# Questions tagged [coherent-sheaves]

The coherent-sheaves tag has no usage guidance.

217
questions

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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 $\mathcal F$ ...

2
votes

0
answers

72
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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 ...

1
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0
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132
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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
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133
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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

136
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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
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0
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125
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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

210
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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
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239
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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 $S$ (...

4
votes

1
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174
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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 ...

2
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0
answers

158
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### Stability and simplicity of tangent sheaf of Grassmannian

Everything is over the complex numbers. Let $ X = \text{Gr}(k,n) $ be a Grassmannian variety and with tangent sheaf $ T_X $.
(1) Is $ T_X $ simple, i.e. is $\text{Hom} ( T_X, T_X) = \mathbb{C} $?
(2)...

2
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0
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129
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### Higgs quotient sheaf of a Higgs bundle

Let $X$ be a smooth complex projective variety of dimension $n\geq2$, let $\mathfrak{E}=(E,\varphi)$ be a Higgs bundle over $X$ of rank $r\geq2$.
Does exists a Higgs quotient sheaf $\mathcal{Q}$ of $\...

2
votes

1
answer

144
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### Semi-orthogonal decompositions over singular schemes

Where can I find any more or less explicit semi-orthogonal decompositions of derived categories of perfect complexes or of bounded derived categories for singular schemes that are proper over a ring R?...

3
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0
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125
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### Bounded derived categories of which smooth projectives possess bounded t-structures whose hearts are categories of modules?

I am interested in $P$ that is smooth and proper over a field and such that the derived category of coherent sheaves $D^b(P)$ possesses a $t$-structure whose heart is the category of finitely ...

7
votes

1
answer

521
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### Relation between ProCoh and solid modules

There are two languages endow the theory of coherent sheaves with a six functor formalism (that I "know" of), one being formulated in $\text{ProCoh}(X)$ by Deligne and the other being $D(\...

3
votes

2
answers

304
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### When are two resolutions of a coherent sheaf homotopic

Let $\mathcal{F}$ be a coherent sheaf on a projective manifold $X$. It is well known that one can construct a resolution of $\mathcal{F}$ by holomorphic vector bundles (locally free sheaves).
Are two ...

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0
answers

69
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### Is the resolution of a sub-sheaf into complex of holomorphic vector bundles a "sub-resolution"?

Let $F\rightarrow X$ be a coherent sheaf on a projective Kahler manifold. We can resolve it into a complex of holomorphic vector bundles $E^{\bullet}\rightarrow X$. Let $G\subset F$ be a subsheaf of $...

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0
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211
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### Stability of vector bundles and corresponding coherent sheaf

Let $j:Y\hookrightarrow X$ be an embedding of projective complex manifolds. Let $E\rightarrow Y$ be a vector bundle and $S=j_*E$ the corresponding coherent sheaf on $X$ (see Push forward of a Vector ...

9
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1
answer

265
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### Does $X\times Y$ have the resolution property if both $X$ and $Y$ have?

We say a complex manifold $X$ has the resolution property if every coherent sheaf $\mathcal{M}$ on $X$ admits a surjection $\mathcal{E}\twoheadrightarrow \mathcal{M}$ by some finite rank locally free ...

3
votes

0
answers

61
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### Derived category supported in a Serre subcategory of a locally noetherian category

This is a cross-post from math.stackexchange at https://math.stackexchange.com/questions/4251692/derived-category-supported-in-a-serre-subcategory-of-a-locally-noetherian-catego, since I didn't get ...

3
votes

1
answer

115
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### Dual family of torsion-free rank-1 sheaves on Gorenstein curves

Let $X$ be a Gorenstein curve over a field an consider the compactified Jacobian parametrizing torsion-free, rank-1 sheaves on $X$.
Is there a chance that the dual functor $Hom(\_, \mathcal O_X)$ ...

8
votes

0
answers

208
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### Direct summands of a pushforward in the derived category of coherent sheaves

For a Noetherian scheme $X$, let $D^b(X)$ denote the bounded derived category of coherent sheaves on $X$.
Let $X$ be a Noetherian scheme, $i:Y \hookrightarrow X$ a closed subscheme and $F$ an object ...

2
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0
answers

163
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### Generic rank of proper pushforward of the trivial line bundle

Given a proper surjective morphism $f:X\rightarrow Y$ where $X$ and $Y$ are smooth projective varieties. The proper pushforward $f_!$ is the homomorphism that sends the class of a coherent sheaf $M$ ...

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0
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102
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### A question about self-intersecting normal crossing divisors

Let $D=D_1\cup D_2$ be a simple normal crossing (snc) divisor in a smooth complex projective variety $X$. Let $E=\mathcal{O}_X(V_1)\oplus \mathcal{O}_X(V_2)$. Then, obviousely,
$$
c(E)\equiv 1+c_1(E)+...

0
votes

1
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121
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### Support of a coherent sheaf over a fiber product scheme

I'm trying to prove the following fact which I don't know if it is true since I am not able to find a counterexample:
Let $X,S$ be two $K$-scheme of finite type with $K$ an algebraically closed field....

4
votes

0
answers

149
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### Coherent sheaves and space filling curves

This paper constructs smooth space filling curves for smooth varieties over finite fields. Let's say we are working in char $p$ on the variety $X$ then this means that there is smooth curve $C_i$ in $...

4
votes

0
answers

127
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### Bounded derived categories of which smooth projectives possess bounded t-structures whose hearts have enough injectives?

For which smooth projective $P$ over a field there exists a bounded $t$-structure $t$ on the bounded derived category of coherent sheaves $D^b(P)$ such the heart $Ht$ of $t$ has enough injectives? ...

7
votes

2
answers

813
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### Grothendieck group generated by classes of invertible sheaves

Given a smooth, projective (complex) varieties $X$, is it true that the grothendieck group $K_0(X)$ of equivalence classes of coherent sheaves on $X$, is generated by clases of invertible sheaves i.e.,...

7
votes

1
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347
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### Heart of a bounded $t$-structure on the derived category of coherent sheaves

Let $X$ be an elliptic curve and $D(X)$ the bounded derived category of $Coh(X)$, coherent sheaves on $X$. If $(D^{\leq 0}, D^{>0})$ is a bounded $t$-structure, then can we already say that the ...

7
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0
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249
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### What is the category of coherent sheaves on a logarithmic scheme?

I try to learn basic things on logarithmic geometry, and in particular I don't find much on the category of coherent sheaves on a logarithmic scheme: is it a notion that makes sense or differ from ...

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0
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67
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### Two questions regarding double short exact sequences

Two short exact sequences on the same objects is called double short exact sequence. The morphism of double short exact sequences is defined in the same way you'd expect, it is a morphism of the ...

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0
answers

62
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### Lifting section coherent sheaf restriction on $\mathbb{P}^1$

Let $\mathcal{F}$ be a rank $n$ sheaf on $\mathbb{P}^n$ given as the image of a square matrix with linear entries of size $> 2n$. In particular, we have two exact sequences:
$$ V \otimes \mathcal{O}...

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0
answers

104
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### Extending automorphism from an affine

Given a projective variety $X$ and an open affine $U$ in $X$. Is there a way to decide whether a given automorphism of a vector bundle $E$ on $U$, is the restriction of automorphism of some coherent ...

3
votes

1
answer

181
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### On locally 3-syzygy sheaves

This is a question that came up in the comments section of here. A reflexive sheaf $E$ is called "locally $3$-syzygy" if it fits into an exact sequence $0\rightarrow E \rightarrow F_1\...

3
votes

1
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165
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### Kernels of surjections from a vector bundle to a reflexive sheaf

Reflexive sheaves on a regular quasi-projective variety can be characterized by the following property that they are the kernel of a surjection from a vector bundle to a torsion-free sheaf. I wonder ...

3
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148
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### When a reflexive sheaf is flat over its base

Let $X$ be a projective and smooth variety with a codimension 2 closed subvariety $Z$. Let $Y=X\times \mathbb{A}^2$ and $E$ a reflexive sheaf on $Y$ that is a vector bundle outside of $Z\times \mathbb{...

1
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0
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102
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### Finite resolution for non-regular schemes

Over varieties that satisfy resolution property, the fact that every coherent sheaf has a finite resolution by vector bundles is equivalent to the variety to be regular. I was wondering whether the ...

4
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0
answers

230
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### Which derived categories of coherent sheaves are equivalent (or "$t$-related") to derived categories of rings?

As far as I understand, it was Beilinson who proved that the bounded derived category of coherent sheaves $D^b(\mathbb{P}^n)$ is equivalent to the bounded derived category of a certain (non-...

4
votes

0
answers

98
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### Cancellation property of vector bundles on non-proper varieties

Krull-Schmidt theorem for proper varieties over a field implies that given an isomorphism of vector bundles between $E\oplus F$ and $G\oplus F$ we can deduce that $E$ and $G$ are isomorphic. My ...

2
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0
answers

138
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### A question about extending vector bundles from formal neighborhood to a coherent sheaf

I have a question which probably is very straightforward but because of my lack of knowledge of formal schemes I'm asking it here. Let's assume we have a vector bundle $E$ on the formal completion $...

2
votes

0
answers

64
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### Map from the stack of coherent sheaves on a curve to the Grothendieck group

Let $X$ be a smooth, projective curve. We let $Coh(X)$ be the stack of coherent sheaves on $X$. Its Grothendieck group is $Pic(X)\times\mathbf{Z}$. Is the map
$$
Coh(X)\rightarrow Pic(X)\times \mathbf{...

2
votes

1
answer

198
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### Varieties satisfying the extension of vector bundles property

We know if we have a regular variety $X$ with $U$ an open sub-scheme such that $codim(X\setminus U)\geq 2$, then any reflexive sheaf has a unique extension from $U$ to $X$. My question is when a ...

10
votes

2
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### Sheaf of relative Kähler differentials intuitively

Let $f: X \to Y$ be a separated morphism between $k$-varieties or more general schemes
of finite type. The most common way in standard literature on algebraic
geometry to define the sheaf of relative ...

2
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0
answers

71
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### Semicontinuity of length for coherent sheaves

Given a coherent sheaf F over a noetherian scheme Y, a classical result in algebraic geometry states the upper-semocontinuity of the function sending any point $y \in Y$ to $\mathrm{dim}_{k(y)}(F \...

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220
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### On definition of stable vector/Higgs bundle

Recall that the slope of a holomorphic vector bundle $\mathcal{E}$ over a smooth projective variety (or rather a compact Kähler manifold) $X$ is defined as
$\mu(\mathcal{E}) :=\frac{\operatorname{deg}(...

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128
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### Use of flattening stratification (from Nitsure's construction of Hilbert and Quot schemes)

I study Nitin Nitsures paper on the Construction of Hilbert and Quot Schemes and not understand the propetry (F) completely:
In previous chapter (Embedding Quot into Grassmanian) it was
proved that ...

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0
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83
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### Locally freeness of twisted direct images $\pi_* \mathcal{F}(i)$

I have a question about a step in the proof of the
Existence of Flattening Stratification I found in
Nitsure's paper here: https://arxiv.org/abs/math/0504590 This question is closely related to Local ...

4
votes

1
answer

244
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### Yoga on coherent flat sheaves $\mathcal{F}$ over projective space $\mathbb{P}^n$

I'm reading Mumfords's Lectures on Curves on an Algebraic Surface (jstor-link: https://www.jstor.org/stable/j.ctt1b9x2g3)
and I found in Lecture 7 (RESUME OF THE COHOMOLOGY OF COHERENT SHEAVES ON
$\...

0
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0
answers

101
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### Hyperplane which does not contain any associated point of qc sheaf $\mathcal{F}$

I have a question about an argument on $m$-regularity
from 'Fundamental Algebraic Geometry' by Fantechi on page 114, Chapter
5.2: Castelnovo-Mumford regularity. The statement is:
Let $k$ be a field ...

1
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0
answers

103
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### Local freeness of $\pi_*F(r)$ from flatness of $F$

In 'Fundamental Algebraic Geometry' by Fantechi there is a lemma in section 5.3.2, page 119:
LEMMA 5.5 Let $S$ be a noetherian scheme and let $F$ be a coherent sheaf on $\mathbb{P}^n_S$. Suppose there ...

3
votes

1
answer

404
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### Proper and flat over $\mathbb{P}^1_{\mathbb{Z}}$ implies locally free

Let $\pi:X\to \mathbb{P}^1_{\mathbb{Z}}$ be a proper flat morphism with $X$ an integral scheme. Is $\pi_*\mathcal{O}_X$ necessarily locally free?