Questions tagged [ringed-spaces]

For questions on ringed spaces or locally ringed spaces

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Is the "sheaf of derivations" locally free?

Let $k$ be a field. We require all algebras to be associative commutative, and when unital we require morphisms between them to respect the identity element. Let $X$ be a topological space, equipped ...
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Constructing a "sheaf of vector fields" for a flasque sheaf of $k$-algebras

Let $k$ be a field. We require all algebras to be associative and commutative. Unital algebra morphisms are required to preserve the multiplicative identity. Let $\mathcal{O}$ be a sheaf of unital $k$...
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Does the "sheaf of diffeologically-smooth real-valued functions" functor reflect isomorphisms?

(This is a follow-up question to this earlier one.) Setup: let $u : \mathrm{Diff} \rightarrow\mathrm{Set}$ denote the forgetful functor on the category of smooth manifolds. Let $\tilde{X} \subset \...
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Which name receives this "pulled-back" sheaf?

Let $f:X \to Y$ be a continuous map of topological spaces, and let $\mathcal{F}_Y$ be a subsheaf of the sheaf of germs of continuous functions over $Y$, i.e. $\mathcal{F}_Y \subset \mathcal{C}^0_Y$. I'...
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Definition of restriction of morphism of ringed space?

I have a question. Is there a definition of restriction of morphism of (locally) ringed space? Let $(f,f^{\flat}): X \to Y$ bea morphism of ringed spaces ; i.e., $f:X\to Y$ is a continuous map and $f^{...
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Well-definedness of ring operations on stalks

Let $(X, \mathcal{O}_X)$ be a locally ringed space. As a sanity check for myself, I'd like to show that the addition and multiplication of the germs at a point $p \in X$ are well-defined. I was able ...
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Is the ideal product presheaf a sheaf?

Given a ringed space $(X,\mathcal{O}_X)$ and ideal sheaves $\mathcal{I},\mathcal{J}\subset\mathcal{O}_X$, we define the ideal product presheaf $\mathcal{I}\cdot_p\mathcal{J}$ as the ideal presheaf $$ ...
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A detail in the proof that tensor product of sheaves of $\mathcal{O}_X$-modules commutes with pullback

Given a morphism of ringed spaces $f:(X,\mathcal{O}_X)\to(Y,\mathcal{O}_Y)$ and $\mathcal{O}_Y$-modules $\mathcal{M}$ and $\mathcal{N}$, here it is proven that $$ f^*(\mathcal{M} \otimes_{\mathcal{O}...
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Thuillier's beth-analytification: preservation of fibre products

I am reading the article Géométrie toroïdale et géométrie analytique non archimédienne by Amaury Thuillier and I'm having a hard time understanding how his $\beth$ functor exactly works. My main ...
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Stacks Project proof that gluing locally ringed spaces which happen to be schemes gives a scheme

I'm currently reading the Stacks Project section on gluing schemes. I can understand the proof of Lemma 01JB, but it is hard for me to understand the proof of Lemma 01JC. By constructions of the ...
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Reference for characterisation of isomorphisms of ringed spaces

Recall that a morphism of ringed spaces $(X,\mathcal{O}_X)\to(Y,\mathcal{O}_Y)$ can be specified in two ways: either in the form $(f,\psi^{\flat})$ or $(f,\psi^\sharp)$, where $\psi^\flat$ and $\psi^\...
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Characterisation isomorphisms of ringed spaces: $(f,f^\flat)$ iso iff $(f,f^\sharp)$ iso?

So I was trying to understand isomorphisms of ringed spaces, looking for a characterization of them. I'll explain what I've found out already and what I don't know yet. Before, I will set some ...
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The spectrum of a ring minus a prime of height $1$

Let $R$ be a ring (commutative, with unit) and let $Q$ be the localization of $R$ at its regular elements (non zero divisors). Let $\mathfrak{p}$ be a prime ideal in $R$. Let $R[\mathfrak{p}^{-1}]\...
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Why there is no notion ´bijective´ regarding morphism of schemes?

Morphism of schemes is defined as morphism between ringed spaces, and the morphism is not a map (pair of maps), so we cannnot define notion of bijectivity of morphism in the category of schemes, is my ...
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Formally smooth morphism of formal schemes $\mathfrak X \to\mathfrak Y$ induces formally smooth morphisms of schemes $\mathfrak X_n \to\mathfrak Y_n$?

Let $f:\mathfrak Y \to \mathfrak X$ be a morphism of formal schemes. We say that $f$ is formally smooth if it satisfies the infinitesimal lifting property, that is if for every affine $\mathfrak X$-...
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Two possible definitions of composition of morphisms of ringed spaces. Are they equivalent?

I'm trying to understand the definition of composition of morphisms of ringed spaces. I know there are several posts on MSE addressing this issue. But the specific issue I am going to discuss in the ...
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Is there an isomorphism $f_*\mathcal{H}om_{\mathcal{O}_X}(f^*\mathcal{G},\mathcal{F}) \cong \mathcal{H}om_{\mathcal{O}_Y}(\mathcal{G},f_*\mathcal{F})$

Let $f: (X,\mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a homomorphism of ringed spaces, and let $\mathcal{F}$ be a module over $\mathcal{O}_X$, $\mathcal{G}$ a module over $\mathcal{O}_Y$. I wonder ...
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Exercise 3.10 from Hartshorne

This is Exercise 3.10(a) from Hartshorne. Can someone verify my solution? Thanks. Exercise 3.10(a): If $f:X\to Y$ is a morphism, and $y\in Y$ is a point, show that $\operatorname{sp}(X_y)$ is ...
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An elementary (and soft) question on strict open subschemes of the affine plane.

Question: A scheme $(X, \mathcal{O}_X)$ is a pair where $X$ is a topological space and $\mathcal{O}_X$ is a sheaf of rings on $X$, with the property that there is an open cover $U_i$ of $X$ with $U_i \...
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What is the difference between invertible sheaf and sheaf of ring on a ringed space?

According to Hartshorne, an invertible sheaf $\mathcal{L}$ on a ringed space $(X, \mathcal{O}_X)$ is a locally free sheaf of rank 1. So, this means that there exists a cover $\{U_i\}_{i\in I}$ of X ...
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Stacks Project, modules locally generated by sections; is the hypothesis necessary?

In the chapter Schemes of the Stacks project, I am confused about Lemma 4.5, which I state here. "Let $X$, $Y$ be locally ringed spaces, $\mathcal{I}\subset\mathcal{O}_X$ be a sheaf of ideals ...
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3 votes
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Ideal sheaf is quasi-coherent if and only if its generated by local sections.

My confusion is lies in Schemes Lemma 10.1 of the Stacks project. First, Modules Definition 8.1 states that a sheaf $\mathcal{F}$ of $\mathcal{O}_X$-modules is locally generated by sections if for all ...
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8 votes
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243 views

When considering a finite-type scheme as a ringed space, is it enough to look at its $k$-points?

I am reading a set of notes by Michel Brion about automorphism groups of projective varieties. The following claim appears in the proof of a theorem stating that if G is a connected group scheme, $X$ ...
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4 votes
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Sections on locally ringed space as functions

Notation/Introduction: Let $(X, \mathcal{O}_X)$ be a locally ringed space, $U \subseteq X$ an open and $p \in U$. Denote by $\mathfrak{m}_p \lhd \mathcal{O}_{X,p}$ the unique maximal ideal and $k_p=\...
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13 votes
1 answer
265 views

Non-isomorphic locally ringed spaces which represent isomorphic functors $\mathsf{CommRing} \to \mathsf{Set}$.

It's well known that the restricted Yoneda functor $よ : \mathsf{Schemes} \to \operatorname{Fun}(\mathsf{CommRing},\mathsf{Set})$ is an embedding, so that (in particular) if $X$ and $Y$ are schemes ...
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5 votes
1 answer
61 views

Liu's Algebraic Geometry Ex 2.12

The exercise to be shown is Let $f:X \rightarrow Y$ be a morphism of ringed topological spaces. Let $V$ be an open subset of $Y$ containing $f(X)$. Show that there exists a unique morphism $g:X \...
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Do we get a section on a ringed space $\mathcal{A}$ by mapping each element $x$ to the multiplicative unit element $1_x$ of $\mathcal{A}_x$?

Suppose $(X, \mathcal{A})$ is a ringed space (where $\mathcal{A}$ is assumed to be a sheaf of unital, commutative rings over $X$) and consider the map \begin{eqnarray} \psi \colon X \to \mathcal{A} \...
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1 answer
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Is a module structure over a sheaf of rings the same as a module structure on each stalk?

The definition of an $O$-module of Rotman textbook, where $O$ is a sheaf of comutative rings over a space $X$ is: an $O$-module is a sheaf $F$ of abelian groups over $X$ such that (i) $F(U)$ is an $O(...
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16 votes
1 answer
483 views

What is the "dimension" of a locally ringed space?

Let $(X,\mathscr{O}_X)$ be a locally ringed space. If it is a scheme, the natural notion of dimension is the dimension of the subjacent topological space (the size of the biggest chain of irreducible ...
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1 vote
2 answers
203 views

Sheaf of rings on a discrete set.

I was reading through some notes for an exam and one exericse asks me to prove the following There is a unique sheaf of rings making a topological set $X$ with discrete topology a ringed space. I ...
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2 votes
1 answer
120 views

The morphism of ringed spaces $\operatorname{Spec}A\rightarrow\operatorname{Spec}B$ is a morphism of locally ringed spaces

Suppose $\varphi: B\rightarrow A$ is morphism of rings. This induces a morphism of ringed spaces as follows: We get a continuous map of topological spaces $\pi:\operatorname{Spec}A\rightarrow\...
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4 votes
0 answers
164 views

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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6 votes
3 answers
638 views

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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2 votes
1 answer
209 views

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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1 vote
0 answers
176 views

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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Why is the ring of local-ring valued points of a ring scheme a local ring.

I'm confused on a supposedly easy claim: Let $T$ be a base scheme, and let $\mathbf{R}$ be a ring scheme over $T$, i.e. a scheme $\mathbf{R} \to T$ such that for all $E \in \operatorname{Sch}_{/T}$ $\...
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1 vote
0 answers
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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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0 answers
62 views

Computing $\mathscr O_X (X − \{x\})$ for $X$ an affine scheme

Let $R=\Bbb Z[T], X = Spec\ R, \mathfrak m := (T − 2, 3)$ where $\mathfrak m$ is a maximal ideal of $R$. Compute $\mathscr O_X (X − \{x\})$ where $x\in X$ corresponds to $\mathfrak m$. We have $X − \...
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1 vote
2 answers
163 views

Characterisation of isomorphisms of ringed spaces

An isomorphism of ringed spaces is a morphism of ringed spaces $(f,\theta) :(X, \mathcal{O}_{X})\rightarrow (Y,\mathcal{O}_{Y})$ $(f:X\rightarrow Y$ continuous, $\theta: \mathcal{O}_{Y}\rightarrow ...
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5 votes
0 answers
78 views

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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1 vote
1 answer
286 views

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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  • 556
2 votes
1 answer
144 views

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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0 votes
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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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2 votes
1 answer
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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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2 votes
0 answers
101 views

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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5 votes
0 answers
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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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3 votes
1 answer
166 views

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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1 vote
1 answer
155 views

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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0 votes
1 answer
39 views

Bijective $\mathcal{O}_X$-Module Homomorphisms

Let $(X,\mathcal{O}_X)$ be a ringed space, $\mathcal{F}$ and $\mathcal{G}$$\,\,\,$$\mathcal{O}_X$-modules, and $\varphi:\mathcal{F}\to\mathcal{G}$$\,$ an $\mathcal{O}_X$-module homomorphism. If $\...
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