Skip to main content

All Questions

Filter by
Sorted by
Tagged with
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 $\mathbf{P}(\mathcal{E}):=\mathrm{Proj}(\mathrm{Sym}(\mathcal{E}))$ ...
rfauffar's user avatar
  • 663
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 ...
Hamed Khalilian's user avatar
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 ...
Yikun Qiao's user avatar
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....
John117's user avatar
  • 395
8 votes
0 answers
351 views

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 ...
hennlu's user avatar
  • 333
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 ...
user avatar
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 ...
user267839's user avatar
  • 6,038
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 ...
user267839's user avatar
  • 6,038
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 $...
prochet's user avatar
  • 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 ...
Luke's user avatar
  • 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, ...
Stabilo's user avatar
  • 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 ...
user24453's user avatar
  • 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 ...
Sasha's user avatar
  • 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 $...
Joel Dodge's user avatar
  • 2,799