# Questions tagged [ringed-spaces]

For questions on ringed spaces or locally ringed spaces

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### Natural example of a ringed space that is not a locally ringed space

It wouldn't be too difficult to cook up some contrived examples of ringed spaces that aren't locally ringed spaces; however, are there any such examples that appear "in the wild," i.e. that ...
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### Ring theory conventions - Zero ring, local homomorphisms

Just wondering about conventions dealing with the zero ring and the zero scheme. Does the category of schemes have an inital object? Is the zero ring considered local? For the purposes of scheme ...
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### Non zero sections of sheaves which vanish on all fibers

I was stuck on the following question and I was wondering if someone more familiar with sheaf theory might be able to help me with it. $\newcommand{\F}{\mathcal{F}}\newcommand{\O}{\mathcal{O}}$ ...
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### Existence of Quotients in Locally Ringed Spaces

This is part of an exercise (2.14) in Qing Liu's book on Algebraic Geometry. I'm going to be verbose to see if there's something fundamental I'm missing. Actual questions will be numbered. Let $G$ act ...
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### Map of global sections is inclusion map

I have a question related to the answer to this question: Morphism of ringed spaces not induced by homomorphism of rings. I understand the entire argument but I'm having trouble how I can see that ...
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### Locally ringed point space is not a scheme?

Let $X$ be the point space $\{*\}$, and let the structure sheaf $\mathcal{O}_X$ be given by: $$\mathcal{O}_X (X) = \mathcal{O}_X (\{*\}) := \mathbb{Z}_p$$ $$\mathcal{O}_X (\varnothing) := 0$$ Why is ...
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### Is $\textbf{MAN}$ a full subcategory of $\textbf{LRS}$? [duplicate]

Let MAN be the category of differentiable manifolds with smooth maps as morphisms. Let LRS be the category of locally ringed spaces. Now I know that there exists a functor $\textbf{MAN}\to\textbf{LRS}$...
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### A closed subset of a prevariety is a prevariety

My question comes from Gathmann's notes https://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2014/alggeom-2014.pdf on page 42 Exercise 5.13. Let $Y$ be a closed subset of a prevariety $X$, ...
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### Condition of closed embedding

One condition, (2) of Definition 25.4.1, for a morphism of ringed spaces $i:Z\rightarrow X$ to be a closed immersion is that $$O_X \rightarrow i_*O_Z$$ is surjective. I have two confusions (a) ...
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### The spectrum of a not necessarily quasi-coherent sheaf of Algebras and a related vague question.

See this answer on Mathoverflow and this wikipedia section. These links claim that one can construct $Spec \mathcal{A}$ for any sheaf of algebras over a scheme(and even for any locally ringed space). ...
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### When the contravariant Hom functor of two finite type integral $k$-schemes are isomorphic on points

Let $k$ be an Algebraically closed field. Let $\mathcal C_k$ be the category of integral $k$-schemes of finite type over $k$ (the morphisms between two objects being the morphism of schemes that also ...
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### Is there a “natural” categorical description of (pre-)sheaves of modules?

I've been wondering about the following: Is there a $\textit{neat}$ description of the category $[\mathcal P]\mathcal{Mod}(\mathcal O)$ of [pre]sheaves of modules on a sheaf of rings $\mathcal O$? ...
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### Is flatness condition for proper base change actually needed?

In his 1988 Paper on Resolutions of Unbounded Complexes, Spaltenstein proves that $Lg^*Rf_! \cong RF_! LG^*$ where $\require{AMScd}$ \begin{CD} A @>F>> B\\ @V G V V @VV g V\\ C @&...
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### Proper base change for ringed spaces

$\newcommand{\Oo}{\mathcal{O}}$ Let $f:X\to Y$ be a proper map of topological spaces. Then we can consider the basechange along any continuous map $g:Y'\to Y$. Denote by $X'=X\times_Y Y'$ the base ...
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### What is the point of the topological space of a (locally) ringed space?

My main motivation is in trying to intuitively understand schemes. As I currently understand them, schemes are tools which allow us to recover points (in the traditional sense) as morphisms or encode ...
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### Open vs Closed Immersions of Locally Ringed Spaces

I'm reading Qing Liu's book at the moment and I'm trying to figure out why open immersions of locally ringed spaces are required to be isomorphisms on stalks, but closed immersions are only required ...
### What properties single out $\operatorname{Spec}(\mathbb{k})$-schemes that are quasi-projective varieties over $\mathbb{k}$?
I have a question in algebraic geometry that I would like to ask: Let $\mathbb{k}$ be an algebraically closed field. Is there a property $P$, phrased in the language of schemes, such that every ...
I am trying to look at smooth manifolds in the context of locally ringed spaces. Vector bundles have a characterization in terms of sheaves of modules: if $X, \mathcal{O}_X$ is a topological (or ...