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Questions tagged [sheaf-theory]

For questions about sheaves on a topological space. Usually you think of a sheaf on a space as the data of functions defined on that space, although there is a more general interpretation in terms of category theory. Use this tag with the broader (algebraic-geometry) tag.

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Sheaves of sections of vector bundles

If I have two sheaves $E$ and $G$ over a smooth manifold $M$ and I want to prove something like say $\mathcal{O}(E) \otimes_{\mathcal{O}} \mathcal{O}(G) \cong \mathcal{O}(E\otimes G)$, where $\mathcal{...
Tim's user avatar
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Are locally free sheaves (mayn't be of finite rank) reflexive? [duplicate]

Suppose $X$ is a scheme, and $\mathcal{F}$ be a locally free sheaf (mayn't be of finite rank). Definition of reflexive is given here: https://stacks.math.columbia.edu/tag/0AVU Note that in the ...
Ramandeep Singh Arora's user avatar
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1 answer
32 views

Group actions on ringed topological spaces

Letting $(X,\mathcal{O}_{X})$ be a ringed topological space, and $G$ a group of automorphisms on $X$, I'm confused on how an element $g\in G$ is supposed to act on an abelian group $\mathcal{O}_{X}(U)$...
Noah 's user avatar
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3 votes
1 answer
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Well-definedness of the projection associated to the sheaf of germs of a presheaf

I'm currently reading Izu Vaisman's Cohomology and differential forms ($1973$) having never studied sheaf theory before, so I will briefly write down the definitions in case they don't match with ...
Bruno B's user avatar
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2 votes
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Sheaf $\mathscr{F}_S$ for which $\mathscr{F}_S(U)$ consists of holomorphic sections that vanish on $S \cap U$ isomorphic to $\mathscr{O}(L^*_Y)$

Let $M$ be a complex manifold and $Y \subset M$ a closed hypersurface and $L_Y$ the holomorphic line bundle associated with $Y$. How can I show that the sheaf $\mathscr{F}_Y$ for which $\mathscr{F}_Y(...
Elena's user avatar
  • 63
2 votes
0 answers
61 views

Map of global sections surjective under a local condition

Let $M$ be a compact complex manifold and $p \in M$. Let $L \to M$ be a line bundle on $M$ and $\mathcal{F}_{\{p\}}$ be the sheaf of holomorphic sections of $L$ that vanish at $p$. Locally around a ...
Elena's user avatar
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Qing Liu's Algebraic Geometry, Chapter 2, Exercise 2.13.

Part (a) of Exercise 2.13: Let $f:X\mapsto Y $ be a continuous map of topological spaces. Let $\mathcal{F}$ be a sheaf on $X$, and $\mathcal{G}$ be a sheaf on $Y$. (a) Show that there exist canonical ...
Noah 's user avatar
  • 63
1 vote
0 answers
37 views

Is there a notion of smooth sheaf which corresponds with smooth etale bundles?

It is well-known and quite useful that there is an equivalence of categories between sheaves of sets on a space $X$ and etale bundles over that space, meaning spaces $E$ and continuous maps $\pi:E\to ...
FShrike's user avatar
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Can we pass between sheaves of suitable type and fibre bundles (not ! etale bundles)?

Today I was forced to think about vector bundles (on manifolds), but in the context of a textbook "otherwise" focused on sheaf theory. I'm interested to what extent this really is an "...
FShrike's user avatar
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Gluing of sheaves via sheafication

Setting. Let $X$ be a topological spaced, and let $\{U_i\}_{i\in I}$ be an open covering. Denote by $U_{ik}$ any intersection $U_i\cap U_k$. Suppose that on each $U_i$ there is a ring sheaf $O_i$ and ...
Ezio Greggio's user avatar
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1 answer
42 views

Rank $1$ locally free quotient sheaves of the cotangent bundle restricted to a smooth generic curve

Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$. Let $X$ be a smooth projective variety over $\mathbb{K}$ such that $K_X$ (the canonical bundle) is ample. For $m\gg0$ there ...
Armando j18eos's user avatar
2 votes
1 answer
66 views

The Grassmannian $G(k,n) \subset G(k,n+1)$ as the zero locus of the tautological sub-bundle

Following my question Pushforward of structure sheaf with respect to canonical inclusion of Grassmannian I have a follow up question I would like to ask. I would have put this as a comment, but I feel ...
Sunny Sood's user avatar
1 vote
1 answer
45 views

Why sheaves of abelian groups form an abelian category

I know it is an elementary result. But some details confuse me. In Vakil's The Rising Sea (version 2022) Theorem 2.6.2 writes: Sheaves of abelian groups on a topological space $X$ form an abelian ...
HIGH QUALITY Male Human Being's user avatar
1 vote
1 answer
25 views

morphisms between the sheaves restricted to open subset

Let $\mathscr F,\mathscr G$ be both sheaves on $X$ and $\varphi$ be morphisms between sheaves $\mathscr F,\mathscr G$. Let $U\subset X$ be an open subset( maybe we do not even need that $U$ is open in ...
Ziqiang Cui's user avatar
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0 answers
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Exercise 2.2.J in Vakil's The Rising Sea (version 2022) [duplicate]

The question is stated below: If $(X,\mathscr{O}_{X})$ is a ringed space, and $\mathscr{F}$ is an $\mathscr{O}_X$-module, describe how for each $p\in X$, $\mathscr{F}_p$ is an $\mathscr{O}_{X,p}$-...
HIGH QUALITY Male Human Being's user avatar
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1 answer
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Pushforward of structure sheaf with respect to canonical inclusion of Grassmannian

Let $j: Gr^{k,n} \rightarrow Gr^{k,n+1}$ be the canonical inclusion of Grassmannians (working over a field for simplicity). I am interested the pushforward $j_{*}\mathcal{O}_{Gr^{k,n}}$. Specifically, ...
Sunny Sood's user avatar
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0 answers
44 views

Glueing of morphism between schemes

On page 70 of Wedhorn& Görtz's Algebraic Geometry I:Schemes, there is one result which I do not know how to prove. Let $X,Y$ be schemes and let $X=\cup_i U_i$ be an open covering. Then morphisms $...
Ziqiang Cui's user avatar
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0 answers
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definition of closed immersion of schemes

The definition that I found on books for definition of closed immersion of schemes is the following: A closed immersion $i:Z\hookrightarrow X$ is a morphism which satifies: (1) The underlying ...
user11695417's user avatar
2 votes
1 answer
37 views

Proper direct image of sheaves along the inclusion of a locally closed subset $W$ is f.f. with essential image equal to sheaves with supp. $\subset W$

$\def\F{\mathcal{F}} \def\Supp{\operatorname{Supp}} \def\E{\mathcal{E}}$In B. Iversen, Cohomology of Sheaves, Section II.6, we find the definition of the proper direct image functor $h_!$ (aka direct ...
Elías Guisado Villalgordo's user avatar
1 vote
1 answer
47 views

Need Help Proving That These Functions are Natural Transformations

In MacLane's Sheaves in Geometry and Logic, it is stated the following. Now, consider a given presheaf $P$ and the sheaf $\Gamma \Lambda_P$ of sections of the bundle $\Lambda_P \rightarrow X$. For ...
babu's user avatar
  • 315
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$f^{-1}(\varphi)_{x} = \varphi_{f(x)}$ ( Stalk of the inverse image )?

It seems easy question but don't find rigorous argument about it until now. Let $f : X\to Y$ be a continuous map. Then we obtain a functor $f^{-1}$ from the category of presheaves on $Y$ to the ...
Plantation's user avatar
  • 2,644
2 votes
1 answer
46 views

How to show that a sheaf is itself a sheaf of modules?

I am currently writing the proof for the following proposition: **Show that the sheaf of sections on a vector bundle $V$ over $X$ is a sheaf of modules over a sheaf of continuous function on $X$. ** I ...
SourBiscuit's user avatar
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0 answers
21 views

For a scheme $Z$, let $z\in Z, t\in \mathscr O_Z(Z)$, what does $t(z)$ mean?

For a scheme $Z$, let $z\in Z, t\in \mathscr O_Z(Z)$, let $Z_t:=\{z\in Z:t(z)\not= 0\}$. The author claims $Z_t$ is open subset of $Z$. But what does $t(z)$ mean?
user11695417's user avatar
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1 answer
64 views

The inverse image functor induced from morphism of ringed spaces commutes with the image?

Let $ f: (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces and let $ w : \mathcal{F} \to \mathcal{F}'$ be a homomorphism of $\mathcal{O}_Y$-modules. Then Q. $f^*(\...
Plantation's user avatar
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1 vote
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Equivalent ways to define structure sheaf on $Spec(A)$.

How do we show that the following two ways of defining the structure sheaf on $Spec(A)$ are the same? Definition 1 $$\mathcal{O}(U) = \left\{s \in \prod_{\mathfrak{p} \in U} A_{\mathfrak{p}} : s \text{...
Babai's user avatar
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1 vote
1 answer
95 views

Correspondence between sheaves of ideals and closed immersion

Let $(X,\mathcal O_X)$ be a locally ringed space. Let $\mathcal J$ be a sheaf of ideals over $\mathcal O_X$. Then we can construct a closed immersion of locally ringed spaces: $$(Z(\mathcal J),i^{-1}(\...
Ezio Greggio's user avatar
4 votes
1 answer
84 views

How can I check that for a quasi-compact morphism $f:Z\to X$, the kernel of $\mathcal{O}_X\to f_*\mathcal{O}_Z$ is quasi-coherent?

If $f:Z\rightarrow X$ is a quasi compact morphism of schemes and $\mathcal J :=\ker(\mathcal O_X \rightarrow f_* \mathcal{O}_Z)$, then $(\operatorname{Supp}( \mathcal O_X/ \mathcal J)$,$i^{-1}(\...
user11695417's user avatar
1 vote
2 answers
53 views

For any open set $U\subset X$, $X$ an integral scheme, $p\in U$, $\mathscr O_{X}(U)\rightarrow \mathscr O_{X,p}$ is injective

In the case that $U$ is affine, the statement is trivial because the stalk at $p$ is the localization of $\mathscr O_{X}(U)$ except for elements in prime ideal $p$ and the map identifies with the map $...
张耀威's user avatar
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1 vote
0 answers
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Why $\chi(\mathscr{F}_n)=n$ in Goss´ Basic structures of Function Fields?

I´m reading Basic Basic structures of Function Fields, and I'm trying to understand Proposition 6.2.3, which talks about a propiety of a torsion-free coherent $\mathscr{O}_X$-module $\mathscr{F}$ on a ...
John Andrew's user avatar
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1 answer
31 views

Stalk of composition of morphisms of sheaves and composition of stalks of morphisms of sheaves

Given a topological space $X$. Let $f:F\to G$ be a morphism of sheaves of abelian groups. Naturally, $f$ induces a morphism of stalks $f_x: F_x\to G_x$ for any $x\in X$. Additionally, there exists ...
msecauchy's user avatar
1 vote
1 answer
53 views

Injection of sheaves and pullbacks

Let $X,Y$ be smooth complex projective varieties, and let $f:X\to Y$ be a morphism. Let $F$ be a vector bundle of rank $k$ on $Y$, and let $E$ be a coherent subsheaf of $F$. Consider the pullbacks: $f^...
ark's user avatar
  • 61
0 votes
0 answers
36 views

Sections of an algebraic tensor product of sheaves

Let $\mathcal{O}_X$ be the sheaf of functions on a smooth affine variety $X$. Then, for every Zariski open set $U$ in $X$, there is an isomorphism: \begin{equation} \mathcal{O}_X(U)\otimes_{\mathbb C}\...
Flavius Aetius's user avatar
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0 answers
62 views

Equivalence of the Definition of closed immersion and closed subscheme

In wedhorn and Görtz‘s algebraic geometry book Algebraic Geometry I:Schemes page 86 I found the following definition. (1)A closed subscheme of $X$ is given by a closed subset $Z\subset X$ and an ideal ...
张耀威's user avatar
  • 121
0 votes
1 answer
52 views

Uniqueness of Harder-Narasimhan filtration for coherent sheafs

I am reading the following lecture notes https://people.math.harvard.edu/%7Elurie/205notes/Lecture20-HarderNarasimhan.pdf on the Harder-Narasimhan filtration. I am having trouble understanding the ...
ASHoa BSGa's user avatar
1 vote
1 answer
39 views

Mayer-Vietoris for Sheaf with support vs Usual Mayer-Vietoris

In Hartshorne Chapter III, Exercise 2.4, (for a topological space $X = Y_1\cup Y_2$) one is asked to prove the exactness of $$ \ldots \to H^i_{Y_1\cap Y_2} (X,\mathcal{F}) \to H^i_{Y_1} (X,\mathcal{F})...
fish_monster's user avatar
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0 answers
31 views

Intermediate extension of arbitrary constructible sheaves

In Achar (and other ressources), the intermediate extension functor along a locally closed embedding $h\colon Y \to X$ for varieties $X,Y$ is defined as \begin{equation} h_{!\ast}\colon \text{Perv}(Y)\...
Pandir's user avatar
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3 votes
0 answers
158 views

How to verify a scheme is a fibre product

Above is one proposition that I found in Wedhorn& Görtz's Algebraic Geometry I in page 103. I do not understand two parts: (1) He uses the assumption II that the induced maps on stalk at any ...
Ziqiang Cui's user avatar
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0 answers
59 views

What is $\mathscr{F}(-P)$ in Goss´ Basic structures of function fields?

I´m studying Goss´ book "Basic structures of function field arithmetic" , chapter 6 about Shtukas, page 183. However, he uses a expression $\mathscr{F}(-P)$, where $\mathscr{F}$ is a torsion-...
John Andrew's user avatar
1 vote
0 answers
74 views

The reason why scheme theory emerged.

In any history, there is a cause-and-effect relationship. So I became curious about the situation in which the scheme theory came to appear. In other words, I'm curious about what problem was left ...
jhzg's user avatar
  • 301
2 votes
0 answers
37 views

Pre-Sheaf on locally compact abelian group

Let $G$ be a locally compact abelian group. We can define a pre-sheaf $\mathcal{F}$ on G by $$\mathcal{F}(U) := \widehat{\langle U\rangle}$$ for any open set $U$, where $\widehat{\langle U\rangle}$ ...
Pedro Lourenço's user avatar
1 vote
1 answer
43 views

The symbol of de Rham cohomology on Stacks Project

In the chapter of de Rham cohomology on Stack Project, there is a symbol $H^{i}(R\Gamma(X, \Omega_{X / S}^{\bullet}))$. What does $R\Gamma$ mean?
jhzg's user avatar
  • 301
4 votes
2 answers
67 views

Which representation of $\mathrm{SL} (2; \mathbb{C})$ does the tautological bundle on $ \mathbb{CP}^1 $ correspond to?

If I understand correctly, by Borel-Weil-Bott, for each weight of $\mathrm{SL} (2; \mathbb{C})$ there is a corresponding (holomorphic) line bundle / invertible sheaf on $ \mathbb{CP}^1 $. Denoting the ...
smitke6's user avatar
  • 699
1 vote
0 answers
43 views

The stalk of a presheaf is isomorphic to the stalk of its inverse image

Let $f:X\to Y$ be a continuous map of topological spaces, and let $\mathcal O$ be a presheaf of (commutative unitary) rings over $Y$. I'm trying to understand how the fact that colimits commute with ...
Ezio Greggio's user avatar
0 votes
0 answers
25 views

Extending sheaves from a locally closed subset

Let $X$ be a topological space and $\mathcal{F}$ a sheaf over $A$ where $A\subset X$ is locally closed. Godement theorem II.2.9.2 states there exists a unique sheaf $\mathcal{F}_X$ on $X$ whose ...
amd1234's user avatar
  • 349
0 votes
0 answers
34 views

Equivalent description of conormal sheaf of immersion

This question describes several equivalent definitions of conformal sheaf of closed immersion, but I still have some doubts about conformal sheaf of immersion. Let $i: Z \rightarrow X$ be a immersion,...
jhzg's user avatar
  • 301
4 votes
1 answer
110 views

Making clear the definition of 'affine variety' in Mumford's book.

I am reading "The Red Book of Varieties and Schemes" by Mumford. In section 4 the author defines the term affine variety: An affine variety is a topological space $X$ plus a sheaf of $k$-...
Toni's user avatar
  • 55
1 vote
1 answer
49 views

Definition of degree of a vector bundle?

Sorry if this question is extremely basic. I'm trying to understand the definition of degree of a vector bundle over a smooth projective curve over $\mathbb{C}$, using the additivity of degree on ...
Anthony Lee's user avatar
1 vote
0 answers
44 views

Decomposition theorem for resolution of surface singularities

In the section 3.1 of the paper Intersection forms,topology of maps and motives decomposition for resolution of three folds by de Cataldo and Migliorini: https://arxiv.org/abs/math/0504554 They prove ...
TaiatLyu's user avatar
  • 151
1 vote
1 answer
36 views

Existence of the restriction of a global section on a sheaf.

Let $\mathcal{A}$ be a sheaf of groups on $X$ An $\mathcal{A}$-sheaf on $X$ is a sheaf of sets $\mathcal{F}$ on $X$ together with a morphisms of sheaves of sets $$ a: \mathcal{A} \times \mathcal{F} \...
Micheal Brain Hurts's user avatar
2 votes
0 answers
82 views

Name of the direct image functor

For any map, we have the inverse image with two adjoints: the image which is the left adjoint, and the "forall" operator which is the right adjoint. This applies pretty generally. However, ...
Trebor's user avatar
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