Questions tagged [ringed-spaces]
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16 questions
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K-flat complexes and unbounded derived tensor product over a non-commutative sheaf of rings
$\def\R{\mathscr{R}}
\def\O{\mathcal{O}}$Let $X$ be a topological space and let $\R$ be a sheaf of unital non-commutative rings over $X$.
When $\R$ is commutative, there is much literature on ...
2
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77
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A K-flat complex is acyclic for the pullback functor. Does the converse hold?
$\def\F{\mathscr{F}}
\def\O{\mathscr{O}}
\def\G{\mathscr{G}}
\def\H{\mathscr{H}}$Let $X$ be a ringed space, and let $\F\in K(X):=K(\O_X\text{-Mod})$ be a complex of $\O_X$-modules. If $\F$ is K-flat, ...
3
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1
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333
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Is $\mathcal{O}(-1)$ strictly perfect?
The standard definition (e.g. in Gortz-Wedhorn II Definition 21.133, the Stacks Project http://stacks.math.columbia.edu/tag/08FK etc.) of a strictly perfect complex on a ringed space/site is (...
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Sheaves of modules over a topologically ringed space
Topologically ringed spaces turn up a few places in the literature (most notably in EGA 1's construction of formal schemes), but I can't find much general information about them (e.g. the nLab page is ...
- 1,474
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Is an infinite direct sum of quasi-coherent $\mathcal{O}_X$-modules quasi-coherent on a complex manifold?
On any ringed space $(X,\mathcal{O}_X)$ we can define quasi-coherent $\mathcal{O}_X$-modules: A sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ is quasi-coherent if for every point $x\in X$ there ...
- 9,297
8
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Who introduced the notion of ringed spaces?
My question is very concise, please forgive it.
Who introduced the concept of ringed space?
My first try would be that they were introduced by Cartan in his study of analytic functions with sheaves. ...
- 914
3
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References on topological ringed spaces
This is a follow up to this question of mine.
First of all, let me fix some terminologies, which may or may not be standard:
Definition: A topological ringed space is a pair $X := (|X|, \mathcal{O}_X)...
- 1,728
2
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3
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Localification of a ringed space
Call a ringed space local it if it lies in the image of the obvious faithful, non-full functor from locally ringed spaces to ringed spaces.
Given a ringed space, is there a map $f$ from it to some ...
user141346
11
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1
answer
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About the relation between the categories $\text{Sch}$, $\text{LRS}$ and $\text{RS}$
I've asked this question https://math.stackexchange.com/questions/1407451/about-the-relation-between-the-categories-textsch-textlrs-and-text on math.stackexchange , however I don't think I will ...
- 2,275
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432
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Clifford algebras for quadratic modules over ringed spaces
What is the earliest possible reference for definition and basic properties of Clifford algebras associated to quadratic modules over a ringed space? The ringed space does not need to be locally ...
- 17.7k
11
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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 ...
- 65.2k
3
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2
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569
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Smooth submanifolds defined by Subrings
To be honest, I don't really know, whether or not the following is a research level
question:
Let $M$ be a smooth manifold, $C^\infty(M)$ the smooth function ring on $M$ and
suppose $R\subset C^\...
- 634
5
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2
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730
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Coherence for pull-backs and push-forwards
Let $p:X \to S$ and $q:Y\to S$ be two objects in the category of ringed spaces over the ringed space
$S$, and let $f:X \to Y$ be a morphism over $S$.
Given a sheaf $\mathcal{F}$ of $\mathcal{O}_Y$-...
- 1,598
8
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1
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Sheaves of $\mathbb Z$-modules = sheaves of abelian groups
In his "Algebraic Geometry", Hartshorne proves that for any ringed spaces $(X,\mathcal O_X)$, category $Mod(X)$ of sheaves of $\mathcal O_X$-modules has enough injectives. If we take $\...
- 1,378
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Cohomology of Structure Sheaves: Algebraic, Constructible and more
I am not an algebraic geometer, but I am a topologist who uses sheaves. I have studied some algebraic geometry and am interested in what happens as I reduce the amount of rigidity in the structure ...
- 2,714
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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 ...
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