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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26 views

Can I define a site as a category endowed with a pretopology instead of a topology?

If $K$ is a pretopology on a category $\mathcal{C}$ and $J$ the topology it induces, are the Grothendieck toposes $\text{Sh}(\mathcal{C},K)$ and $\text{Sh}(\mathcal{C},J)$ the same in general? As I ...
5
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41 views

Why don't we use the constant sheaf as the dualizing sheaf in Verdier Duality?

The Verdier dual of a complex of sheaves $F$ on a complex manifold $X$ is defined in the derived category $D_b(X)$ as $\mathcal{RHom}(F,\mathbb{D}_X)$, where $\mathbb{D}_X$ is the dualizing "sheaf" ...
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21 views

Prove extension by zero is a special case of lower shriek?

The lower shriek functor is defined by $$f_{!}F(U)=\{s\in\Gamma(f^{-1}(U),F)\;:\; f|_{\mathrm{supp}(s)}:\mathrm{supp}(s)\rightarrow U\text{ is proper}\}$$ On the other hand, if $j:V\subset X$ is the ...
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1answer
21 views

Does the inverse image sheaf have a left adjoint for $\mathsf{Set}$-valued sheaves?

It's known that for sheaves with values in modules, the inverse image sheaf functor $j^\ast$ for $j:U\subset X$ an inclusion of an open set has a left adjoint which is extension by zero. Is there any ...
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40 views

What is meant by $\mathbb{C}^*$?

In Brylinski (Loop Spaces, Characteristic Classes and Geometric Quantization), the symbol $\mathbb{C}^*$ is used quite a lot, and I'm really not sure what it means. I've no shortage of ideas: the ...
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1answer
14 views

Restriction of a sheaf equals zero

I'm reading online lectures notes on sheaves, and I'm confused about the meaning of the equation $F|_U=0$ below. $F|_U$ is defined as the inverse image sheaf $f^\ast F$ along the inclusion of some ...
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49 views

What are some properties of the sheaf of distributions?

In a course on measure theory, the lecturer proved that distributions (on a locally convex space I think) form a sheaf $\mathcal D$. He isn't interested in sheaves, so he didn't elaborate. Afterwards, ...
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1answer
63 views

Additivity of the first Chern class

I have a highly elementary question: is the first Chern class additive? More specifically, given a short exact sequence of coherent sheaves on a nonsingular curve $X$ $$ 0 \to \mathscr F'\to \mathscr ...
2
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1answer
57 views

Is the category of (pre)sheaves over a singleton isomorphic to the category of sets?

I'm wondering whether $\mathsf{PSh}(\{x\})$ or $\mathsf{Sh}(\{x\})$ are equivalent to the category of sets. For sheaves, since the value on an initial object is a terminal object, It seems like the ...
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0answers
90 views

Simple question in “Sheaves in geometry and logic”

There's an argument I don't understand in "Sheaves in geometry and logic" by Mac Lane and Moerdijk, that seems a priori easy but I can't see it. Page 174, diagram (10) (involving the power ...
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29 views

Normal cone and specialization

This question is from the Kashiwara and Schapira's book: Sheaves on Manifolds Let M be a closed submanifold of X and let S be a locally closed subset of X, prove that C$_M$(S) = ...
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1answer
67 views

Finite flat pushforward of a constant sheaf

Let $A$ be an abelian group and consider the associated constant sheaf $A$ on a (smooth projective) variety $Y$ (over a field). Let $f: Y \to X$ be a surjective finite flat morphism. Is $f_*A$ also ...
2
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1answer
54 views

Opposite directions of adjunction between direct and inverse image in $\mathsf{Set}$ and $\mathsf{Sh}(X)$

Let $f:X\rightarrow Y$ be an arrow in $\mathsf{Set}$. $f$ induces three functors: $$\exists (f),\forall (f):\mathcal P(X)\rightarrow \mathcal P(Y),\; \mathsf I(f):\mathcal P(Y)\rightarrow \mathcal ...
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1answer
189 views

Grothendieck's yoga of six operations - in relatively basic terms?

I'm reading about the basic interactions between sheaves over topological spaces and arrows in $\mathsf{Top}$, in particular, about the inverse/direct image functors $f^\ast \dashv f_\ast$, the proper ...
3
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1answer
34 views

Inverse image sheaf functor: proving $(f^\ast P)_x=P_{f(x)}$

Let $P$ be a $\mathsf{Set}$-valued presheaf and let $f^\ast:\mathsf{PSh}(Y)\rightarrow \mathsf{PSh}(X)$ be the (topological) inverse image sheaf functor, defined on objects as the filtered colimit ...
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41 views
+50

How do the direct and inverse image sheaf functors interact with homotopy?

The direct image sheaf functor $f_\ast$ and inverse image sheaf functor $f^\ast$ (here I mean the usual inverse image sheaf functor often denoted by $f^{-1}$) form a well-known adjunction for ...
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102 views

Why is the sheaf $\mathcal{O}_X(n)$ called the “twisting sheaf” (where $X=\operatorname{Proj}(S)$ for a graded ring $S$)?

Basically my question is why the sheaf $\mathcal{O}_X(n)$ is called the twisting sheaf, here $X=\operatorname{Proj}(S)$ and $S$ any graded ring.
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28 views

How is a connection the same as Morphism between different pullbacks to the first infinitesimal neighbourhood

This question arose from the first pages of Delignes Equations Differentielles. There he defines a connection on a vector bundle $V$ on an complex analytic space $X$ as the following Data: For all ...
4
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0answers
52 views

Abstract definition of a differential operator

In Kolar, Michor, and Slovak, a differential operator is said to be a rule transforming sections of a fibred manifold $Y \to M$ into sections of another fibred manifold $Y' \to M'$. Is this is a ...
3
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1answer
32 views

Globally generated vector bundle

Could someone give me a definition of globally generated vector bundle? A rapid search gives me the definition of globally generated sheaves, but I am in the middle of a long work and don't really ...
4
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1answer
63 views

Computing the sheaf of 1-forms on a toric variety

Consider projective space $P^{2}$ and its corresponding fan. We have the affine opens defined by $U_{\sigma_{0}} = Spec(\mathbb{C}[x,y])$, $U_{\sigma_{1}} = Spec(\mathbb{C}[x^{-1},x^{-1}y])$ and ...
2
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3answers
73 views

Presheaf that do not satisfy: If $\{U_i\}$ is an open cover of $U \subseteq X$ and $s \in \mathcal{F}(U)$, then $s=0$ iff $s|_{U_i}=0$ $\forall i$

Let $X$ be a topological space. Find an example of a presheaf $\mathcal{F}$ that do not satisfy: If $\{U_i\}_{i \in I}$ is an open cover of $U \subseteq X$ and $s \in \mathcal{F}(U)$, then ...
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1answer
30 views

What does the notation $\mathcal{O}_{\mathbb{P}^n}(1)$ mean?

I have tried looking at my sheaves notes but couldn't find anything.
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12 views

Sheafification and restriction to open subset

Let $X$ be a topological space and $\mathcal{F}$ be a presheaf on $X$. We denote by $\mathcal{F}^+$ the sheafification of $\mathcal{F}$. Let $U\subset X$ be an open subset. We denote by ...
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11 views

Tensor product of sheaf

Let $M$ and $N$ be sheafs on a space $X$ then what is the relation of $\pi_1^* M\otimes \pi_2^* N$ and $\pi_1^* N\otimes \pi_2^* M$, where $\pi_i$ is the $i$ projection of $X\times X$
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1answer
37 views

Definition of a sheaf: What is $s\rvert_{V_i}$ if $V_i\not\subseteq U$?

I am reading Hartshorne's book on algebraic geometry, which defines a sheaf to be a presheaf $\mathscr F$ on a topological space $X$ such that: For all open sets $U$ and open coverings $\{V_i\}$ of ...
5
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1answer
101 views

Calculating global sections of sheaves

Consider the usual projective space $\mathbb{P}^{1} = \mathbb{C} \cup \{\infty\}$, and the Weil divisor $D = \{0\} \subset \mathbb{C} \subset \mathbb{P}^{1}$. Writing projective space as the union of ...
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2answers
41 views

Where is sheafification in the definition of exact sequence of sheaves?

I am reading Andreas Gathmann's notes on Algebraic geometry,http://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/main.pdf Def 7.1.14(iv)says the following As usual, a sequence of sheaves ...
3
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67 views

sections of higher direct image sheaf

Let $f:X \rightarrow Y$, be a proper birational morphism of projective algebraic varieties with $X$ smooth. Denote by $R^if_* \mathcal{O}_X$, the higher direct image sheaves. Do exists a simple way ...
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64 views

Can someone explain presheaf to a calculus student?

From reading some stuff online, there is the claim that continuous functions are presheafs, so that the toy functions I play with in class such as $f(x) = x^2$ can also be thought of as presheafs. ...
3
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1answer
40 views

Step in the construction of the global spec of a sheaf of algebras

I'm working my way through the construction of the global spec of a sheaf of algebras. Here is the setup. Let $ Y $ be a scheme. Let $ \mathscr{A } $ be a quasi coherent sheaf of $ ...
2
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1answer
58 views

Sheafification as a Kan Extension of the Identity?

How can the sheafification functor be described in terms of a Kan extension of the identity on the category of $\mathsf{Set}$-valued sheaves (over some topological space)? How about general $\mathsf ...
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87 views

Calculation with Leray spectral sequence

The Leray spectral sequence is a cohomological spectral sequence of the form $$H^p(Y;R^q f_*(F)) \Longrightarrow H^{p+q}(X;F)$$ for abelian sheaves $F$ on a site $X$ and morphisms of sites $f : X \to ...
3
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1answer
51 views

Subsheaf generated by one section is coherent

I'm working on exercise II.5.15 in Hartshorne's book. I need to prove the following bit. Let $ X $ be a noetherian scheme. Let $\mathscr {F }$ be a quasi coherent sheaf on $ X$. Then the subsheaf ...
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1answer
31 views

Decomposing Sheaves into a direct sum

I have two questions about sheaves inspired by representation theory: Is there a theorem that tells me when a sheaf of vector spaces can be decomposed into the direct sum of two "smaller" sheaves? ...
2
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34 views

“Étalé space almost looks like a sheaf”

In Daping Weng's A Categorical Introduction to Sheaves, in the paragraph preceding section 3.2 (page 8), the author says A sheaf space [= Étalé space] almost looks like a sheaf... I would very ...
4
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1answer
101 views

What is the definition for a fine sheaf/ a partition of the unity on a sheaf?

From Griffiths and Harris, p. 42: A sheaf $\mathcal F$ on $M$ is called fine, if for $U=\bigcup_i U_i\subseteq M$ there is a partition of the unity subordinate to the cover $(U_i\mid i\in I)$. By this ...
4
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0answers
50 views

Determining a scheme $X$ is affine from $Qcoh(X)$

My question is a subquestion of this question. I do not want to use full reconstruction theorems. The settings is the following. Let $X$ be a scheme. Assume that the adjunction in TAG01BH of the ...
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1answer
25 views

Canonical group structure of germs

Given a topological space $X$ and a presheaf of abelian groups on $X$, $A$, we can construct the set of germs at a point $x\in X$ by taking $\mathscr{A}_x=\lim\limits_{\rightarrow} A(U)$ where $x\in ...
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1answer
36 views

Characterisation of closed subschemes of projective spaces

I'm trying to understand a step in the proof of corollary 5.16 in Hartshorne. The statement Let $ A $ be a ring. if $ Y$ is a closed subscheme of $\Bbb{P}_{A}^{r} $ then there is a homogeneous ...
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1answer
36 views

Pushforward and sheaf-hom

If $S$ is a surface over the complex numbers $\mathbb{C}$, and $C$ is a curve in $S$, and $i:C\longrightarrow S$ is the inclusion morphism. If $A$ is a line bundle over $C$, then is it true that ...
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1answer
82 views

Is this sheaf simple?

Let $S$ be a surface and $C$ be an effective divisor in $S$. That is $C$ is a curve in $S$ and $i:C\longrightarrow S$ is the inclusion morphism. Let $E$ be line bundle over $C$, so $i_*E$ is a ...
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1answer
32 views

Is any reflector from a presheaf category $PSh(K)$ to a topos $C$ necessarily left exact?

Let $C$ be a topos, $K$ a small category and $$ PSh(K) \leftrightarrows C $$ a reflective subcategory with inclusion $i\colon C\hookrightarrow PSh(K)$ and reflector $T$. Is $T$ left exact? $D$ ...
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32 views

Compute some sheaf cohomology supported at a point

This question is from the Kashiwara's book: Sheaves on Manifolds Ex3.1 Let $X=\mathbb{R}$,$Z=\left\{\frac{1}{n}; n\in\mathbb{N}\setminus\{0\}\right\}\cup\{0\}$ and $Y=X\setminus Z$. Prove that ...
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0answers
52 views

Abstract interpretation of isomorphism between tensor product with dual and hom

I'm interested in the following statement, coming from Remark 6.4.21 of Qing Liu's Algebraic Geometry and Arithmetic Curves: Let $\mathcal{F}, \mathcal{G}$ be quasi-coherent sheaves on a scheme ...
2
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1answer
91 views

Why are sections of sheaves called sections?

Generally, a section is a right inverse. On the other hand, if $F$ is a ($\mathsf{Set}$-valued) sheaf, then the elements of $FU$ are usually also called sections. Why is this terminology justified ...
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30 views

Subfunctor of a sheaf which is not a sheaf

The following is proposition 1 from Maclane & Moerdijk's Sheaves in Geometry and Logic, part II, section 1. Proposition 1. If $F$ is a sheaf on $X$, then a subfunctor $S\subset F$ is a subsheaf ...
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26 views

Theory around the Cellular Sheaf

I have lately stumbled upon cellular (co)sheaves, which look very interesting. To understand them better, I would like references that systematically develop the theory behind them (preferably in ...
1
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1answer
61 views

Ideal sheaf of intersection of two surfaces in $\mathbb P^3$

Let X be an intersection of 2 surfaces of degree $d_1,d_2$ in $\mathbb P^3$. Is it true that there is a short exact sequence $$ ...
3
votes
2answers
70 views

An example of a coproduct of sheaves in the category of presheaves that is not a sheaf

For a site $C$ there is an adjunction $a\colon PSh(C)\leftrightarrows Sh(C)\colon \iota$ with sheafification $a$ and inclusion $i$. $a$ preserves finite limits. The inclusion $\iota$ does not preserve ...