The coherent-sheaves tag has no wiki summary.
1
vote
0answers
61 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
104 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
171 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
1answer
139 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 ...
2
votes
1answer
173 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 ...
1
vote
1answer
190 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
156 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
0answers
105 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
40 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
160 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
142 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 ...
13
votes
5answers
687 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
165 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
212 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
178 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
93 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
117 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
172 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
508 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
180 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 ...
4
votes
2answers
403 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
96 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
199 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
121 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
131 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
124 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
499 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
211 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
531 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
461 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 ...
6
votes
1answer
504 views
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 ...
0
votes
0answers
160 views
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 $H^\ast (Z,i^{-1}S)$ ...
2
votes
1answer
268 views
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 ...
4
votes
2answers
478 views
Local freeness of direct images
This question arose from an unsuccessful attempt to settle another question of mine: Vector fields on complete intersections
Let $X\to Y$ be a smooth projective morphism of noetherian schemes and let ...
9
votes
3answers
978 views
Derived categories of (coherent) sheaves of modules: exceptional images, gluing, and proper descent?
I am interested in the properties of (the derived categories) of various categories of (coherent) sheaves of modules (over varieties). I would like to understand in what extent these properties are ...
1
vote
1answer
294 views
Do some of the local cohomology groups of the structure sheaf on the singular locus vanish?
Ok, so this is where I reveal my ignorance as an algebraic geometer: I had previously asked about pushforward of line bundles from the smooth locus of a variety to the whole thing. I think I ...
8
votes
3answers
792 views
Is the pushforward of a line bundle on the smooth locus of a terminal singularity again a line bundle?
In algebraic geometry, it is a sad fact of life that pushforward doesn't preserve being a coherent sheaf; for example, the pushforward by the complement of a divisor of the structure sheaf (or more ...
7
votes
2answers
559 views
coherent sheaves on affine formal schemes
Let $\hat{X} = \text{Spf} \hat{A}$ be obtained as the formal completion of an affine scheme $X = \text{Spec} A$ where $A$ is an adic noetherian ring. Given a coherent sheaf $\mathfrak{F}$ on ...
3
votes
1answer
442 views
Coherent sheaves on projective space over a general ring
Let $K$ be a field and $n \geq 0$. Serre proved that $\text{Qcoh}_f(\mathbb{P}^n_K)$ is equivalent to the localization of $\text{grMod}_f(K[x_0,...,x_n])$, in which the inclusions $M_{\geq a} \to M$ ...
3
votes
1answer
1k views
question about torsion sheaf
Hi, i'm stuck on the following, please can someone help?
Let $E$ be a complex holomorphic vector bundle of rank r over a compact kahler
manifold $M$, let me indicate $\mathcal{E}$ the associated ...
3
votes
1answer
225 views
Semicontinuity and cohomological flatness for algebraic spaces
Let $f \colon X \to S$ be a proper morphism form a seperated algebraic space $X$ to an affine noetherian scheme $S$.
Given a coherent sheaf $F$ on $X$, we know from Knutson's book, that the ...
10
votes
3answers
1k views
Is it true that if the pushforward of a coherent sheaf is locally free, then the original sheaf is locally free?
I think the title says it all. If I have a finite map $p:X\to Y$ between schemes, and $F$ is a coherent sheaf on $X$ such that $p_*F$ is locally free, can I conclude that $F$ is locally free?
...
4
votes
2answers
508 views
Deformations of sheaves via automorphisms. How to express $Ext^1$?
Let $X$ be a complex manifold (for example $\mathbb CP^n$), let $v$ be a holomorphic vector field on $X$, and let $F$ be a coherent sheaf (for example a vector bundle or a structure sheaf of a point). ...
10
votes
3answers
952 views
Are schemes that “have enough locally frees” necessarily separated
Let me motivate my question a bit.
Thm. Let $X$ be a locally noetherian finite-dimensional regular scheme. If $X$ has enough locally frees, then the natural homomorphism $K^0(X)\longrightarrow ...
9
votes
2answers
496 views
What are the merits of the different finiteness conditions on quasi-coherent sheaves?
It's my understanding that there's no disagreement about the right way to define a quasi-coherence for a sheaf $F$ of $O_X$-algebras (over a scheme, locally ringed space, or even locally ringed ...
3
votes
1answer
533 views
Quasi-coherent sheaves of O_X-algebras
Let $X$ be a quasi-compact scheme and let $\mathcal{A}$ be a quasi-coherent sheaf of $\mathcal{O}_X$-algebras on $X$. $X$ being quasi-compact, we can write $X = U_1 \cup \dots \cup U_n$ with each ...
3
votes
1answer
263 views
How to characterize Abelian sheaves that are quasi-coherent?
Let $X$ be a scheme. Suppose you are given a sheaf of Abelian groups $\mathcal{A}$ over $X$. How can you determine if $\mathcal{A}$ is the underlying Abelian sheaf of a sheaf of $O_X$-modules? In ...
5
votes
2answers
459 views
Is there a presentation of the cohomology of the moduli stack of torsion sheaves on an elliptic curve?
Let $E$ be your favorite elliptic curve, and let $Tor^m$ be the moduli stack of torsion sheaves of degree $m$ on $E$. This sounds horrible, but it's not so bad; it's a global quotient of a smooth ...
