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Serre duality for arbitrary sheaves?

Let $X$ be a connected proper Cohen-Macaulay scheme over a field $k$ of dimension $n$. Is there an example of an $\mathcal{O}_X$-module $\mathcal{F}$ on $X$ such that Serre duality $$ \operatorname{...
fish_monster's user avatar
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For quasi-coherent sheaves $\mathcal{F}, \mathcal{G}$ on a proper scheme, $\operatorname{Ext}^2(\mathcal{F},\mathcal{G}) = 0$

In their book, Görtz and Wedhorn, claim that the derived category of quasi-coherent sheaves on a proper scheme has the following properties: abelian exact coproducts for all $\mathcal{F},\mathcal{G} \...
fish_monster's user avatar
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A crystal on $\operatorname{Cris}(X/S)$ is the same as giving a projective system of crystals on $\operatorname{Cris}(X/S_n)$ with compatability?

I'm currently reading Berthelot- Cohomologie cristalline des schémas de caractéristique p>0 and I'm having a question on page 228. He claims that if $X=\operatorname{Spec}(k)$ for a perfect field $...
Enki's user avatar
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coherent sheaves annihilated by ideal sheaves and morphisms between them

Let $X$ be a Noetherian scheme and $\mathcal I\subseteq \mathcal O_X$ be a coherent ideal sheaf defining a closed subscheme $Z$ of $X$. Let $i:Z\to X$ be the closed immersion. I have the following ...
Alex's user avatar
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Stalk of coherent sheaves and pushforward/pullback

Let $X$ be a Noetherian scheme. Let $\mathcal F$ be a coherent $\mathcal O_X$-module. Let $x\in X$. There is a natural morphism Spec $\mathcal O_{X, x}\xrightarrow{j} X$. Define $\mathcal F(x):=j_*(...
Alex's user avatar
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surjective Ox-mod endomorphism on quasi-coherent finite type sheaf is bijective

Exercise 7.22 in Görtz-Wedhorn "Algebraic Geometry I: Schemes" goes as follows: "Let $X$ be a scheme, $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module of finite type. Show that ...
Absent mind's user avatar
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Sections of $F(D(s))$ on a qcqs scheme extend to $(F \otimes L^{\otimes n})(X)$.

Vakil's Foundations of Algebraic Geometry, exercise 15.4.N. Let $F$ be a quasicoherent sheaf on a qcqs scheme $X$, and $L$ is an invertible sheaf with global section $s$. Then, show that $(\oplus_{n \...
David Lui's user avatar
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1 answer
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Non-split exact sequence of locally free sheaves on $\mathbb{P}^1$

$\newcommand{\ra}{\rightarrow}$ On $\mathbb{P}^1_k$, This question gives that $Ext^1(O_X(2), O_X) = H^1(O(-2)) \neq 0$, so there should be a non-split exact sequence of the form $O \ra O_X \ra F \ra ...
David Lui's user avatar
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Universal property of global spec of a quasicoherent sheaf of algebra

I am reading Vakil's FOAG (july31, 2023). Now in 17.1.C he gives an exercise: Exercise. Given an $X$-morphism $\gamma:W\to\mathcal{S}pec \mathscr{R}$ where $\mathscr{R}$ is a sheaf of $\mathcal{O}_X$-...
Mizutsuki's user avatar
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Global version of an ideal being "finitely generated up to radical"

Let $X$ be a quasicompact, quasiseparated scheme and $Z$ a closed subset of $X$ with quasicompact complement. Is there a finite type quasicoherent ideal sheaf $\mathcal{J}$ on $X$ which cuts out $Z$? (...
Brendan Murphy's user avatar
2 votes
1 answer
51 views

Definition of an $\mathcal{O}_X$-module generated by its global sections in Liu

In Liu's Algebraic Geometry and Arithmetic Curves, the author says that an $\mathcal{O}_X$-module is generated by its global sections at $x \in X$ if the canonical homomorphism $\mathcal{F}(X) \...
stillconfused's user avatar
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On the right adjoint of the derived pushforward of a proper birational morphism of Noetherian quasi-separated schemes

Let $f: X \to Y$ be a proper birational morphism of Noetherian quas-separated schemes. Let $a: D(QCoh(Y))\to D(QCoh(X))$ be the right-adjoint of the derived pushforward functor $Rf_*: D(QCoh(X))\to D(...
strat's user avatar
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Vakil's Generalization of QCQS Lemma

In the most recent notes of Vakil, this is problem 15.4.Y (which I think is just Hartshorne Lemma 5.14): Let $X$ be qcqs, $\mathscr{L}$ invertible sheaf on $X$, $s\in \Gamma(X, \mathscr{L})$, and $\...
Teddy's user avatar
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If $U\cong\operatorname{Spec} B\subset \operatorname{Spec} A$ is an affine open, then $\widetilde{M}(U)\cong M\otimes_A B$

I want to prove the theorem. Let $A$ be a ring and $X = \operatorname{Spec} A$. Furthermore, let $M$ be an $A$-module and $F$ an $O_X$-module that is associated to $M$. Then, for any affine open ...
lee's user avatar
  • 405
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1 answer
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Closed immersion of locally ringed spaces vs closed immersion determined by ideal sheaf

Let $f: X \to Y$ be a closed immersion of locally ringed spaces, that is, $f$ is a homeomorphism onto a closed subset of $Y$, $f^{\#}:\mathscr{O}_Y \to f_*\mathscr{O}_X$ is surjective, $\mathscr{I} = ...
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On free direct summand locus of finitely generated modules over commutative Noetherian rings

Let $M$ be a finitely generated module over a commutative Noetherian ring $R$. Assume that there exists an injective $R$-linear morphism $f: R\to M$. Consider the sets $$U:=\{\mathfrak p\in \text{Spec}...
Alex's user avatar
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Can we explicitly describe the derived pullback $\mathbf L\pi^* \widetilde M$ for a closed immersion $\pi$ of affine schemes?

Let $I$ be an ideal of a commutative Noetherian ring $R$. Let $M$ be a finitely generated $R$-module and $\widetilde M$ be its associated sheaf on $\text{Spec} (R)$. We have the closed immersion $\pi:...
uno's user avatar
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1 answer
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$\det (f_{*} \mathcal{O}_D) \simeq \mathscr{L} (f_{*} D)$ (Hartshorne Exercise IV.2.6)

I am stuck at the end of exercise IV.2.6 and I would appreciate a hint on how to conclude. We have $f:X\rightarrow Y$ a finite morphism of degree $n$ between curves and $D$ an effective divisor over X....
Akhalbing's user avatar
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Do quasi-coherent sheaves form a reflective subcategory?

Let $X = $ Spec $A$ be an affine scheme. I know that there is an inclusion of categories from $A$-modules to sheaves of $\mathcal O_X$-modules on $X$, which is exact and fully faithful. It seems to me ...
Adelhart's user avatar
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1 answer
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Simple objects in Quasi-coherent sheaves are isomorphic to structure sheaves of a closed point

Need to prove that simple objects in Quasi-coherent sheaves are isomorphic to structure sheaves of a closed point. "Simple object" means that there is no non-trivial subobject. I encountered ...
Acoustica's user avatar
1 vote
1 answer
174 views

The stalk of a point on a scheme is a localization of ring of affine open?

Let $(X,\mathcal O_X)$ be a Noetherian scheme. For every affine open subset $U$ of $X$, it holds that $U=\text{Spec}(\mathcal O_X(U))$. Let $x \in X$, and let $U$ be an affine open subset of $X$ ...
Muni's user avatar
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1 answer
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Is relative spectrum a left adjoint?

Given a scheme $X$ and a quasicoherent sheaf of algebras $\mathscr{R}$ on it. Vakil's FOAG, section 17.1.2, page 470 says that the relative spec $\beta: Spec \mathscr{R} \to X$, representing the ...
onRiv's user avatar
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Quasi-coherent sheaves on a complex manifold form an abelian category?

Quasi-coherent modules on a ringed space is defined in Stacks Project. Is there a complex manifold $X$ such that the category of quasi-coherent $O_X$-modules is not an abelian subcategory of the ...
Doug's user avatar
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4 votes
1 answer
244 views

Classify all coherent sheaves on $\mathbb{A}^1_k$

Given a field $K$, I want to classify all coherent sheaves on $\mathbb{A}^1_k$, and moreover saying if there exist locally free sheaves that are not free on $\mathbb{A}^1_k$. I am following Gathmann's ...
Aron's user avatar
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1 answer
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Double dual of locally free $\mathcal{O}_X$-module of finite rank

This question has been asked by many people and the answers are mostly suggesting verifying the isomorphism on stalks. But I really have no idea how to proceed: we define $\alpha:\mathcal{E} \to (\...
Mizutsuki's user avatar
  • 494
2 votes
1 answer
177 views

Hartshorne proposition II.5.4

I am really confused on the proof of Proposition II.5.4 on Hartshorne's algebraic geometry book. Especially, I am not sure how does the previous Lemma II.5.3 work on this proposition. Let me state the ...
Mizutsuki's user avatar
  • 494
6 votes
4 answers
844 views

Why quasi coherent sheaves?

So i was wondering why one considers quasi-coherent sheaves in algebraic geometry. I have read a lot that they are closely linked to the geometric properties of the underlying space. This means that ...
Adronic's user avatar
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2 answers
158 views

What is the universal property of the thickening $Y[\varepsilon]$?

Given an $S$-scheme $Y$, let $Y[\varepsilon]$ denote thickening $Y[\varepsilon]=Y \times_S D_S$ of $Y$. Here $D_S$ is the $S$-scheme $D\times_\mathbb Z S\to S$ where $D = \operatorname{Spec} \mathbb{Z}...
Nico's user avatar
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2 votes
1 answer
143 views

Example of a morphism of schemes whose kernel sheaf is not quasi coherent

I am trying to think of an explicit example of a morphism $\varphi: X\longrightarrow Y$ of schemes for which the kernel sheaf $ker(\mathcal{O}_Y\longrightarrow \varphi_*\mathcal{O}_X)$ is not quasi ...
Sam's user avatar
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0 answers
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Quasi-coherent sheaves on projective space.

I would like to better understand the characterization of quasi-coherent sheaves on $X=\mathbb{P}_A^d = \operatorname{Proj}A[x_0,...x_d]$, where I call $B = A[x_0,...x_d]$ We define $\mathcal{F}[k] = \...
user avatar
3 votes
1 answer
99 views

Quasi-coherent $\mathcal{F} \cong \tilde{M}$ on $\operatorname{Proj}B$ does not determine $M$

I would like to understand in detail the fact that given $\tilde{M}\cong \tilde{N}$ on $\operatorname{Proj}B$, then it is not always true in general that $M \cong N$. I know this has to do with ...
user avatar
6 votes
1 answer
97 views

Reconciliation of definition of presentable module/quasi-coherent sheaf with the fact that all modules have free presentation

Presentable module is defined in nLab (https://ncatlab.org/nlab/show/presentable+module) as “the cokernel of a homomorphism of free modules”. What I’m confused about is that, according to https://en....
lukemassa's user avatar
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1 answer
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Hartshorne Chapter 2 Exercise 5.6. d)

In this exercise 5.6. d) in chapter 2 from "Algebraic Geometry" by Hartshorne one has to show that for any ideal $\mathfrak a \subset A$ of a noetherian ring $A$ and $A$-module $M$ the sheaf ...
linkja's user avatar
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1 vote
0 answers
138 views

How to show saturation map (w.r.t quasicoherent sheaves) isnt always injective / surjective

This question is motivated by problem 15.4.D(a) in Vakil, but to give some setup since the notation / terminology may differ: let $S_\bullet$ be a nice graded algebra (finitely generated, generated in ...
cdsb's user avatar
  • 417
2 votes
1 answer
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Internal hom of equivariant sheaves

Let $G$ is affine algebraic group over $\mathbb{C}$ acting on a smooth scheme $X$ over $\mathbb{C}$, let $\mathcal{F},\mathcal{G}$ be two quasi coherent equivariant sheaves on $X$. Is there a natural ...
frogorian-chant's user avatar
17 votes
2 answers
977 views

Reference request: Bundles in Algebraic Geometry

I heard many times that quasi-coherent sheaves of $\mathcal O_X$-modules are morally the same thing as the sheaves of sections of a bundle $V\to X$ over $X$. We think of a ring $A$ as of the ring of ...
Nico's user avatar
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5 votes
1 answer
134 views

sheaves on a scheme

I work within the framework of Demazure & Gabriel`s book. (My schemes are all functors). Let $X$ be a scheme. I can define an underlying topological space $|X$| of $X$. Its points are equivalence ...
Nico's user avatar
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1 vote
1 answer
337 views

Transition functions on invertible sheaves

I'm studying "Foundations of algebraic geometry" by Ravi Vakil. Chapter 14.1 is about invertible sheaves on $\mathbb{P}^1_k$. It says between two affine subsets $\operatorname{Spec}k[x_{1/0}]...
nafise modaresi's user avatar
2 votes
1 answer
488 views

Vanishing of Ext groups of Coherent sheaves over Noetherian regular scheme

Let $(X,\mathcal O_X)$ be a Noetherian regular scheme of dimension $1$. Then, for any coherent sheaf $\mathcal F$ and any quasi-coherent sheaf $\mathcal G$, it holds that $\mathcal Ext^i(\mathcal F, \...
Snake Eyes's user avatar
1 vote
0 answers
29 views

Globalisation of varieties

I am trying to understand the way to globalize the notion of variety. Let $\mathcal A$ be a quasi-coherent sheaf locally of finite type over an affine variety $X$. We define a variety $Y=\mathrm{Specm}...
Sergio's user avatar
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0 answers
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How to think about the two quasi-coherent algebras $\bigoplus\limits_{n}\mathscr{E}^{\otimes n}, \,\, \bigoplus\limits_{n}\mathscr{E}(n)$?

Let $X$ be a scheme and $\mathscr{E}$ be a quasi-coherent $\mathcal{O}_X$-module. Take $X$ to be projective with a very ample bundle $\mathcal{O}_X(1)$ for a structure morphism $\pi: X \to S$, so that ...
Somatic Custard's user avatar
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0 answers
107 views

Linear system of conics passing through $3$ points

I would like to understand why the set of classes of divisors corresponding to the conics of $\mathbb P^2$ passing through $3$ non-colinear points is a linear system of dimension $2$. Thank you very ...
user avatar
0 votes
1 answer
182 views

Euler characteristic of a curve

Let $C$ be an irreducible curve on a surface $S$. We have the classical exact sequence $0\rightarrow\mathcal O_S(-C)\rightarrow\mathcal O_S\rightarrow\mathcal O_C\rightarrow 0$. Why can we deduce that ...
user avatar
0 votes
1 answer
328 views

Is a sheaf, which is flasque when restricted to any open affine necessarily flasque?

This is something I got stuck thinking about while trying to solve a problem in Hartshorne: if $\mathcal{F}$ is a quasi coherent sheaf on a noetherian scheme $X$ such that $\mathcal{F}|_U$ is flasque ...
frogorian-chant's user avatar
3 votes
0 answers
159 views

Hartshorne problem II.8.7

Again I'm stuck on a problem in Hartshorne. The situation is as follows: Let $X=\text{Spec}(A)$ be an affine, non-singular scheme which is finite of some field $k$. Let also $\mathcal{F}$ be a ...
frogorian-chant's user avatar
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0 answers
114 views

It seems to me that quasi-coherent sheaf is a sheaf of sheaves.

In Wikipedia, I was looking for the definition of quasi-coherent sheaf. However I got confused. It seems to me that quasi-coherent sheaf is a sheaf of sheaves. Though, indeed, it says quasi-coherent ...
MAS's user avatar
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Proving that the stalk of the sheaf of homomorphisms of a module of finite presentation is isomorphic to the module of homomorphism of stalk

Consider $\mathscr{F}$ to be a module of finite presentation over a ringed space $X$. I want to prove that, for any sheaf of modules $\mathscr{G}$ and $x \in X$, the canonical morphism $\mathscr{H}om(\...
user480840's user avatar
3 votes
0 answers
87 views

Quasicoherent sheaves over manfiolds

For any locally ringed space $\left(X,O_X\right)$, a quasicoherent sheaf is a sheaf of $O_X$-modules which are locally the quotient of free modules. Considering a manifold (Haussdorf, second-countable)...
Chetan Vuppulury's user avatar
3 votes
0 answers
132 views

Equivalent definitions of flat morphism

Suppose $\pi : X \to Y$ satisfies that pullback on quasi-coherent sheaves is exact, how do I prove that $\pi$ is flat via the local definition; i.e stalkwise $O_{X,p}$ is a flat $O_{Y,q}$ module ...
Andy's user avatar
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1 answer
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Theorem 7.4.16 (Riemann–Hurwitz formula) in Qing Liu's Algebraic Geometry and Arithmetic Curves

I have a question about a part from the proof of Theorem 7.4.16 on page 290 from Liu's "Algebraic Geometry and Arithmetic Curves". The claim is Theorem 7.4.16 Let $f : X \to Y$ be a finite ...
user267839's user avatar
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