Skip to main content

Questions tagged [coherent-sheaves]

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a specific class of sheaves having particularly manageable properties closely linked to the geometrical properties of the underlying space.

Filter by
Sorted by
Tagged with
0 votes
0 answers
56 views

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
0 votes
0 answers
88 views

Derived category of $\mathbb A^n/{\mathbb G_m}$

Beilinson proved that the bounded derived category $D^b(\mathbb P^n)=<\mathcal O, \mathcal O(1),...\mathcal O(n)>$ as a semiorthogonal decomposition. I would like to know what happens for $D^b(\...
Angry_Math_Person's user avatar
1 vote
1 answer
77 views

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
  • 423
1 vote
0 answers
33 views

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
  • 423
1 vote
1 answer
36 views

Pullback of a section though a point vs. value of the section at that point

Let $X$ be a projective, integral variety over a field $k$, and let $L$ be a line bundle (i.e. invertible sheaf) on $X$. Now take a local section $s\in L(U)$ and a point $x\in X$ and look at the ...
manifold's user avatar
  • 1,905
1 vote
0 answers
31 views

Restriction of a vector bundle to a nodal curve

Let $S\to B$ be an elliptic surface with one nodal singular fiber C (a nodal projective curve $C$ of genus $g=1$). Let $\mathcal F$ be a slope-semistable rank-$2$ vector bundle on $S$. What can we say ...
Conjecture's user avatar
  • 3,270
0 votes
0 answers
10 views

Is a finite locally-free sheaf free on all open affine covering?

If $\mathscr{F}$ is a rank $r$ finite locally-free sheaf over a smooth irreducible curve $X$ and $U_i$ any covering of $X$ by affine open sets, is true that $$\mathscr{F}|_{U_i} \cong \mathcal{O}|_{...
Leonardo Soares Moço's user avatar
2 votes
1 answer
52 views

Sections are determined by their germs at associated points

This comes from the exercise 6.6.W of Vakil's FOAG. I guess it is saying that a function (section) is determined by its germs at associated points. And this sort of motivates the introduction of ...
Mizutsuki's user avatar
  • 494
1 vote
1 answer
88 views

Support of the pushforward of structure sheaf of a smooth scheme along proper birational morphism

Let $k$ be a field of characteristic $0$ and $R$ be a finite type $k$-algebra. Let $X$ be a smooth $k$-scheme and $f: X \to \text{Spec}(R)$ be a proper birational morphism. Then, is the $R$-module $f_*...
strat's user avatar
  • 341
0 votes
0 answers
42 views

Finding a good homogeneous coordinate ring for a smooth projective variety

TL;DR Given a smooth projective variety $X\subseteq \mathbb{P}^m$ (I guess this means that the homogeneous coordinate ring $S(X)$ is regular whenever localizing at a non-maximal graded prime ideal). ...
Noto_Ootori's user avatar
1 vote
1 answer
74 views

Pushforward of structure sheaf on quotient by finite group acting freely

Let $G$ be a finite group acting freely on a smooth projective complex variety $X$. This induces an action on $\mathrm{Coh}(X)$ by pullback. Let $\pi\colon X \rightarrow Y=X/G$ denote the quotient. ...
Conic3264's user avatar
1 vote
1 answer
76 views

2.3.A Zariski's construction (PAG1 - R. Lazarsfeld), big and nef divisor which is not finitely generated

I'm reading Lazarfeld's book «Positivity in Algebraic Geometry I» and I'm stuck on the construction of a big and nef divisor on a variety $X$ such that its canonical ring/algebra $R(X,D) = \bigoplus_{...
NaNoS's user avatar
  • 555
2 votes
1 answer
78 views

Properties of the Lazarsfeld-Mukai vector bundle - Dualizing an exact sequence of vector bundles

I'm trying to understand a passage from an article by R.Lazarsfeld concerning properties of the Lazarsfeld-Mukai bundle for my reseach. Here is the paper of Lazarsfeld giving the construction (p.3 of ...
NaNoS's user avatar
  • 555
5 votes
0 answers
77 views

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
  • 423
5 votes
0 answers
209 views

Coherent sheaves, Serre’s theorem and ext groups

Let $X$ be a smooth projective variety over an algebraically closed field $k$ (if necessary we assume that $\operatorname{ch}(k)=0$). Let $O_X(1)$ be a very ample invertible sheaf on $X$. Then, the ...
Walterfield's user avatar
1 vote
0 answers
94 views

Euler Characteristic of Sheaves

Suppose my space $X$ is a K3 surface, hence has trivial canonical bundle. Given two coherent sheaves $F, E$ on $X$, we may define $\chi(E,F) = \sum_i (-1)^i \mathrm{ext}^i(E,F)$. I've read without ...
Slim Shady's user avatar
2 votes
0 answers
104 views

Computing the Exceptional inverse image functor for some simple closed immersions

Let $f: X \rightarrow Y$ be a morphism between schemes. Then, under mild hypothesis on $f, X$ and $Y$, we have Grothendieck duality. This gives an isomorphism $\mathcal{R}\mathcal{H}om_{Y}(\mathcal{R}...
Sunny Sood's user avatar
2 votes
1 answer
96 views

Lazarsfeld's proof of Mumford's regularity Theorem 1.8.3

I want to apologize first of all if my English is bad, I hope my message will still be understandable. I’m currently reading Lazarsfeld’s book "Positivity in Algebraic Geometry I" and I’m ...
NaNoS's user avatar
  • 555
3 votes
0 answers
61 views

Sheaves of modules isomorphic after pullback - when isomorphic in general?

Let $f: X \to Y$ be a morphism of schemes and $\mathcal{M}, \mathcal{N}$ be $\mathcal{O}_Y$-modules. Suppose $f^*\mathcal{M} \cong f^* \mathcal{N}$. Under which assumptions can I conclude that already ...
Matthias's user avatar
  • 797
3 votes
0 answers
27 views

Is a Noetherian sheaf of rings stalkwise Noetherian?

Let $X$ be a scheme such that $O_X$ is a coherent $O_X$-module. Assume that for every open subset $U\subset X$ and every family of coherent ideal sheaves $\{I_i\}_i$ of $O_U$, $\sum_iI_i$ is a ...
Doug's user avatar
  • 1,308
1 vote
1 answer
58 views

A property of invertible sheaves

Let $f:X \to Y$ be a morphism of schemes, and let $\mathscr{F}$ be an invertible sheaf on $Y$. It is clear that if $\mathscr{F}^{\otimes n} \cong \mathscr{O}_Y$ then $(f^*\mathscr{F})^{\otimes n} \...
mathfan24's user avatar
  • 612
1 vote
0 answers
56 views

Characterizing pushforwards of sheaves under Galois covers

Let $\pi: Y \to X$ be a Galois cyclic cover with automorphism group $G$ generated via $g$, that arises from a line bundle $L$ on $X$ that is $n$ torsion. I will give lots of context, but my question ...
user135743's user avatar
2 votes
1 answer
84 views

Why non-existence of Harder-Narasimhan filtrations for flat families of sheaves?

Consider over $\mathbb{C}$. Let $X$ be a projective scheme and let $\mathcal{O}(1)$ be an ample line bundle on $X$. Consider Gieseker (semi)stability. For a coherent pure sheaf $\mathcal{F}$ on $X$, ...
Display Name's user avatar
  • 1,409
2 votes
1 answer
100 views

Show the fibers are geometrically connected

This exercise is the 5.3.9 of Liu's famous book about algebraic geometry. Let Y be a normal, locally Noetherian, integral scheme, and let $f : X \mapsto Y$ be a projective dominant morphism with $X$ ...
Analyse300's user avatar
2 votes
1 answer
50 views

Kernel of an epimorphism of coherent sheaves on Noetherian schemes

Let $f\colon\mathcal{E}_1\to\mathcal{E}_2$ be an epimorphism of locally free sheaves on a Noetherian scheme $X$. Then also $\ker(f)$ is a locally free sheaf. Proof. For all $x\in X$ one has a short ...
Armando j18eos's user avatar
0 votes
0 answers
72 views

What do we know about moduli spaces of sheaves on $\mathbb{P}^n$?

I want to know some examples of moduli schemes of (geometrically) stable sheaves over a higher dimensional base scheme. The simpliest base schemes are the projective spaces $\mathbb{P}^n$ for $n\geq2$....
Display Name's user avatar
  • 1,409
1 vote
1 answer
97 views

Short exact sequence of vector bundles over $\mathbb{P}^1_k$.

$\newcommand{\C}{\mathbb{C}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\P}{\mathbb{P}} \newcommand{\db}[1]{D^b(\mathrm{Coh}(#1))} \newcommand{\O}{\mathcal{O}} $ I am reading ...
J.V.Gaiter's user avatar
  • 2,534
4 votes
1 answer
245 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
  • 263
3 votes
1 answer
119 views

Serre: coherent $\iff$ locally finitely presented, when sheaf coherent over itself

Note: this question is relevant but doesn't answer my question. I want to prove the following proposition from (an English translation of) Jean-Pierre Serre's article Faisceaux algébriques cohérents. ...
Jeremy Lindsay's user avatar
4 votes
1 answer
204 views

Pushforward of the Segre embedding in K-theory

Fix $n$, $m\ge 1$, and let $d=\binom{m+n}{m}$ and $N=mn+m+n$. Consider the Segre embedding $\sigma:\mathbb{P}^m\times \mathbb{P}^n \hookrightarrow \mathbb{P}^{N}$, which has degree $d$. I'm trying to ...
Alvaro Martinez's user avatar
1 vote
0 answers
143 views

Vanishing of $\text{Ext}^i_{X}(E, F)$ vs. $\text{Ext}^i_{\mathcal O_x}(E_x, F_x)$ for $E,F \in \mathcal D^b(\text{Coh } X)$

Let $X$ be a Noetherian scheme and $E,F \in \mathcal D^b(\text{Coh } X)$. Let $x\in X$ be a closed point. Then, is there any connection between $\text{Ext}^i_{\mathcal O_x}(E_x, F_x)$ and $\text{Ext}^...
Alex's user avatar
  • 423
1 vote
2 answers
100 views

Autoequivalences of $\operatorname{Coh}(X)$

Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic zero. Is there a description of $\operatorname{Aut}(\operatorname{Coh}(X))$, i.e. the autoequivalences ...
freeRmodule's user avatar
  • 1,872
2 votes
1 answer
282 views

Hartshorne Exercise II 6.11 (c)

Exercise II 6.11: Let $X$ be a nonsingular curve over an algebraically closed field $k$. (c) If ${\mathscr{F}}$ is any coherent sheaf of rank $r$ (means that its stalk at the generic point has ...
Eric's user avatar
  • 521
3 votes
1 answer
217 views

Locally free resolution of coherent sheaves on nonsingular curves

This question is from Exercise II 6.11 of Hartshorne. Let $X$ be a nonsingular curve over an algebraically closed field $k$. For any coherent sheaf $\mathcal{F}$ on $X$, show that there exist ...
Eric's user avatar
  • 521
1 vote
0 answers
20 views

Do analytic coherent sheaves remain coherent in the analytic Zariski topology?

Let $X$ be a compact Kähler manifold. Let $F$ be a coherent $O_X$-module. Is there always an open cover $\{U_i\}_i$ for $X$ such that for every $i$: 1.$X\setminus U_i$ is an analytic subset of $X$; ...
Doug's user avatar
  • 1,308
1 vote
1 answer
158 views

Support of the direct image sheaf equals the image?

$\def\sO{\mathcal{O}} \def\supp{\operatorname{Supp}} \def\sI{\mathcal{I}} \def\sC{\mathcal{C}} \def\colim{\operatorname{colim}}$I am studying complex spaces using Grauert, Remmert, Coherent Analytic ...
Elías Guisado Villalgordo's user avatar
1 vote
1 answer
153 views

Support of a section of an $\mathcal{O}_X$ module

Given a coherent sheaf $F$ over a space $X$, I understand from https://en.wikipedia.org/wiki/Support_of_a_module that the support of the sheaf of modules is all those points of $X$ such that the stalk ...
lukemassa's user avatar
  • 724
1 vote
1 answer
172 views

Trace map on Ext groups of coherent sheaves

Let $ X $ be a projective variety and $ \mathcal{F} $ a coherent sheaf on $ X $. I'm kind of stumped at how the trace map $ \operatorname{Ext}^i( \mathcal{F}, \mathcal{F} ) \rightarrow H^i(X, \mathcal{...
Cranium Clamp's user avatar
0 votes
1 answer
124 views

Find $H^1(I_X(r))$ for all $r\geqslant0$ where $I_X$ is a four-point set's sheaf of ideals

Let $X\subset\mathbb P^n$ be a set of $4$ points and let $I_X\subset\mathcal O_{\mathbb P^n}$ be it's sheaf of ideals. I'd like to compute $H^1(I_X(r))$ for all $r\geqslant0.$ I've tried to consider a ...
Maxim Nikitin's user avatar
2 votes
1 answer
97 views

Describe $H^0(\mathcal O_X(m-2))\to H^1(I_X(m-2))$ induced by exact short sequence of sheaves

Let $i:X\subset\mathbb P^1$ be a set of $m$ points and let $I_X\subset\cal O_{\mathbb P^n}$ be its sheaf of ideals. Then the sequence $$(1)0\to I_X(m-2)\to\mathcal O_{\mathbb P^1}(m-2)\to\mathcal i_*...
Maxim Nikitin's user avatar
2 votes
2 answers
350 views

A question on varieties and morphisms of varieties.

I apologize in advance for making a similar question to my previous one; as they pointed out in the comments, the question was not too clear because I am not used to the notions of locally ringed ...
Jerry Scott's user avatar
0 votes
0 answers
57 views

Is the pushforward of an analytic coherent sheaf still coherent?

Let $f:X\to Y$ be a morphism of Stein manifolds. Let $F$ be a coherent $O_X$-module, $\mathcal{A}=f_*O_X$. The question is: do we know that $f_*F$ is a coherent $\mathcal{A}$-module? In fact, there is ...
Doug's user avatar
  • 1,308
0 votes
0 answers
47 views

Understanding coherent sheaf obtained via sheaf injections of holomorphic vector bundles on $T\mathbb{C}P^1$

My problem involves holomorphic vector bundles $E,F$ of the same rank on $T\mathbb{C}P^1$. I have a short exact sequence of sheaves $$0\rightarrow E\rightarrow F\rightarrow Q\rightarrow 0.$$ I want to ...
AlgGeoNoob's user avatar
0 votes
0 answers
40 views

On the cohomology group of kernel of $\mathcal{F} \to \mathcal{F}(d)$

This is from Mumford-Oda's Algebraic Geometry 2, And here is pdf of chapter 7-8. https://www2.math.upenn.edu/~chai/624_08/mumford-oda_chap7-8.pdf My question is on page 243, I can't see why $H^{i+1}...
Mugenen's user avatar
  • 1,111
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
0 votes
1 answer
193 views

When is an extension a vector bundle?

Let $ X $ be a smooth threefold and $ C \subset X $ be a smooth (but not necessarily irreducible) curve with ideal sheaf $ \mathcal{I_C} $. I am looking for an answer to the question of when an ...
Cranium Clamp's user avatar
2 votes
0 answers
71 views

Hilbert scheme of $\mathbb{P}^2$ is not a product?

Fixing a connected component in the Hilbert scheme of $\mathbb{P}^2$ is the same as fixing the Hilbert polynomial $ax+b$. Taking a primary decomposition of any subscheme in this connected component ...
Mathmop's user avatar
  • 478
2 votes
1 answer
99 views

Is there a direct proof that $\mathcal O(n) \in \text{thick}_{D^b(\mathbb P^1)}\{\mathcal O, \mathcal O(1)\} $ for all $n\in \mathbb Z$

Let $k$ be an algebraically closed field, and let $\mathbb P^1$ denote $\mathbb P^1_k$. Let $D^b(\mathbb P^1)$ be the bounded Derived Category of Coherent Sheaves on $\mathbb P^1$. Let $\text{thick}_{...
uno's user avatar
  • 1,560
3 votes
1 answer
315 views

Global sections of pushforwards

Let $X$ and $Y$ be projective schemes over $\mathrm{Spec}A$, where $A$ is a ring. Let $\pi:X\rightarrow Y$ be a morphism. Let $\mathscr F$ be a coherent sheaf on $X$. When is it true that $\Gamma(X,\...
Pickle Liobe's user avatar
0 votes
0 answers
37 views

Analytic sheaves for Cartan's theorem

When reading about Cartan's theorems for Stein manifolds (theorems A & B), about half of the sources state this in terms of coherent sheaves and half of them uses coherent "analytic" ...
NDewolf's user avatar
  • 1,713

1
2 3 4 5
8