$ where X and Y are schemes and $\mathcal{G}$ a quasi-coherent sheaf of $\mathcal{O}_{Y}$- modules, that ...
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1answer
47 views

Understanding the Gluing axiom of the Structure Sheaf on $Spec(R)$

Let $X = Spec(R)$ be an affine scheme for some commutative ring $R$. The structure sheaf $\mathscr{O}_{X}$ is a contravariant functor (I think) $\text{Open}(X) \leadsto \text{Ring}$ from the category ...
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33 views

Is there a reasonable Grothendieck topology on the category of modules over a ring?

How about over a field (i.e. f.d vector spaces)? Can these categories be considered as a site?
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24 views

vector bundles of $\mathbb{P}^2$ [closed]

Where I can find a complete description of the vector bundles of $\mathbb{P}^2$?
2
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1answer
33 views

Explicit procedure for integrating densities (twisted $n$-forms)?

I'm aware of several slightly different (yet equivalent) definitions for the density bundle of a non-orientable manifold. Unfortunately, I've been unable to make sense of integration in any of them. ...
0
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1answer
23 views

Sheaves of spaces?

I was skimming some of Lurie's work and he mentioned "sheaves of spaces." Is this a functor from Top^op -> Top? Or is it meant to just mean a sheaf on a space?
2
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1answer
41 views

Cech cohomology commuting with colimits? (Non noetherian confusion.)

Suppose that $X$ is a quasicompact, separated $A$-scheme, and $I$ is some directed poset. Suppose that $F_i$ is a system of sheaves on $X$ over $I$. I am having difficulties proving the claim that ...
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36 views

$C^k$-maps between manifolds is a sheaf?

I know that the functor from the category of open subsets of a manifold $M$ to the Sets, taking an open set $U$ and associating to it the collection of $C^k$ maps to $\mathbb{R}$ is a sheaf. My ...
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29 views

Interesting sites without pull-backs

I've seen at least a couple of sources (e.g. the nLab) that do not require the covering families in a coverage to be closed under pull-backs. The question is simple: are there any interesting ...
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31 views

$G$ equivariant quasicoherent sheaves on $X$ as compatible $G$ actions on the total spaces?

Let $G$ be an algebraic group, and $X$ a scheme on which $G$ acts: i.e the $S$ points of $G \times X \to X$ is a group for each affine $S$. Let $F$ be a quasicoherent sheaf on $X$. There is a notion ...
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65 views

Nonstandard construction of sheafification

Let $F$ be a presheaf on a topological space $X$ of some category of "sets with structure." In Borel's Linear Algebraic Groups, he gives the following explanation for how to construct the associated ...
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1answer
67 views

If $X$ is a closed set in $\mathbb{A}^n$, is $\mathcal O_X$ the inverse image sheaf of $\mathcal O_{\mathbb{A}^n}$?

Let $k$ be algebraically closed, and let $X$ be a closed subset in $k^n$ with corresponding radical ideal $I$. Let $\mathcal O$ be the standard sheaf associated with the space $k^n$, and let ...
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31 views

Verification of example to show surjective maps of sheaves need not surject onto sections in all open sets

As an exercise in understanding the notion of surjectivity in the category of sheaves, I came up with this example, slightly modifying the standard ones given in my textbooks. I feel like this one is ...
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0answers
36 views

Monomorphism from a sheaf to a flasque sheaf: determining the stalk.

Let $M$ be a topological Hausdorff space. We use the following definitions (as they may vary): A presheaf $\mathcal{F}$ is a collection of vector spaces $\mathcal{F}(U)$ for each open subset $U$ of ...
2
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1answer
16 views

Why does sheaf $\pi_0$ of a simplicial presheaf determine maps to sheaves?

This question refers to an argument from Section 6, p. 22 of Freed–Hopkins, "Chern–Weil forms and abstract homotopy theory." There's something presumably straightforward I'm just not ...
2
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1answer
54 views

Composition of morphisms of locally ringed spaces

I have a specific question about defining the composition in (locally) ringed spaces. The definition I had formulated myself while reading Hartshorne, since he conveniently neglected to suggest any ...
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0answers
22 views

Extending a $\mathcal{B}$-sheaf via inverse limits vs stalks.

Let $X$ be topological space and $\mathcal{B}$ a basis. A $\mathcal{B}$-sheaf is a contravariant functor $\mathcal{O}_{\mathcal{B}}$ from $\mathcal{B}$ to the category of abelian groups that satisfies ...
3
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0answers
22 views

Moving from sheaves over spaces to sheaves over sites

The first example of a sheaf that I have consciously come across is the sheaf of continuous (real) functions on some topological space. The fact it is a sheaf is equivalent to the pasting lemma, which ...
0
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1answer
19 views

Understanding square-root sheaf and similar sheaves

Let $n$ be a positive integer, and $\mathbb C^* = \mathbb C -\{0\}$. Let $f: \mathbb C^* \rightarrow \mathbb C^*$, sending $z$ to $z^n$. Let $\mathcal F$ be the constant sheaf on $\mathbb C^*$ with ...
5
votes
1answer
50 views

Orientation on a manifold as a sheaf

I am thinking about orientation of a connected manifold $M$ of dim $n$ as a sheaf. There are two definitions I could use, the first is the sheaf associated to the presheaf $$U\mapsto H_n(M,M-U;R).$$ ...
6
votes
1answer
77 views

Is representability of Zariski sheaves local on the base?

Let $F: \mathsf{Sch_{/S}}^{op} \to \mathsf{Set}$ be a Zariski sheaf on the category of $S$-schemes. $F$ being a sheaf means it satisfies the following property: Sheaf condition: For every ...
0
votes
1answer
28 views

Surjective morphism to Quotient sheaf, nonsurjective on sections

Let $k$ be an algebraically closed field, $X=\mathbb{P}^1$ the projective line. Let $P=(1:0)$ and $Q=(0:1)$ be points on $X$, and $\mathscr{F}$ be the sheaf of regular functions on $X$. Define a ...
1
vote
1answer
59 views

Kähler differential and higher derivations (geometric interpretation of diagonal here)

I am studing Kähler differentials and I tried to understand the geometric motivation behind these settings. What I do not understand is the role which plays the diagonal in all these theory. The ...
2
votes
0answers
77 views

Sheaves on a Simplicial Complex

I came across a video the other day which discussed practical applications of sheaves. However, rather than defining a sheaf on the open sets of a traditional topological space, the lecturer outlined ...
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0answers
21 views

Locally presentable sheaves and the associated module functor

Let $R$ be a commutative ring. Any $R$-module has a presentation $R^{(J)}\rightarrow R^{(I)}\rightarrow M\rightarrow 0$. The associated module functor $M\mapsto \tilde M$ is exact and so preserves ...
1
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1answer
38 views

Relationship between smooth manifold and differentiable manifold

Define a smooth manifold to be a ringed space (a Hausdorff, second countable, $n$-manifold) that is locally isomorphic (as a ringed space) to the sheaf of smooth real valued functions on some open $U ...
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0answers
19 views

Map out of flasque sheaf can be specified on global sections?

If I have a map of sheaves $\varphi : \mathscr{F}\rightarrow \mathscr{G}$ on some topological space $X$, where $\mathscr{F}$ is flasque (i.e. all restriction maps are surjective), is it true that the ...
5
votes
1answer
63 views

Is this a valid way to think about sheafification?

This all feels like it should be valid, but I just wanted to get more experienced eyes on it in case I've made a mistake. Take a presheaf $\mathscr{F}$ on a topological space $X$. In order to be a ...
0
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0answers
23 views

Cech Homology and Direct Sums

Does Cech homology for a precosheaf F on a topological space U with open cover {Ui} commute with direct sums? I know that Cech cohomology does but was wondering if that property is preserved through ...
2
votes
1answer
35 views

Functor of section over U is left-exact

I am trying to prove $\Gamma(U,\cdot)$ is a left-exact functor $\mathfrak{Ab}(X)\to\mathfrak{Ab}$. This is Exercise 1.8 in Hartshorne, Chapter II or exercise 2.5.F of Vakil's notes (Nov 28,2015 ...
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0answers
32 views

Relation between stalks of twisted sheaf and structure sheaf

Let $A$ be a ring, $B = A[T_0,\dots, T_d]$, and $X = \textrm{Proj } B$. Then at every point $x \in X$, $$\mathcal{O}_X (n)_x \cong \mathcal{O}_{X,x}$$ Let $x$ correspond to a homogeneous prime ...
1
vote
1answer
22 views

Cosheaf homology Global Sections

Let X be a topological space and U={Ui} be some open covering of X. Let F be a cosheaf of abelian groups on X. Is the 0th cech homology group the same as the global sections on X?
4
votes
1answer
78 views

Vector bundle as locally free coherent sheaves

I am studying coherent sheaves and was looking for a geometric motivation. Hence, in wikipedia and although here is stated that it can be seen as a generalization of vector bundles, which is quite ...
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1answer
24 views

Global Section Functor exactness for Precosheaves [closed]

Is the global sections functor for precosheaves fully exact? Or just right exact?
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27 views

Cosheaf homology

Suppose we have two sheaves F and G that are isomorphic on some open set U or topological space X. We can write this as 0 -> F(U) -> G(U) -> 0 is exact. When we pass onto global sections, suppose that ...
4
votes
2answers
64 views

Gluing together functions on a closed subvariety

I'm trying to get an intuition for what sheafification does. I came across a passage from Perrin's algebraic geometry book about closed subvarieties. If says that if X is an algebraic variety and Y ...
1
vote
1answer
21 views

Global Section Functor Existence for Pre(Co)Sheaves?

I currently have a co-presheaf F : Top -> Vect. I don't know how to cosheafify it. It is well-known that the Global section functor for sheaves is a covariant left exact functor (right exact for ...
2
votes
1answer
29 views

Global section functor on cosheaves

Is the global section functor for cosheaves right exact? For sheaves, this functor is left exact, thus giving rise to sheaf cohomology as a right derived functor, so I was wondering if this ...
0
votes
1answer
34 views

Injective ring homomorphism induces injective morphism on sheaves

I'm really confused on this exercise. People have suggested hints and I've seen online solutions involving modules and tensor products, but I don't see how any of it is related to the problem. Let ...
2
votes
0answers
41 views

Extending sections of quasi-coherent sheaves on locally Noetherian normal schemes across codimension 2 sets?

Question (global): Is it possible to extend sections of quasi-coherent sheaves on locally Noetherian normal schemes across codimension 2 sets? Specifically, if $X$ is a locally Noetherian normal ...
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0answers
94 views

Sheaves of categories

I recently read an answer on this MO post explaining that one reason people are interested in higher category theory is to make reasonable sense of something like a "sheaf of categories" on a ...
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0answers
125 views

When can stalks be glued to recover a sheaf?

Let $\mathcal{F}$ be a sheaf over some topological space. The stalks are $\mathcal{F}_x= \underset{{x\in U}}{ \underrightarrow{\lim}} \mathcal{F}(U)$. Is there a special name for a sheaf that ...
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0answers
26 views

Construction of relative projective space via glueing

I would like to gain further practice on glueing schemes by constructing projective space over a ring. I am considering the following: I wonder how we get that $\mathcal{O}_{X_i} (X_{ij}) = ...
2
votes
0answers
56 views

Sheafification and monomorphisms.

I was showing that a monomorphism of sheaves induces a monomorphism in stalks. I used the classical fact about filtered colimits but I was wondering that if the inclusion is adjoint to sheafification ...
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0answers
53 views

Is there homology in the sections of an infinite exact sequence of injective sheaves of $\mathcal O_X$-modules?

Let $(X, \mathcal O_X)$ be a scheme and the following an infinite, exact sequence of injective sheaves of $\mathcal O_X$-modules: $$ \cdots \overset{f_5}\longrightarrow I_5\overset{f_4}\longrightarrow ...
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0answers
8 views

For every abelian sheaf, $F(\phi)=\{ 0 \}$?

In the book 'Sheaf theory' by Tennison, exercise 1.9 asks to prove that if $G$ is an abelian sheaf then $G(\phi)=\{ 0 \}$. I have been unsuccessfully trying to prove this and the following seems to be ...
1
vote
1answer
35 views

Let $i : U \to X$ be an open inclusion. Is $ i_* i^{-1} $ an exact functor?

Let $i : U \to X$ be an open inclusion. Is $ i_* i^{-1} $ an exact functor? I think possibly not, but I am kind of convinced yes (because the maps on stalks are either the same or made into zero). ...