All Questions
14 questions
1
vote
1
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226
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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
3
votes
0
answers
255
views
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 ...
- 930
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
8
votes
0
answers
351
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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 ...
- 333
0
votes
0
answers
220
views
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 ...
- 6,028
0
votes
0
answers
89
views
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 ...
- 6,028
0
votes
1
answer
550
views
Completed stalks of the pushforward of the structure sheaf
Let $\pi:X\to S$ be a proper morphism of Noetherian schemes. Is it possible that $\pi_*\mathcal{O}_X\neq \mathcal{O}_S$ but the natural map $\mathcal{O}_s^\wedge\to (\pi_*\mathcal{O}_X)_s^\wedge$ is ...
user158636
3
votes
0
answers
155
views
Bass theorem on non-affine scheme
A famous theorem of Bass tells that over a noetherian ring $A$, with $\operatorname{Spec}(A)$ connected, every projective module of infinite type is free.
Now, consider a connected noetherian scheme $...
- 3,472
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
2
votes
1
answer
234
views
Glueing modules over $\{x\}\times \operatorname{Spec} R$
Let $k$ be a field and $(C,\mathcal{O}_C)$ be a smooth geometrically irreducible projective curve over $k$ of function field $k(C)$ and let $x$ be a closed point on it. From Laszlo-Beauville's lemma, ...
- 1,479
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
8
votes
1
answer
1k
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 ...
- 5,562
3
votes
1
answer
1k
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 $...
- 2,799