# Questions tagged [coherent-sheaves]

The coherent-sheaves tag has no usage guidance.

257
questions

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### Resolution property in rigid analytic geometry

I am not a rigid analytic geometrer, so I apologise if the question is trivial, but I can't find an answer anywhere myself. I'm trying to understand in what ways (rigid) analytic geometry compares to ...

3
votes

0
answers

116
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### Trace map on Ext group

Let $R$ be a (possibly non-commutative) unital ring and $M$ be a perfect left $R$-module. Then, we have the trace map
$$
\operatorname{Tr}\colon \mathrm{Hom}_R(M,M)\to R/[R,R]\,.
$$
According to the ...

7
votes

1
answer

575
views

+100

### Converses to Cartan's Theorem B

Here is a phrasing of some Cartan Theorem B statements:
Consider the following conditions:
$X$ is a {Stein manifold, affine scheme, coherent analytic subvariety of $\mathbb{R}^n$, contractible ...

7
votes

1
answer

408
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### Smooth analogue of Cartan's Theorem B

Cartan's Theorem B can be stated as follows:
Let $X$ be a space let $\mathcal{F}$ be a sheaf on $X$.
Consider the following three conditions:
$X$ is "simple";
$\mathcal{F}$ is "nice&...

5
votes

1
answer

232
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### On the bounded derived category of sheaves with coherent cohomology

Let $(X,\mathcal{O}_X)$ be a locally ringed space such that $\mathcal{O}_X$ is locally notherian, and let $\operatorname{Coh}(\mathcal{O}_X)$ be the category of coherent $\mathcal{O}_X$-modules. The ...

5
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0
answers

152
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### Description of pull-back of coherent sheaves under a smooth morphism of Artin stacks

I am new to these formalisms, so pardon me if the question is basic. Let $\mathscr{X}$ be an Artin stack (you can take it to be Deligne-Mumford stack if it helps). By a coherent sheaf on $\mathscr{X}$ ...

1
vote

0
answers

35
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### (Quasi)-coherence of the weight $\theta$-sheaf

In this paper, the author defined the weight $\theta$-sheaf as follows: Let $A^{k}_{X}$ be the sheaf germs of real smooth $k$ forms on the smooth manifold $X$, we perform a complexification on this ...

1
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0
answers

67
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### Trace map for universal bundle of Grassmannian

Let $G := G(k,V)$ denote the Grassmannian of $k$-linear subspaces in a $\mathbb{C}$-vector space $V$ of dimension $n$. Let $S$ denote the tautological bundle over $G$. There is a canonical map on ...

1
vote

0
answers

104
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### Monomorphism/Isomorphism of $C_4$-tangent cones for complex varieties

Suppose that $(M,\mathcal{O}_M)$ is a reduced complex analytic space (or complex algebraic variety if you prefer). The tangent linear fiber space $TM$ associated to $M$ is defined as the analytic ...

4
votes

1
answer

616
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### 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 ...

2
votes

1
answer

233
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### Compatibility of Beck Chevalley condition: sheaves

Given a (not necessarily Cartesian) square of spaces
$$\require{AMScd}\begin{CD}
X @>g>> \overline{X} \\
@VVfV @VV\overline{f}V \\
Y @>\overline{g}>> \overline{Y}
\end{CD}$$
does the ...

2
votes

1
answer

229
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### Is any "relative support" for (complexes of) quasi-coherent sheaves known?

Let $f:X\to S$ be a morphism of Noetherian schemes; in the case I am interested in $S=\operatorname{Spec}R$ is affine and $f$ is proper. For a complex $C$ a complex of quasi-coherent sheaves on $X$ I ...

2
votes

0
answers

293
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### The definition of the determinant of a coherent sheaf

Let $ X $ be a smooth (projective) variety and $ \mathcal{F} $ a torsion-free coherent sheaf of rank $ r $ on $ X $. The determinant $ \det \mathcal{F} $ can be defined by
(1) $ \det \mathcal F := ( \...

6
votes

0
answers

282
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### Is there a sheaf of categories $\text{QCoh}_X(1)$ analogous to $\mathcal{O}_X(1)$?

Given a scheme $X$ and sum of divisors $D$, you can take the line bundle
$$\mathcal{O}_X(D)\ =\ \{\text{functions }f\text{ with [conditions on zeroes/poles]}\}\ \subseteq\ j_*\mathcal{O}_\eta\ =\ \...

2
votes

0
answers

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### Pullback of coherent sheaves on Stein manifolds

Let $i:X\to Y$ be a closed embedding of Stein spaces, $G$ be a coherent $O_Y$-module. Set $I=\ker(i^*:O(Y)\to O(X))$. Then $I$ is an ideal of the ring $O(Y)$. Is that true that $\Gamma(X,i^*G)=G(Y)/...

1
vote

0
answers

57
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### What is the semistable reduction for sheaves?

Let $\Bbbk$ be an algebraically closed field with characteristic zero. Let $X$ be a projective scheme over $\Bbbk$ and let $L$ be an ample invertible $\mathcal{O}_X$-module. Fix a Hilbert polynomial $...

3
votes

0
answers

116
views

### Obstruction to the existence of a deformation of a subvariety compatible with the given deformation of a variety

Let $X$ be a smooth projective variety over a field $k$ of characteristic 0,
and let $A$ be a local Artinian $k$-algebra, say, $A=k\oplus I$
where $I$ is an ideal such that $I^2=0$.
Let $\frak X$ be a ...

1
vote

0
answers

85
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### Slope-stability, tilt-stability, and Bridgeland stability

Following the definition of slope-stability ($\mu$-stability) and tilt-stability ($\nu$-stability) on page 8 of https://arxiv.org/abs/1410.1585, does an object's tilt-stability imply its slope-...

2
votes

0
answers

121
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### Canonical basis and perverse coherent sheaves on the nilpotent cone

In the paper of Ostrik, he introduced a canonical basis of $K^{G\times {\mathbb C}^*}(\mathcal N)$, where $\mathcal N$ is the nilpotent cone for the group $G$. Question: does this canonical basis ...

2
votes

0
answers

108
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### Unicity of modifications of vector bundle on a regular base

I think that I have overheard the following statement, and would be grateful for either a reference or an explanation about why hearing must be slightly faulty and a clarification about what must ...

3
votes

1
answer

125
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### Finer classification of semistable sheaves

Usually in the moduli space of semistable sheaves, two semistable sheaves correspond to one point if and only if they are S-equvialent, i.e. the graded objects associated to their Jordan-Holder ...

4
votes

1
answer

299
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### Derived pushforward of a projection

Given two smooth projective varieties, $X,Y$, consider their derived categories $D^b(X), D^b(Y)$. Let $\mathcal{F}$ a complex of coherent sheaves in $D^b(X \times Y)$, why the derived pushforward of ...

4
votes

0
answers

289
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### Are $\mathcal{O}_X$-modules "more actual" then quasicoherent sheaves in some sense?

In the Stacks project and in a book of Brian Conrad the "main" derived category of a scheme is the one of $\mathcal{O}_X$-modules. I would like to understand whether $D(\mathcal{O}_X)$ is ...

9
votes

1
answer

903
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### Is the functor from the unbounded derived category of coherent sheaves into the derived category of quasi-coherent sheaves fully faithful?

Let $X$ be a Noetherian scheme. Is the obvious functor $D(\operatorname{Coh}(X))\to D(\operatorname{QCoh}(X))$ fully faithful?
If this is true then $D(\operatorname{Coh}(X))$ is equivalent to the full ...

1
vote

1
answer

191
views

### flatness of restriction of structure sheaf over ring of global sections

Let $X$ be an affine scheme. $U \subseteq X$ open. Then I want to show that $\mathcal{O}_X(U)$ is flat over $\mathcal{O}_X(X)$.
But I want to prove it only by knowing the definition of structure sheaf ...

1
vote

1
answer

119
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### Families of torsion-free sheaves whose length jumps

For a long time, I had a false belief that the space/stack $\text{Coh}^{tf}_{c_1,c_2}S$ of torsion-free sheaves $\mathcal{E}$ on a smooth algebraic surface $S$ was not connected, since if you take its ...

2
votes

1
answer

294
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### Maps from $\mathbb A^1/ \mathbb G_m$ to Coherent sheaves

I am reading this paper https://arxiv.org/abs/1608.04797
Let $\Theta$ be the stack $\mathbb A^1/{\mathbb G_m}$. Let $X$ be a smooth projective curve of genus $g$ over a field $k$. Let $Coh_P$ be the ...

1
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0
answers

66
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### Understanding coherent sheaf obtained via sheaf injections of holomorphic vector bundles on TCP^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 ...

4
votes

2
answers

620
views

### Basic question on projective bundles

Let $\mathcal{E}$ be a coherent sheaf on an irreducible scheme $S$ ($S$ can be pretty nice, say noetherian of finite type), and let $\mathbf{P}(\mathcal{E}):=\mathrm{Proj}(\mathrm{Sym}(\mathcal{E}))$ ...

2
votes

1
answer

274
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### Resolving complexes of coherent analytic sheaves

Background
Throughout, let $X$ be a smooth complex manifold.
It is a classical fact that a coherent analytic sheaf admits a local resolution by locally free sheaves (also known as a local syzygy). ...

2
votes

1
answer

140
views

### Could certain closed covering determine a coherent sheaf?

We know that a coherent sheaf on a scheme is determined by its restriction on certain open coverings (satisfying compatibility condition). Now I wonder how about a closed covering. To do so I started ...

3
votes

0
answers

91
views

### Explicit example of wall-crossing for sheaves

I would like to see an explicit example of a coherent sheaf $\mathcal{E}$ on a projective complex threefold $X$ which crosses the wall of stability. That is, I would like some $1$-parameter family $\...

3
votes

0
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93
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### Analogous tensor product operation for reflexive sheaf

Suppose now $(X,\mathcal O_X)$ is a normal complex space, and $\mathcal F$ is a coherent analytic sheaf on it.
Product the reflexive sheaf $$\mathcal F^{[p]}:=(\mathcal F^{\otimes p})^{**},$$ where $\...

1
vote

0
answers

91
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### Chern class of rank one sheaves supported on subvarieties

Let $X$ be a smooth, quasi-projective variety of dimension $n$ and $\mathcal{F}$ be a globally generated coherent sheaf supported on a codimension two subvariety $V \subset X$. Is $c_2(\mathcal{F}) \...

9
votes

1
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314
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### Are the tensor-invertible coherent sheaves on an algebraic space (Zariski) locally free of rank one?

On a scheme, the coherent sheaves that are invertible objects for the tensor product (monoid) operation are precisely the coherent sheaves that are (Zariski) locally free of rank one. Is the same ...

0
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0
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197
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### Sheaves of abelian groups over a smooth projective variety

Can someone point some good reference (books or lecture notes) for these topics:
Let $X$ a smooth projective variety over an algebraically closed field
Sheaves of abelian groups over $X$
Quasi-...

3
votes

1
answer

318
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### Existence of rigid objects in the derived category of a smooth projective variety

Let $X$ be a smooth projective variety (say over $\mathbb{C}$). An object $F \in D^b(X)$ is said to be rigid if $\mathrm{Ext}^1(F,F)=0$. I was wondering if we can always find a rigid object on a ...

0
votes

1
answer

240
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### Stability of sheaves of non-constant rank

Let $E\to X$ be a coherent sheaf over a compact (projective) Kahler manifold. The definition of stability of sheaves as stated in Huybrechts-Lehn (Definition 1.2.12) says that $E$ is stable if for all ...

3
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0
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### Does a torsion-free coherent sheaf embed into a locally free sheaf?

Let $ X $ be a Noetherian integral regular scheme and $ \mathcal{F} $ be a torsion-free coherent sheaf. (One definition of torsion-free is that the natural map $ \mathcal{F} \rightarrow \mathcal{F} \...

2
votes

1
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239
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### Understanding spaces is the same as understanding (sheaves of) functions on the space

I'm trying to understand Ravi Vakil's FOAG. In chapter 2 it is written:
[...] understanding spaces is the same as understanding (sheaves of) functions on the spaces, and
understanding vector bundles (...

3
votes

1
answer

193
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### Is this a true weakening of the quasi-coherence property?

Let $R$ be a commutative Noetherian ring, and $X=$Spec$(R)$ the associated affine scheme. Let $F$ be a sheaf of $O_X$-modules. Consider the following condition
(#) For all containments $V \subseteq ...

3
votes

1
answer

279
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### Can we recover the sheaf from the functor?

Let $k$ be an algebraically closed field of characteristic zero. Let $S$ be a scheme of finite type over $k$. Let $\mathrm{Sch}/S$ be the category of schemes of finite type over $S$. Let $\mathcal F$ ...

4
votes

0
answers

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### Connectedness of moduli spaces of semistable sheaves on K3

In 1987/88, Mukai described the moduli spaces of (semi-)stable sheaves on a K3 surface $X$, showing that they consist of smooth pieces $M(v)$ of dimension $\langle v , v \rangle + 2$, for every ...

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0
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### Projectivization in the derived category of coherent sheaves

Let $X$ be a compact Kahler manifold. There exists a notion of projectivization of holomorphic vector bundles and coherent sheaves over $X$. Does that concept extend to objects in the derived category ...

3
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0
answers

233
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### For a family of short exact sequences of coherent sheaves, can we define the splitting subscheme?

This question has been asked in SE.
Let $k$
be an algebraically closed field of characteristic zero. Let $X$ be a projective scheme over $k$. We can talk about short exact sequences of coherent ...

4
votes

1
answer

195
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### Higher direct image of coherent sheaf and rigid analytification

Let $k$ be a non-archimedean field of characteristic zero. Then let $$f:X \rightarrow Y$$
be a (proper) morphism of smooth projective varieties over $k$. The GAGA functor (for rigid analytic spaces) ...

2
votes

0
answers

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### Projectivization of a coherent sheaf using resolution by vector bundles

Let $\mathcal{F}\to X$ be a coherent sheaf over a compact Kahler manifold and let $E^{\bullet}\to \mathcal{F}$ be a resolution of $\mathcal{F}$ by holomorphic vector bundles.
Is there a way to ...

3
votes

1
answer

265
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### When is a sheaf $\mathcal{L}_1 \subset \mathcal{F} \subset \mathcal{L}_2$ sandwiched between two line bundles also a line bundle?

This question is in the interest of answering one part of this question, but I think it is distinct enough to warrant a separate question.
Let $X$ be a regular 2-dimensional Noetherian scheme, for ...

3
votes

1
answer

266
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### Is local freeness open for curves?

Let $X$ be a complete nonsingular curve and $S$ a scheme over $k$ algebraically closed, and $\cal{F}$ a coherent sheaf on $X \times S$, generated by finitely many global sections and flat over $S$ (...

5
votes

1
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361
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### Which complexes of coherent sheaves can be presented as countable homotopy limits of perfect complexes?

Let $X$ be a noetherian scheme (actually, I need the case where $X$ is proper over an affine scheme), $C$ is an object of the derived category $D_{coh}(X)$ of coherent sheaves on $X$. Under which ...