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2
votes
0answers
69 views

Are injective modules flabby on basic open sets?

In order to give a simple proof of a basic fact about quasi-coherent modules (see below), I'm interested in knowing whether the following statement holds: Statement: If $A$ is a commutative ring and ...
6
votes
1answer
544 views

Is every module the colimit of its finitely generated submodules? (for algebraic spaces or stacks)

For (quasi-compact and quasi-separated) schemes there is a categorical way to characterise quasi-coherent sheaves of finite type using purely the abelian category $\operatorname{QCoh}(X)$. In an ...
7
votes
0answers
230 views

Colimits of quasi-coherent sheaves on a ringed space

Recall from the stacks project that a sheaf of modules $F$ on a ringed space $X$ is called quasi-coherent if there is an open covering $\{U_i\}$ such that each $F|_{U_i}$ has a presentation, i.e. is ...
2
votes
1answer
214 views

Projection formula for immersions

Let $i : Y \to X$ be a quasi-compact immersion of schemes and let $M$ be a quasi-coherent sheaf on $X$. There is a canonical homomorphism $M \otimes i_* \mathcal{O}_Y \to i_* i^* M.$ Question: Is ...
5
votes
2answers
429 views

Basics(?) about quasi-coherent modules on projective schemes

EDIT. (05-04-12) I have revised and improved the questions. Let $A$ be a commutative $\mathbb{N}$-graded $R$-algebra, which is finitely generated by $A_1$ as an $A_0$-algebra. You may also assume ...
2
votes
1answer
182 views

Is the pushforward via a proper map of a finite presentation module of finite presentation?

It's true that the pushforward of a coherent sheaf is coherent via a proper morphism: but do proper morphisms preserve a finite presentation? Under some assumptions perhaps? Does it change if we are ...
3
votes
0answers
211 views

Flatness of a homogeneous quasi-coherent sheaf on the formal plane from flatness on lines through the origin?

(This is a follow-up to a previous question; as often happens, I didn't include enough hypotheses, so I'm asking a new, more-likely-to-be-true question). Consider the formal plane ...
1
vote
1answer
233 views

Flatness on the formal plane from flatness on lines through the origin?

Consider the formal plane $\operatorname{Spec}\mathbb C[[t,h]]$, and let $\mathcal F$ be a quasi-coherent sheaf. Now assume that $\mathcal F$ is flat over infinitely many lines through the origin of ...
3
votes
0answers
518 views

quasi-coherent modules outside algebraic geometry?

Let $X$ be a ringed space. A quasi-coherent module on $X$ is a module which has locally a presentation, i.e. locally on $X$, it is the cokernel of a map between free modules. If $X$ is a scheme, then ...