The coherent-sheaves tag has no wiki summary.

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### Strong form of Grothendieck's algebrization theorem

Let $R$ be a Henselian discrete valuation ring with algebraically closed residue field $k$ ($R$ is not necessarily complete), $X$ a regular surface over $\mathrm{Spec}(R)$ and a sequence of locally ...

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### Sheaves whose restriction maps are monomorphisms?

When the restriction maps of a sheaf of $\mathcal O_X$-modules are epimorphisms,
the sheaf is flasque and we have a whole theory of that. Is there a detailed study of the opposite phenomenon, i.e., ...

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### Do I know what “coherent sheaf” means if I know what it means on locally Noetherian schemes?

I've been trying to convince myself that "coherent sheaf" is a natural definition. One way I might be satisfied is the following: for modules over a Noetherian ring $A$, coherent and finitely ...

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### Extending a section of a coherent sheaf and homomorphism

Let $X=\mathrm{Spec}(A)$ be an affine integral scheme of finite type over $\mathbb{C}$ and $\phi:\mathcal{F} \to \mathcal{G}$ be a surjective morphism of coherent sheaves on $X$. Let $f \in A$, ...

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### Algebraicity of the stack of coherent sheaves

I am trying to understand the proof of Theorem 4.6.2.1 in the book on algebraic stacks by Laumon and Moret-Bailly. The setting is this: $S$ is a Noetherian scheme, $f\colon X \rightarrow S$ is a ...

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### Is a general extension of general stable sheaves on $\mathbb P^2$ stable?

Theorem 2 in this paper by Bhosle gives a nice condition on slopes for when a general extension of general stable bundles on curves is stable. Does anyone know whether there is an analogous result for ...

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### Pull-back of reflexive sheaves

Let $X$ be a noetherian, projective scheme, $\mathcal{F}$ be a reflexive sheaf on $X$ pure of dimension $\dim(X)$ and $Y \subset X$ be a closed subscheme of $X$. Is it possible that the pull-back of ...

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### Singular projective variety where the Cartan homomorphism is not an isomorphism? [duplicate]

Please delete. The question linked is sufficient. --- OP

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### Locally free sheaves and flat families of projective scheme

Let $f:X \to Y$ be a flat proper morphism of noetherian projective schemes and $\mathcal{F}$ is a coherent sheaf on $X$. Suppose for all $y \in Y$, $\mathcal{F} \otimes_{\mathcal{O}_Y} \mathcal{O}_y$ ...

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### K-theory of coherent sheaves on complex manifolds: references and gamma-filtration?

For a complex manifold $X$ one has an exact category of locally free coherent sheaves; so it seems to be no problem to define certain $K$-theory (I do not know whether the $K$-groups given by the ...

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### Hilbert polynomial for any invertible sheaf

Let $X$ be a projective scheme over algebraically closed field $k$, $L$ is invertible sheaf on $X$ and $\mathcal F \in \operatorname{Coh}(X)$ , we define Hilbert polynomial $P_{\mathcal ...

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### Help understanding the proof of a theorem about Cohomology of vector Bundles

I am trying to understand a paper called Betti tables of graded modules and cohomology of vector bundles, but i am stuck in Proposition 6.8 which states:
Let $\mathcal{E}$ be a vector bundle on ...

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### Obstruction to the existence of global resolution of coherent sheaf

It is well known that any coherent sheaf on a complex manifold (or more generally a complex space) admits locally a resolution with locally free sheaves. It is also well known that for non-algebraic ...

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### G-equivariant coherent sheaves on Bott-Samelson Resolutions

Let $G$ be a Lie group, $B$ be a Borel subgroup. $G/B$ is the corresponding flag variety.
Let $w$ be an element of the Weyl group $W$ with a reduced expression
$w = s_1 \cdots s_n$. Let $X_w$ be ...

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### modular forms, invertible sheaves, and quotients

I'm very confused about some contradicatory statements, and I hope someone can help me clarify this.
Let $\Gamma$ be a congruence subgroup. It is well known that modular forms of weight $k$ for ...

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### Can Kuranishi families glue together to give a moduli space?

In his article "Moduli spaces of vector bundles on K3 surfaces and symplectic manifolds", Mukai constructs moduli of simple sheaves on a K3 surface through a theorem that asserts the existence of a ...

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### A canonical algebra of type $(2,2,r)$ is derived equivalent to a path algebra of type $\tilde{D}_{r+2}$ (references)

According to several articles I could find, a canonical algebra of type $(2,2,r)$ is derived equivalent to a path algebra of type $\tilde{D}_{r+2}$, where $r \geq 2$.
I don't know how to obtain this ...

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### K-theory of ringed spaces (including henselian and formal schemes); excision and Mayer-Vietoris

Given a ringed topological space $S$ one can easily define its K and K'-theory as the K-theories of the categories of locally free sheaves and of the category of coherent sheaves on it, respectively. ...

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### Iterated extensions of quotients of vector bundles

Let $X$ be a noetherian scheme. Consider the smallest class $\mathcal{S}$ of coherent sheaves on $X$ which has the following closure properties:
Every locally free sheaf is in $\mathcal{S}$.
If $A,B ...

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### Some questions on vanishing of Ext sheaves

Let $X$ be a complex manifold of dimension $3$ and $\mathcal{E}$ be a coherent sheaf such that $\dim supp(\mathcal{E})=1$. In this situation, I would like to know why we have the Ext sheaf ...

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### Cech cohomology.

Let us consider the scheme $X$ and the coherent sheaf $\cal F$ on it. We consider finitely many affine open covers $U_{\lambda}$ with $\lambda \in \Lambda$. When I calculate Cech cohomology with ...

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### Non-separatedness of moduli space of sheaves

The moduli space of sheaves over a smooth variety is in general not separated. That is, there exists a flat family of coherent sheaves over a punctured disk which extends to a flat family of coherent ...

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### Global to local for Ext groups and Sheaves

Let $X$ be a projective variety. The sheaf $\mathcal{E}xt^{1}(\Omega_{X},\mathcal{O}_{X})$ is supported on $Sing(X)$.
Now, there should be a theorem (perhaps by Schlessinger) that says that if $X$ ...

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### Extend morphism between coherent sheaves in $\mathbb{P}^n$

Let $\mathcal{F}_1, \mathcal{F}_2$ be coherent sheaves over $\mathbb{P}^n_{\mathbb{C}}$ for $n \ge 3$. Now, $\Gamma_*(\mathcal{O}_{\mathbb{P}^n})=\mathbb{C}[X_0,...X_n]$. Denote by $U_0$ the affine ...

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### Grothendieck duality for stacks

Let $\mathcal{X}$ be a smooth, proper and separated Deligne-Mumford stack and let $\pi:\mathcal{X}\rightarrow X$ be its coarse moduli space. Does Grothendieck duality hold for the morphism $\pi$ ?
In ...

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### Convolution of DQ-Modules

On page 92 of Deformation Quantization Modules Kashiwara and Schapira define two different convolution products for DQ-modules that differ by whether one uses $Rp_{13*}$ or $Rp_{13!}$ to push ...

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### what is the zero locus of a morphism of the derived category?

For a morphism of quasi-coherent sheaves $v\colon E \to F$, on a scheme $X$, one can ask about the locus where $v =0$.
When $F$ is a vector bundle, it's easy to see that this locus is closed.*
Is ...

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### Do there exist equivariant sheafs that are not equivariant vector bundles?

For $F \subset G$ two algebraic groups, consider a homogeneous space $H$ of the form $G/F$. Now every vector bundle over $H$ is a coherent sheaf, but the converse is not true. What happens in the ...

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### is the category of coherent sheaves some kind of abelian envelope of the category of vector bundles?

This might be obvious to experts, but I'm not sure where to look for the answer. On a reasonably nice, at least noetherian, scheme (or variety, algebraic space, stack), can the category of coherent ...

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### Admissible subcategories of $D^b(\mathbb{P}^n)$

Recall that a triangulated subcategory $\mathcal{A}$ of a triangulated category $\mathcal{B}$ is called admissible if the inclusion functor has both left and right adjoints.
Is it true that all ...

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### Global sections of a coherent sheaf in terms of a presentation

Let $A$ be a graded ring satisfying the usual finiteness conditions of EGA II (for example $A_0$ is noetherian, $A_1$ is finite over $A_0$ and $A$ is generated by finitely many elements of $A_1$ as an ...

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### Base change for complex of coherent sheaves

I wonder if the base change for a flat complex of coherent sheaves works. Namely, let $f : X \rightarrow Y$ be a proper morphism of varieties over $k$. Let $\mathcal{F}^{\bullet}$ be a bounded complex ...

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### Realization of a formal duality isomorphism via integration

This is a more precise version of my previous question. Let $X$ be a smooth variety of dimension $n$ over $\mathbb{C}$ and $Z$ a proper sub-scheme. We denote by $\tilde{X}$ the formal completion of ...

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### The support of a finite type module on an algebraic space

I'd like to ask this question to make sure I understand a very basic thing about supports. Let $X$ be an algebraic space and F a quasi-coherent sheaf on it of finite type.
In here the schematic ...

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### what are mutations of sheaves all about?

Suppose I have a smooth projective variety $X$ and a semi-orthogonal decomposition of its bounded derived category of coherent sheaves $D^b(X)$. Then I can apply right or left mutations to the full ...

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### Coherent Sheaves and Holomorphic Vector Bundles

For a complex manifold $M$, I'm trying to understand (from a differential geometry point of view) what its category of coherent sheaves is. If I understand correctly, then the sheaf of holomorphic ...

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### The locus where a sheaf is supported in a certain dimension

I am trying to understand a particular case of this question.
Let $T$ be an affine scheme of finite type over the ground field C. Let $X \to T$ be a morphism. I'll assume this to be the base change ...

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### English reference for the Grauert–Riemenschneider vanishing theorem

What is the best reference in English for the following theorem of Grauert–Riemenschneider:
Theorem:
Let $\phi:X \to Y$ be a proper bi-rational morphism of algebraic varieties over characteristic ...

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### coherent sheaves

suppose $X$ is a smooth variety, with a finite open covering $U_i, i=1,....,k$.
$F$ and $G$ are coherent sheaves on $X$, and $G$ is locally free.
Suppose on each $U_i$, there is a sheaf morphism ...

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### restriction of sheaf

suppose X is a smooth variety and F is a locally free sheaf on X. Let U be an open subset of X and i denote the inclusion map.
Is i_*i^*F equal to F ?
thanks.

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### Surjective and injective criteria via Hilbert polynomials

Let $X$ be a projective complex manifold. Is it true that any coherent sheaf $\mathcal{E}$ on $X$ whose Hilbert polynomial is a constant has a surjective morphsim $\mathcal{O}_X\rightarrow ...

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### Def-Obs theory of sheaves with fixed determinant on CY3.

Let $\mathcal{E}$ be a stable sheaf on a smooth complex projective threefold $X$ and $Ext^k_0(\mathcal{E},\mathcal{E})$ be the traceless Ext groups, defined by the kernel of the trace map
$$
...

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### Some bounded theorem of algebraic stack of coherent sheaves

Let $X$ be a connected projective scheme. Let $U$ be a finite type open substack of the algebraic stack of coherent sheaves on $X$ with a fixed Hilbert polynomial. Can one take $p>0$ such that ...

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### got any tricks to build up t-structures on derived categories?

Are there any good tricks to construct a heart of a t-structure? (I'm thinking on the derived category of coherent sheaves of some variety)
I'll start with the only one I know. If $(T,F)$ is a ...

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### Total exterior Product

In his paper Gaussian maps and plethysm, Manivel uses the term "total exterior product" of line bundles, appearing for the first time on page 3. Given two (projective) varieties $V_1$ and $V_2$ and ...

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### Can one prove vanishing of higher direct images fiber-wise?

Let $\pi:X\to Y$ be a proper map of algebraic varieties (over $\mathbb C$) which is a bi-rational equivalence.
are the following statements equivalent?
The derived direct image of $O_X$ is $O_Y$.
...

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### Is the bounded derived category of coherent sheaves of a variety a small category?

The question is in the title.
I am trying to apply the Mitchell (Freyd-Mitchell?) embedding theorem, which states that for every small abelian category $A$, there exists a ring $R$ such that A ...

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### Proper morphism sending coherent to coherent

Hello,
Is there a proof that the push forward by a proper morphism of Noetherian schemes sends coherent sheaves to coherent ones, without passing in the argument through projective morphisms?
Thank ...

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### Do infinitesimal neighbourhoods help to compute the inverse images of coherent sheaves?

Let $i:Z\to X$ be a closed embedding of (projective) varieties; $S$ is a coherent sheaf on $X$. How could one compute $H^*(Z,i^\ast S)$ (I don't know whether I should write $H^\ast (Z,i^{-1}S)$ ...

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### can one define the pullback between stacks of coherent sheaves for non-flat morphisms?

Consider a morphism $f: Y \to X$ between two varieties and consider the stacks parametrizing coherent sheaves on them $\mathcal{M}_X, \mathcal{M}_Y$.
Does one have for free an induced pullback ...