Questions tagged [locally-ringed-spaces]

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3
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1answer
169 views

Sheaf of chain complexs glued by chain homotopy equivalences

Let $(X,\mathcal O_X)$ be a locally ringed space with an open covering $\mathscr U$. Suppose: For any $U\in\mathscr U$, we have a chain complex $(C_U, d_U)$ such that $C_U$ is an $\mathcal O_X(U)$-...
2
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0answers
49 views

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)...
7
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0answers
231 views

Does a field extension define an effective descent morphism for locally ringed spaces?

Let $K'/K$ be an extension of fields and set $X=\operatorname{Spec}(K)$ and $X'=\operatorname{Spec}(K')$. As the category of locally ringed spaces has fibre products (see arXiv:1103.2139 or here) we ...
5
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1answer
206 views

Hakim's definition of a locally ringed topos

In Hakim's book "Topos annelés et schémas relatifs", Chap. III, Def. 2.3 states that a ringed topos $(X,A)$ is a locally ringed topos when two equivalent conditions are satisfied: (i) For ...
2
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0answers
53 views

For which locally ringed spaces is the structure sheaf given by LRS morphisms to the real line?

Let $\mathsf{LRS}_{\mathbb R}$ denote the category of locally $\mathbb R$-ringed spaces. Given a locally ringed space $(X,\mathcal O_X)$, write $C_{(X,\mathcal O_X)}^p$ for the hom-sheaf on $X$ of ...
2
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3answers
304 views

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 ...
4
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2answers
293 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 ...
2
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0answers
196 views

Immersions of locally ringed spaces and locally closed image

Let $f:X\to Y$ be a morphism of locally ringed spaces. In this MSE answer, the first definition below is suggested. Say $f:X\to Y$ is an $R$-immersion of locally ringed spaces if it's a topological ...
6
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1answer
499 views

Morphisms of locally ringed spaces into affine schemes

In Görtz and Wedhorn's Algebraic Geometry I, there's the following proposition: Proposition 3.4. Let $(X,\mathcal O_X)$ be a locally ringed space. If $Y$ is an affine scheme then the natural map ...
10
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1answer
670 views

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 ...
1
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0answers
181 views

Embedding dimension: local finiteness & intuition for more general spaces

Can every complex analytic space be covered by Stein spaces of finite embedding dimension? I am almost sure that ought to be true. Here the definition of embedding dimension I have in mind is $$ \...
13
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1answer
1k views

Which local ringed spaces are schemes?

(This was originally asked on math.stackexchange, but didn't get any responses. I figured it might be worthwhile to move it here and try again.) This paper gives a proof that the underlying ...
4
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3answers
557 views

Are schemes pushouts of neighbourhoods and formal neighbourhoods?

Hello, I have two questions, the first less important. Let $X$ be a scheme, $x \in X$ a schematic point. What is an elegant way of defining/characterizing the map $\operatorname{Spec}(O_{X,x}) \...
22
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0answers
1k views

Riemannian manifolds etc. as locally ringed spaces?

There are, among others, three general ways of equipping a "space" (which for the purposes of this question could be a topological space or a differentiable manifold, according to the case) with ...
12
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3answers
3k views

Justification of the term "invertible sheaf"

Let $X$ be a locally ringed space (or a scheme) and $M,N$ two $\mathcal{O}_X$-modules such that $M \otimes N \cong \mathcal{O}_X$. Does it follow that $M$ is invertible in the usual sense, namely that ...
9
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1answer
911 views

Examples of locally ringed spaces

I want to know more classes of examples of locally ringed spaces. The reason is that when I want to prove/disprove something about locally ringed spaces, my examples are often not eclectic enough. ...
5
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2answers
3k views

Closed subschemes and pulling back the structure sheaf via the inclusion map

I would just like a clarification related to closed subschemes. If $(X,{\cal O}_X)$ is a locally ringed space and $A\subset X$ is any subset with the subspace topology then $i^{-1}{\cal O}_X$ will be ...
4
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2answers
663 views

Given a morphism from X to Y, when is the morphism from O_Y to the pushforward of O_X injective

I would like to know under what condition the morphism $\mathcal{O}_Y\longrightarrow f_\ast \mathcal{O}_X$ induced by a morphism $f:X\longrightarrow Y$ of schemes is injective. Let me give an example ...
30
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3answers
3k views

What is the right version of "partitions of unity implies vanishing sheaf cohomology"

There are several theorems I know of the form "Let $X$ be a locally ringed space obeying some condition like existence of partitions of unity. Let $E$ be a sheaf of $\mathcal{O}_X$ modules obeying ...