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4
votes
0answers
143 views

object in D^bCoh(P^2) with prescribed RHom's

Let $\mathbb{C}P^2$ denote the projective plane. From reading the section of http://homepages.math.uic.edu/~coskun/gokova.pdf which surveys Gieseker stable sheaves, I have understood that there are ...
1
vote
1answer
182 views

Will a tilting sheaf over Z which is a generator over Q be a generator modulo every prime?

Assume you have a smooth quasi-projective scheme $X$ (you can actually assume $X$ is projective over an affine scheme of finite type) defined over $\mathbb Z$ (or if you prefer, a discrete valuation ...
6
votes
1answer
176 views

Do general sheaves on P^2 have cohomology governed by their Euler characteristic?

Suppose $\xi$ is chern character on $\mathbb P^2$. Then there is a moduli space $M(\xi)$ of semistable sheaves of chern character $\xi$. If $\xi$ has Euler characteristic 0, then apparently there is ...
1
vote
1answer
213 views

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 ...
0
votes
0answers
70 views

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., ...
14
votes
1answer
749 views

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 ...
0
votes
1answer
155 views

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 $f \in A$, ...
3
votes
1answer
391 views

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 ...
0
votes
0answers
58 views

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 ...
1
vote
1answer
170 views

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 ...
0
votes
0answers
90 views

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

Please delete. The question linked is sufficient. --- OP
0
votes
0answers
101 views

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 $y \in Y$, $\mathcal{F} \otimes_{\mathcal{O}_Y} \mathcal{O}_y$ ...
3
votes
0answers
107 views

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 ...
0
votes
1answer
198 views

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 ...
1
vote
0answers
101 views

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 ...
5
votes
0answers
61 views

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 ...
0
votes
0answers
116 views

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 ...
7
votes
2answers
439 views

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 ...
2
votes
0answers
221 views

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 ...
3
votes
1answer
137 views

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 ...
2
votes
0answers
95 views

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. ...
2
votes
1answer
118 views

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 ...
2
votes
2answers
243 views

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 ...
2
votes
0answers
229 views

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 ...
3
votes
1answer
214 views

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 ...
0
votes
1answer
262 views

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 $X$ ...
1
vote
1answer
197 views

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 ...
3
votes
1answer
248 views

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 ...
3
votes
0answers
53 views

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 ...
2
votes
0answers
196 views

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 ...
1
vote
1answer
172 views

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 ...
16
votes
5answers
1k views

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 ...
3
votes
0answers
219 views

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 ...
2
votes
0answers
244 views

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 ...
1
vote
0answers
214 views

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 ...
1
vote
0answers
105 views

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 ...
3
votes
0answers
160 views

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 ...
3
votes
1answer
216 views

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 ...
5
votes
1answer
632 views

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 ...
0
votes
1answer
212 views

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 ...
6
votes
2answers
619 views

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 ...
0
votes
0answers
103 views

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 ...
1
vote
3answers
273 views

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.
1
vote
1answer
133 views

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 ...
6
votes
0answers
157 views

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 $$ ...
2
votes
1answer
138 views

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 ...
7
votes
1answer
658 views

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 ...
2
votes
2answers
216 views

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 ...
4
votes
2answers
699 views

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 $O_Y$. ...
5
votes
1answer
518 views

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 ...