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.
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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 ...
- 23
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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}...
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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 ...
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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 ...
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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 ...
- 1,216
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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} \...
- 458
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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 ...
- 371
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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 $X$, ...
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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 $X$ ...
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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 ...
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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 $n\geq2$....
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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 ...
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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 ...
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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. ...
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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 ...
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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}^...
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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 ...
- 1,832
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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 ...
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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 ...
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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 $X$;
...
- 1,216
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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 ...
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75
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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 ...
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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{...
- 2,603
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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 ...
- 1,303
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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
$$(1)0\to I_X(m-2)\to\mathcal O_{\mathbb P^1}(m-2)\to\mathcal i_*...
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Given $\mathcal E$ finite locally free, which sub-line bundles are dual of quotients of $\mathcal E^\vee$?
Let $X$ be a scheme. Let $\mathcal E$ be a finite locally free $\mathcal O_X$-module. We call $$q:\mathcal E\to \mathcal Q$$ an invertible quotient of $\mathcal E$ if $q$ is surjective and $\mathcal Q$...
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Is the pushforward of an analytic coherent sheaf still coherent?
Let $f:X\to Y$ be a morphism of Stein manifolds. Let $F$ be a coherent $O_X$-module, $\mathcal{A}=f_*O_X$. The question is: do we know that $f_*F$ is a coherent $\mathcal{A}$-module?
In fact, there is ...
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Understanding coherent sheaf obtained via sheaf injections of holomorphic vector bundles on $T\mathbb{C}P^1$
My problem involves holomorphic vector bundles $E,F$ of the same rank on $T\mathbb{C}P^1$. I have a short exact sequence of sheaves
$$0\rightarrow E\rightarrow F\rightarrow Q\rightarrow 0.$$
I want to ...
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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}...
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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 ...
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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 ...
- 2,603
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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 ...
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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}_{...
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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,\...
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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" ...
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Kernel of sheaf morphism of coherent sheaves is coherent in the general case
I'm trying to see that if $\mathcal{F}, \mathcal{G}$ are coherent sheaves of $\mathcal R$-modules over some topological space and $\phi: \mathcal{F} \rightarrow \mathcal{G}$ is a sheaf morphism, then $...
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Cokernel of Morphism of Free Sheaves with Infinite Rank
From exercise II.5.4 of Hartshorne, we know that $\mathcal{O}_X$-module $\mathcal{F}$ of Noetherian scheme $X$ is coherent if and only if it is cokernel of morphism of free sheaves with finite rank. ...
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Global sections of a torsion sheaf
Let $X$ be a smooth projective curve, $\mathcal{F}$ a torsion sheaf on $X$, $\Gamma(X,\mathcal{F})$ finitely generated by $ \{s_1,\ldots,s_n\}$ (or $n:= \dim \Gamma(X, \mathcal{F}) $). Is $\mathcal{F}$...
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Sections of a torsion sheaf
I am trying to get a better understanding of torsion sheaves over projective schemes.
Is it true that if $\mathcal{F}$ is torsion over a projective scheme $X$, then $$\Gamma(X,\mathcal{F}) \cong \...
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Simple vector bundles are stable
It is clear that stable vector bundles over a projective scheme are simple, i.e. $\operatorname{Hom}(F,F) \cong \mathbb{C}$.
Let $X$ now be an elliptic curve.
Then every simple vector bundle is stable....
- 522
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Kernel of $F\to F^{**}$ has rank $0$ [closed]
I want to show that every coherent sheaf over a curve is a direct sum of a torsion sheaf and a locally free sheaf via the following:
Let $X$ be a smooth projective $1$-dimensional scheme. Let $F$ be a ...
- 522
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$H^0$-stability and slope-stability
Definition: $E$ is called $H^0$-semistable if for all line bundles $L\subset E$ we have
$$
h^0(L) \leq \frac{h^0(E)}{2}
$$
$E$ is called slope-semistable if for all subbundles we have
$$
\frac{\deg L}...
- 522
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Reference request: Slope Stability, Bridgeland Stability
Just wanted to ask for references about these topics:
Smooth projective curves
Coherent & Quasicoherent sheaves and their connection to bundles
Degree of a line bundle on a curve
Derived category ...
- 25
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32
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Principal fiber space associated to singular linear fiber space
I am wondering if the construction of principal bundles associated to vector bundles can be generalized to linear fiber spaces in the category of complex/(real) analytic spaces. Linear fiber spaces ...
- 474
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146
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Hartshorne problem III.10.5
Here is a problem I thought I solved but now I think it can't be right.
The problem is as follows: Let $X$ be a scheme and $\mathcal{F}$ a coherent sheaf such that every $x\in X$ has an étale ...
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147
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Is the derived category of coherent sheaves idempotent closed?
I am wondering whether the bounded derived category $D^b Coh(X)$ is idempotent complete, i.e. whether every idempotent morphism splits. This is true for $Perf(X)$ since perfect complexes are the ...
- 1,372
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75
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Homology of the diagonal sequence of 3x3 commutative diagram of modules
Suppose we have modules $M_{i,j}$ over a commutative ring $R$ (or members of some abelian category, like quasi-coherent sheaves of modules),
and suppose that we have a 3x3 commutative diagram, where ...
- 1,024
6
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2
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521
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Example of torsion-free sheaf which is not locally free
What is a standard example of a torsion-free sheaf, say on the complex projective plane, which is not locally free?
- 301
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Hartshorne exercise III.10.2
This problem has been discussed already (Hartshorne ex III.10.2 on smooth morphisms) but I cannot understand the solution and since that questions is 8 years old or so I decided to make a new question ...
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When is the anticanonical sheaf ample, on an arithmetic surface of genus 0?
Let $\pi:X \to S$ be an arithmetic surface, i.e. a flat, projective, scheme of relative dimension 1 over a Dedekind scheme $S$. So $X$ is a 2-dimensional excellent scheme.
Suppose that the generic ...
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