All Questions
Tagged with sheaf-theory coherent-sheaves
27 questions
2
votes
1
answer
232
views
existence of a coherent sheaf
I am doing algebraic geometry. My question is the following: Here $X=\mathbb{A}^1-0$, $\mathbb{A}^1 = Spec A[t]$, $A$ is a commutative, noetherian ring with unity, consider $\mathcal{O}_{Spec A}$ as $\...
- 613
7
votes
1
answer
607
views
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 ...
- 1,109
2
votes
1
answer
245
views
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 ...
- 5,701
1
vote
1
answer
226
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 ...
4
votes
2
answers
642
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 $\mathbf{P}(\mathcal{E}):=\mathrm{Proj}(\mathrm{Sym}(\mathcal{E}))$ ...
- 663
2
votes
1
answer
257
views
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 (...
- 61
3
votes
1
answer
201
views
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 ...
- 1,802
0
votes
1
answer
188
views
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....
- 395
4
votes
1
answer
289
views
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
$\...
- 6,038
0
votes
0
answers
157
views
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 ...
- 6,038
2
votes
0
answers
144
views
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 ...
- 6,038
1
vote
0
answers
262
views
Devissage lemma (Mumford's & Oda's AG II)
This question is part II of my proof reading of Lemma of devissage
from Mumford's & Oda's Algebraic Geometry II, findable on page 81; Theorem 6.12:
Theorem 6.12 (“Lemma of devissage”). Let $K$...
- 6,038
5
votes
0
answers
268
views
Coherent cohomological dimension and affine morphisms
For simplicity, all varieties in this question are quasiprojective varieties over an algebraically closed field of characteristic $0$.
The coherent cohomological dimension $cd(X)$ of a variety $X$ is ...
- 59
2
votes
0
answers
645
views
Direct image functor commuting with infinite direct sum of sheaves
Normally I would think this kind of question doesn't belong on overflow, but I haven't been able to find an answer anywhere else, so perhaps it is not so trivial.
Let $f: X \rightarrow Y$ be a ...
- 453
4
votes
3
answers
473
views
Locally ringed space with noetherian stalks and a non-coherent structural sheaf
I am looking for a locally ringed space the stalks of which are noetherian and such that the structural sheaf is not coherent over itself. Can you provide me an example of this?
Notice that one may ...
- 529
1
vote
0
answers
104
views
A sheaf for factorization
Let $R$ be a commutative ring with $1$ and let $X$ be the space of connected componens of $Spec (R) $ with Zariski topology ( The boolean spectrum of $R $ )and let for each $x\in X$ there exists a ...
- 11
2
votes
0
answers
363
views
Singularities of reflexive sheaves
I am studying reflexive sheaves (on $\mathbb{P}^3$) by the Hartshorne's paper ''Stable reflexive sheaves''. As far I understood, reflexive sheaves fail to be locally free at a finite number of points (...
- 556
4
votes
0
answers
536
views
When is a coherent subsheaf determined by its global sections
I am reading an article in which a proof is based on defining a subsheaf by only giving its global sections.
The exact setting is that, one has a surjective finite morphism $f:Y\to X$ between ...
- 333
2
votes
0
answers
272
views
Double dual of ample sheaf
Let $X$ be a projective manifold. Then we can define ample sheaves on $X$, and many results of ample vector bundles still hold in this more general case (See K. Kubota, Ample sheaves).
Now I was ...
- 147
2
votes
0
answers
102
views
Are there Coherent Cosheaves?
Is there a well-defined notion of coherent cosheaves in a similar sense to coherent sheaves? If so, what properties do they hold?
- 913
20
votes
1
answer
2k
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 ...
- 118k
2
votes
0
answers
306
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 $U_{\...
- 181
0
votes
1
answer
788
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 $X$ ...
- 8,998
4
votes
0
answers
110
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 forward....
- 41
0
votes
0
answers
186
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 $H^\ast (Z,i^{-1}S)$ ...
- 16.9k
17
votes
2
answers
1k
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 topos)....
- 9,408
4
votes
1
answer
383
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 ...
- 23.3k