# Questions tagged [locally-ringed-spaces]

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The locally-ringed-spaces tag has no usage guidance.

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Let $X$ be a locally ringed space. We have for a point $p$ the exact sequence
$$0\to \mathfrak{m}_p^2\to \mathfrak{m}_p\to \mathfrak{m}_p/\mathfrak{m}_p^2 \to 0$$
where $\mathfrak{m}_p$ is the maximal ...

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This coming fall, I will be teaching a course on differential topology to a small group of strong students. In preparation for it, I'm trying to find a category $\mathrm{GDiff}$ with the following ...

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It is known that the natural functor of smooth functions from the category of smooth manifolds into the category of locally ringed spaces is a full embedding (see, for example, here).
Is a similar ...

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1
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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 $\mathcal O_X(U)$-...

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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)...

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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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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 ...

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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 ...

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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 ...

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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 ...

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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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1
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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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1
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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 ...

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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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1
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(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 ...

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577
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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}) \...

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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 ...

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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 ...

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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. ...

6
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2
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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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2
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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 ...

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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 ...