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3
votes
1answer
48 views

Hom functor of quasi-coherent sheaf maybe not quasi-coherent

I notice that some books say that for arbitrary quasi-coherent sheaves $F$, $G$ over a scheme $X$, the $\mathcal O_X$-module $\mathrm{Hom}_{\mathcal O_X}(F,G)$ maybe not quasi-coherent, who can give ...
4
votes
0answers
69 views

Global generation of $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ and $\mathcal{E}$

I'm trying to prove that $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ is generated by global sections on $\mathbb{P}(\mathcal{E})$ if and only if $\mathcal{E}$ is generated by global sections ($\pi: ...
3
votes
1answer
97 views

Quasi-coherent sheaves on varieties

I am reading Kempf's book "Algebraic varieties". On page 55 the author considers a sheaf of rings ${\mathcal{A}}$ (commutative, unital) on a topological space $X$. An ${\mathcal{A}}$-module ...
3
votes
0answers
84 views

Question from Liu, Chapter 5 Ex 1.16

I have a question from Liu's book on Algebraic Geometry, doing Chapter 5 Question 1.16: Let $f:X\rightarrow S$ be a morphism of schemes, with the following base change. $$\require{AMScd} \begin{CD} ...
1
vote
0answers
69 views

Relative spectrum of a quasi-coherent algebra.

I'm working out the notion of spectrum of a quasi-coherent algebra over a scheme. $\require{AMScd}$ Let $(X,\mathcal O_X)$ be a scheme and $\mathcal A$ a quasi-coherent $\mathcal O_X$-algebra, i.e. a ...
7
votes
1answer
329 views

Deligne's formula

Let $M$ be some $A$-module and $f \in A$. Why do we have an isomorphism $$\varinjlim_n \hom_A(f^n A,M) \cong M_f \text{ ?}$$ Background. Let $X$ be a scheme, $U$ an open subscheme, and $F,G$ ...
3
votes
1answer
158 views

Elementary questions on ample invertible sheaves

Let $X$ be a quasi-compact scheme. Let $\mathcal L$ be an invertible sheaf on $X$. We say $\mathcal L$ is ample if for any finitely generated quasi-coherent sheaf $\mathcal F$ on $X$, there exists ...
7
votes
1answer
172 views

Quasicoherent ideal sheaves on open subschemes

Let $X$ be an open subscheme of an affine scheme $\operatorname{Spec} A$, let $f : X \to \operatorname{Spec} A$ be the inclusion, and let $\mathscr{I}$ be a quasicoherent ideal sheaf on $X$. Since ...
7
votes
1answer
107 views

Definition of prime ideal sheaf

In the chapter I.3.3 of Eisenbud & Harris "The Geometry of Schemes" they give a definition of "prime ideal sheaf": Let $\mathcal{O}_S$ be the structure sheaf of a scheme $S$, and let ...
2
votes
0answers
67 views

Example of a coherent sheaf on an open subset where extension isn't trivial?

Motivation: I am working on problem II.5.15 in Hartshorne's Algebraic Geometry, which is to prove that, given a noetherian scheme $X$, an open subset $U\subset X$, and a coherent sheaf $\mathscr{F}$ ...
4
votes
1answer
286 views

The projection formula for quasicoherent sheaves.

I am looking for a certain way of proving the following : Let $f. X \rightarrow S$ be a morphism of schemes. Suppose that f is quasiseparated and quasicompact, or that X is noetherian. Let ...
8
votes
2answers
341 views

Inverse image of the sheaf associated to a module

In Hartshorne, Algebraic geometry it's written, that for every scheme morphism $f: Spec B \to Spec A$ and $A$-module $M$ $f^*(\tilde M) = \tilde {(M \otimes_A B)}$. And that it immediately follows ...
3
votes
1answer
97 views

Tensor product of the structure sheaf with itsself

Let $\pi : X \to S$ be a morphism of schemes. Is the $\mathcal{O}_X$-module $\mathcal{O}_X \otimes_{\pi^{-1} \mathcal{O}_S} \mathcal{O}_X$ quasi-coherent? Here, $\mathcal{O}_X$ acts on the tensor ...
3
votes
1answer
120 views

How to compute $H^1(\Bbb P^1,\mathcal{O}_{\Bbb P^1})$ and $H^1(\Bbb P^1,\mathcal{O}_{\Bbb P^1}(n))$

Let $\mathcal{O}_{\Bbb P^1}$ be the structure sheaf of the projective line $X=\Bbb P^1_k$ over some field $k$ (algebraically closed of characteristic $0$). What is a good (and, preferably, easy) way ...
22
votes
1answer
516 views

Quasi-coherent sheaves, schemes, and the Gabriel-Rosenberg theorem

In the context of commutative rings, a ring is completely determined by its category of modules. That is, two commutative rings $R$ and $S$ are isomorphic if and only if the category of $R$-modules is ...
9
votes
2answers
395 views

Stalk of a pushforward sheaf in algebraic geometry

Excuse me if this is a naive question. Let $f : X \to Y$ be a morphism of varieties over a field $k$ and $\mathcal{F}$ a quasi-coherent sheaf on $X$. I know that for general sheaves on spaces not much ...
5
votes
1answer
193 views

Modules over a functor of points

I have a question on the ''functor of points''-approach to schemes and $\mathcal{O}_X$-modules. Please let me first write up a defintion. Let $Psh$ denote the category of presheaves on the opposite ...
7
votes
2answers
368 views

Classification of automorphisms of projective space

Let $k$ be a field, n a positive integer. Vakil's notes, 17.4.B: Show that all the automorphisms of the projective scheme $P_k^n$ correspond to $(n+1)\times(n+1)$ invertible matrices over k, modulo ...
8
votes
1answer
386 views

is the pushforward of a flat sheaf flat?

Let $f:X \to Y$ be a morphism of schemes and let $F$ be an $\mathcal{O}_X$-module flat over $Y$. Is $f_*F$ flat over $Y$? What's wrong with this argument? [EDIT: as Parsa points out, the (underived) ...
5
votes
1answer
181 views

First Ext group of a sheaf

Let $F$ be a quasicoherent sheaf on a scheme $X$, which is supposed to be sufficiently nice. Does one then have a canonical isomorphism $Ext^1(F,F) \simeq H^1(X, \underline{End}(F))$, where with ...
2
votes
0answers
124 views

exact sequence of sheaves

I'm starting with $X=\mathbb{P}^2(\mathbb{C})$ and a cubic curve $B \subset X$ and a flex $P$ on $B$ such that for a hyperplane section $H$ I have $3P \sim dH\vert_B$ (where $d \in \mathbb{N}$). With ...
0
votes
0answers
84 views

Adjunction morphism in a projection of schemes

how can I see quickest that the following holds: Let $X$ and $Y$ be sufficiently nice schemes (e.g. always noetherian or maybe varieties) and denote with $p$ the projection $X\times Y \rightarrow ...
5
votes
2answers
614 views

Arbitrary products of quasi-coherent sheaves?

I have a short question: Does the category of quasi-coherent sheaves on a scheme have arbitrary products? I know that it does if the scheme is affine and I know that they will not be isomorphic to ...
5
votes
2answers
131 views

Cohomological decomposition of tensor sheaves?

My question is similar to this, but not identical. I believe the following to be true, but I'd like a reference. Given (quasicoherent?) sheaves of $\mathcal O_X$ modules $E$ and $F$ on a projective ...
4
votes
1answer
411 views

Classifying Quasi-coherent Sheaves on Projective Schemes

I know some references where I can find this, but they seem tedious(Both Hartshorne and Ueno cover this). I am wondering if there is an elegant way to describe these. If this task is too difficult in ...