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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Map between free sheaf modules not arising from a “matrix”

My question is about this example in the Stacks project. For convenience and completeness, here is the relevant part. Let $X$ be countably many copies $L_1, L_2, L_3, \ldots$ of the real line ...
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Build sheaf from stalks

If I have a topological space $T$ and for each $p \in T$ I have an object $A_p$ in some category $\mathscr{A}$, then how can I define a sheaf out of this? In other words can I build a sheaf with ...
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Inclusion of quotient sheaves restricted to open subset

When introducing sheaf cohomology following for example Chapter 8 of Kempf's book on Algebraic Varieties, we make the following standard definitions. If $\mathcal{F}$ is a sheaf of abelian groups on ...
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A torsion-free sheaf of rank 1 on a surface

Let $X$ be a surface and $E$ be coherent sheaf on $X$. Now there is always a natural map $\mu:E\longrightarrow E^{\vee\vee}$. The kernel of this map is precisely the torsion subsheaf of $E$. Now if ...
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Linear Systems And invertible Sheaf

Hello Fellow Mathematicians/Algebraic Geometer , This is question has two parts one which is more conceptual and the other more straight forward: i) Let $X$ be a non-singular protective variety ...
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Questions about strata of a variety: the nilpotent cone of $\mathfrak{sl}_2$.

I am reading the lecture notes. In the end of page 3, let $X = \{(a, b; c, -a): a, b, c \in \mathbb{C}^3, a^2 + bc = 0\}$. It is said that there are two strata: the regular orbit $U$ and $0$. What is ...
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(Non-)Isomorphism of (pre-)sheaves

I want to prove the following statement: Let $f:F \to G$ be a homomorphism of sheaves. Let $(U_i)_{i \in I}$ be an open coverage of $X$, such that $f|{U_i}: F|U_i \to G|U_i$ is an isomorphism for all ...
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Axiom of glueing: direct limit of sheaves in a noetherian topological space. [duplicate]

I'm trying to prove that in a noetherian topological space the following property is satisfied: Consider a direct system of sheaves and morphisms $\{ \cal{F}_t, f_{ij} \}_t$. Consider the presheaf ...
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43 views

Taking module sheaf commutes with tensor product

I'm trying to prove proposition II.5.2.b in Algebraic Geometry by Hartshorne. The proposition states that for $ A $-modules $ M $ and $ N $ and $X=\text{Spec}\ A$ there is an isomorphism $ ...
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First axiom of sheaves: in noetherian topological spaces the direct limit presheaf is a sheaf.

Consider a topological space $X$ and a direct limit of sheaves and morphisms $\{ \cal{F}_i, f_{ij}\}$. Define the direct limit presheaf by $U \to \varinjlim \cal{F}_i $. In general this is just a ...
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Proof of $\mathcal{O}_{\mathbb{P}^1 \times \mathbb{P}^1}(a,b)$ is ample $\iff$ $a,b >0$.

I would like some help understanding the proof in $(\impliedby)$ direction. Hartshorne on page 156, Example 7.6.2 says: If $\mathcal{L}$ is an invertible sheaf on $\mathbb{P}^1 \times \mathbb{P}^1$ ...
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Inductive limit of sheaves over noetherian topological space

Let $X$ be a topological space. Let $I$ be a poset and let $\mathcal F_i$ for $i\in I$ be sheaves on $X$, and $\{\pi_{ij}\colon \mathcal F_i \rightarrow \mathcal F_j\}_{i,j\in I}$ be an inductive ...
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Analogue of locally constant sheaf in algebraic geometry

If I take just the definition of locally constant sheaves for algebraic varieties I get something pretty trivial (basically due to irreducibility); so how can one make a set up similar to what happen ...
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Direct limit sheaf.

Let $\{ \cal{F}_i, \mu_{ij}\}$ a direct system of sheaves and morphisms on a topological space $X$. Define the direct limit os the system $\{ \cal{F}_i, \mu_{ij}\}$ as the sheaf associated to the ...
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An isomorphism theorem for sheaves.

Let $\varphi: \cal{F} \longrightarrow \cal{G}$ a morphism of sheaves. My goal is to prove that $im\varphi \simeq \cal{F} / Ker \varphi$. My thoughts about this problem: 1) $im \varphi(U) \simeq ...
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Sheafification preserves injectivity.

I want to prove the following: Let $\varphi: \cal{F} \longrightarrow\cal{G}$ a morphism of presheaves such that $\varphi (U): \cal{F}(U) \longrightarrow\cal{G}(U)$ is injective for each $U$. Then the ...
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Hartshorne notation in section III.12

I am reading section III.12 in Hartshorne, the one about the Semicontinuity Theorem. For $f:X \rightarrow Y$, where $Y=\mathrm{Spec}A$ and $\mathcal{F}$ a coherent sheaf on $X$, he writes ...
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must a surjective sheaf map be surjective on some open set?

A surjective (i.e. surjective on stalks) map of sheaves $F\to G$ on a space $X$ need not be surjective on global sections, but is it true that for every $p\in X$, there exists a neighborhood $U$ of ...
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61 views

Acyclicity of Flasque sheaves without A.C.

I say that a sheaf on a space X, is flasque if the restriction maps are surjective, that is any local section extend to a global section. Now it is a fact that if $F_1$ is flasque and if $0 \to F_1 ...
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Interpretation of sheaf flat over a base

I am trying to get an interpretation of what means for a sheaf to be flat with respect to a base. The definition is that, given $f:X \rightarrow Y$ morphism of schemes, $\mathcal{F}$ is flat over $Y$ ...
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Product of affine varieties is the product of topological spaces

Let $k$ be an algebraically closed field, and $A, B$ affine $k$-algebras. We can define a functor $\mathfrak F$ from the category of affine $k$-algebras to that of affine algebraic varieties, by ...
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The higher cohomologies of a quasi-coherent sheaf on the intersection of two affine open subsets.

It is well-known in algebraic geometry that if $X$ is affine and $\mathcal{F}$ is a quasi-coherent sheaf on $X$, then $$ H^i(X,\mathcal{F})=0,~ \forall ~i\geq 1. $$ Now let $X$ be an arbitrary ...
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Is the push-forwad of a quasi-coherent sheaf under open immersion still quasi-coherent?

My question is related to this question: When is the pushforward of a quasi-coherent sheaf quasi-coherent? Hartshorne proof In Hartshorne page page 115 Proposition 5.8 it has been proved that if $X$ ...
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On the definition of locally free module sheaf

A sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ is by definition locally free iff there is an open covering $X = \bigcup_i U_i$ such that each $\mathcal{F}|_{U_i}$ is free as a sheaf of ...
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About the presheaf used to define the inverse image sheaf.

Let $f \colon X \to Y$ be continuous and $\mathcal{F}$ be a sheaf on $Y$. Then the inverse image sheaf $f^*\mathcal{F}$ is defined to be the sheafification of the presheaf on $X$ given by $$ U \mapsto ...
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Injective resolution of abelian sheafs induces cochain complex on global sections.

Given an injective resolution for an abelian sheaf $\mathcal{F}$ on $X$, that is an exact sequence $$ 0 \to \mathcal{F} \xrightarrow{\epsilon} \mathcal{A}^0 \xrightarrow{\delta^0} \mathcal{A^1} ...
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58 views

constant stalks but not constant sheaf?

In the coherent world we have the following: if X is reduced and F is a coherent sheaf on it, then if the rank of all fibres in constant then F is locally free. I thought something similar held in ...
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When does Sheafification commute with direct image?

Given a presheaf $\mathcal{F}$ on a space $X$ and a map $f: X \rightarrow Y$, when does $f_* A(\mathcal{F}) = A(f_* \mathcal{F})$, where $A$ is the associated sheaf/sheafification functor? Since ...
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Action of $Aut(X)$ on $Coh(X)$

I was reading about Bondal and Orlov reconstruction theorem. In particular that for a smooth variety with ample or anti-ample canonical bundle $\mathrm{Aut}(D^b(X)) \cong \mathbb{Z} \times ...
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Extensions of quasicoherent sheaves are quasicoherent.

Harts theorem 5.7: Given an exact sequence $0 \to \mathscr F_1 \to \mathscr F_2 \to \mathscr F_3 \to 0 $ of sheaves on $X = \mathrm{spec} A$, if $\mathscr F_1$ and $\mathscr F_3$ are quasicoherent, ...
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1answer
29 views

Equivalent conditions for a closed immersion of schemes

In Hartshorne, a closed immersion of schemes is defined to be a scheme morphism $\Phi \colon Y \to X$ such that $\Phi$ is a homeomorphism onto $\Phi(Y)$, $\Phi(Y)$ is closed in $|X|$ and $$ (*) ...
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Small question regarding gluing sheaves

Let $X$ be a topological space and $X = \cup U_i$ an open cover of $X$. Suppose we have sheaves $F_i$ on $U_i$ along with isomorphisms $\phi_{ij}: F_i |_{U_i \cap U_j} \rightarrow F_j |_{U_i \cap ...
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cohomology of local system

Let $X_r$ be a finite simplical complex. Let $V_r$ be a sheaf which is a local system on $X_r$. Is it true that: $H^n(X_r,V_r$) i.e the cohomology of the sheaf $V_r$ coincide with the cohomology of ...
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59 views

Pull-back line bundle under morphism of degree $d$

This question is partially related to Direct image of vector bundle Let $f:\mathbb{P}^1\to\mathbb{P}^1$ be a morphism of degree $d$. For $n>0$, how can we compute ...
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What is meant by “A and B represent the same functor whence are isomorphic” in this solution?

While browsing some old questions I came across the following: tensor product of sheaves commutes with inverse image It seemed like something interesting was going on in the answer, but I don't quite ...
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morphism of ringed spaces glue

Suppose $(X,O_X)$ and $(Y, O_Y)$ are ringed spaces, and let $X = U_1 \cup U_2$ be an open cover. Suppose we have morphism of ringed spaces $\pi_i: U_i \rightarrow Y$ such that they agree on the ...
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sections of tensor product of sheaves of modules

I am confused about notation concerning tensor products of sheaves of modules. I know that given a ringed space $X$ and $\mathcal{O}_X$-Modules $\mathcal{F}$ and $\mathcal{G}$ their tensor product is ...
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1answer
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is the property of representability of a sheaf on the big etale site checkable on the small site?

Let $S$ be a scheme and $F$ a sheaf on $(\textbf{Sch}/S)_\text{etale}$, whose restriction to the small etale site $S_\text{etale}$ is representable (in fact in my case this restriction is ...
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Sections of inverse image sheaf of sheaf of sections of vector bundle

Let $\eta \colon Y \to Z$ be holomorphic map between complex manifolds, $E$ holomorphic vector bundle over $Z$, $\eta ^\ast E$ its pullback over $Y$, ${\cal O}_E$ sheaf of sections of $E$, and ...
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1answer
25 views

Local homeomorphisms: terminology

Let $(E,\pi,X)$ be a local homeomorphism, so for any $x\in E$ there is an open set $U\ni x$ such that $\pi|U$ maps $U$ homeomorphically onto $\pi[U]\subseteq X$, and $\pi[U]$ is open in $X$. Is there ...
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When defining a Grothendieck pretopology,can we get away with less than the fibre product axiom?

$\newcommand\restr[2]{{\left.#1\right|_{#2}}}$ I'm fairly new to this whole area, so correct me if there are any technical errors in any of this. The base category for a classical sheaf is the ...
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When the preimage of the family of all open neighborhoods of a point is cofinal

Let $f \colon X \to Y$ be a continuous map of topological spaces. Denote by $O_y(Y)$ the family of all open neighborhoods of a point $y \in Y$. Define $$ f^{-1}(O_y(Y)) = \bigl\{ f^{-1}(U) \colon U ...
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A question about the definition of a stalk

In the definition of the stalk of a presheaf of abelian groups on a topological space, at a point, one uses the fact that the open sets containing that point form a poset, which is directed via ...
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Show that $\mathscr{C}_{X,a}\cong\mathscr{C}_{A^n,a}/ I(X)\mathscr{C}_{A^n,a}$.

Let $X\subset A^n$ be an affine variety, and let $a\in X$ be a point. Show that $\mathscr{C}_{X,a}\cong\mathscr{C}_{A^n,a}/ I(X)\mathscr{C}_{A^n,a}$, where $I(X)\mathscr{C}_{A^n,a}$ denotes the ideal ...
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44 views

Showing that some function is not a rational function on Spec $A$

Let $k$ be a field, $A = k[x,y] / (y^2, xy)$. I know that the only associated points of Spec $A$ are $[(x,y)]$ and $[(y)]$. I want to show that $$ \frac{x-2}{(x-1)x} $$ is not a rational function. ...
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A basic question regarding a rational function on a locally Noetherian scheme

Let $k$ be a field, $A = k[x,y] / (y^2, xy)$. I know that the ideals $P = (x-1)$ and $Q = (x-3)$ are maximal ideals in A, so $U = $ Spec $A \backslash \{ P, Q \}$ is an open subset of Spec $A$. Then ...
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How to show that this presheaf is a sheaf?

I am working with a presheaf of rings and I'm having problems to show that it is in fact a sheaf. Specifically: Let $A$ be a commutative ring with identity. Let $E(A)$ be it's ring of idempotents with ...
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1answer
44 views

Morphism of $\mathscr O_X$ modules induced by gbolal section

Let $A$ be a commutative ring with identity, $X = \text{Spec}A$. Let $\mathscr F$ be a sheaf of $\mathscr O_X$ modules. Let $M = \Gamma(X, \mathscr F)$. Is there a natural way to induce a morphism ...
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1answer
48 views

Exercise in Springer: Linear Algebraic Groups (1.4.8 (i) )

This is a pretty bad book to learn algebraic geometry from if you've never seen it before. I'm trying to verify the following assertion. Let $k$ be an algebraically closed field, and $X, Y$ be ...
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27 views

Do surjective sheaf morphisms have a one-sided inverse?

If $f:A\rightarrow B$ is a surjective function of sets then $\exists$ $g:B\rightarrow A$ such that $f\circ g=1_B$, the identity map on $B$. The same is true for groups and homomorphisms, rings and ...