Questions tagged [ringed-spaces]

For questions on ringed spaces or locally ringed spaces

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Relation between $\text{Aut}_{\mathcal{O}_X}\left(\bigoplus_{i=1}^k \mathcal{O}_X\right)$ and $\text{GL}(k,\Bbb R)$?

Let $(X, \mathcal{O}_X)$ be a ringed space with $X$ a smooth manifold and $\mathcal{O}_X$ the sheaf of continuous functions $X \to \Bbb R$. Do we have some kind of relation between $\text{Aut}_{\...
Nikolai's user avatar
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Definition of a Prevariety - Specifically a covering by ringed spaces

I'm reading Gathmanns's lecture notes on Algebraic Geometry where he defines a Prevariety as ringed space $V$ which has a finite open cover of affine varieties $U_i$. I assume this not just a set-...
Nothingisreallyworking's user avatar
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Closed immersion of locally ringed spaces vs closed immersion determined by ideal sheaf

Let $f: X \to Y$ be a closed immersion of locally ringed spaces, that is, $f$ is a homeomorphism onto a closed subset of $Y$, $f^{\#}:\mathscr{O}_Y \to f_*\mathscr{O}_X$ is surjective, $\mathscr{I} = ...
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Do morphisms of locally ringed spaces which agree on stalks agree on the whole of the locally ringed space?

Let $f,g:X\rightarrow Y$ be morphisms of locally ringed spaces, such that the topological maps $f$ and $g$ are identical, and the maps of sheaves $f^\sharp,g^\sharp:\mathcal{O}_Y\rightarrow f_*\...
Chris's user avatar
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Let $(X,\mathcal{O}_X)$ be a locally ringed space, does there exists a monomorphism $\iota:U\rightarrow X$ for all open $U$?

Let $(X,\mathcal{O}_X$ be a locally ringed space, and suppose that $U\subset X$ is an open set equipped with the sheaf of rings $\mathcal{O}_X|_U$ defined by $\mathcal{O}_X|_U(V)=\mathcal{O}_X(V)$ for ...
Chris's user avatar
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Defining algebraic varieties in general

I have encountered two general notions of algebraic variety when reading different texts in algebraic geometry, and wanted to ask whether they were equivalent or whether one is stronger than another. ...
user0134's user avatar
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If $\mathfrak{p}_y$ the preimage of $\mathfrak{m}_x$ by $\mathcal{O}_Y(Y)\rightarrow\mathcal{O}_X(X)\rightarrow \mathcal{O}_{X,x}$, show that $f(x)=y$

I'm trying to solve the following problem: Let $(X,\mathcal{O}_X)$ be a locally ringed space and $Y$ an affine scheme. Let $f:X \rightarrow Y$ be a morphism of locally ringed spaces. Then, for any $x\...
Gokimo's user avatar
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Pullback of structure sheaf along point inclusion into locally ringed space is residue field?

I'm coming from a differential geometry background (though I'm pretty familiar with category theory) and trying to learn a bit about ringed spaces. Let $(X,\mathscr{O}_X)$ be a locally ringed space, ...
ಠ_ಠ's user avatar
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Closed subvariety is a subvariety

I am attempting to solve Exercise 4.4.7 from this note: Given a variety $X$ and a closed subset $Y \subseteq X$ equipped with the induced topology. For $V \subset Y$ open define $$\mathcal{O}_Y(V)=\{f:...
Mystery girl's user avatar
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Closed subspace cut off by the ideal sheaf coming from a locally closed immersion

$\def\sO{\mathcal{O}} \def\sI{\mathcal{I}}$Given a locally ringed space $Y$ and an ideal sheaf $\sI\subset\sO_Y$, we can consider the closed subspace of $Y$ cut off by $\sI$, i.e., the closed ...
Elías Guisado Villalgordo's user avatar
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When can $\textbf{SpecMax}(R)$ be a scheme?

Let $R$ be a commutative ring, and let $(\text{Spec(R)},\mathcal{O}_R)$ be the affine scheme associated to $R$. Let $\text{SpecMax}(R)$ the subspace the $\text{Spec(R)}$ of the maximal ideals. My ...
Luis Antonio Sanchez's user avatar
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Finite product of supermanifold.

I am trying to prove that finite product exist in $sMan$ (the category of supermanifolds). the product of supermanifolds that mention is defined as: Take $M,N$ supermanifolds, $F:\operatorname{sMan}\...
Mousa Hamieh's user avatar
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In what sense the "pullback at $ x $" map if functorial?

Let $ X $ and $ Y $ be differentiable manifolds, and let $ f\colon X\to Y $ be a smooth map. Given $ x\in X $ one can define the canonical pullback at $ x $ map $$ f_x^*\colon \mathscr C_{Y,f(x)}^\...
GeometriaDifferenziale's user avatar
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A surjecive homomorphism of $ \mathbb R $-algebras

The main question Let $ A $ and $ B $ be two algebras over the real numbers, and let $ J $ be an ideal of $ B $. Let $ f\colon A\to B $ be a homomorphism of $ \mathbb R $-algebras, and suppose $ f^{-1}...
GeometriaDifferenziale's user avatar
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Existence of a sheafification for every presheaf $ \mathcal{O} '$

I need to show that the existence of a sheafification for every presheaf $\mathcal{O}'$. I know that it's enough to show that the covariant functor $$Hom(\mathcal{O}' , ·) :\textbf{Sh}_X\...
Mousa Hamieh's user avatar
3 votes
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More examples of morphisms of ringed spaces that aren't local?

$\def\Spec{\operatorname{Spec}}$All questions and answers that I've found in MSE regarding a morphism of ringed spaces between affine schemes that isn't a morphism of locally ringed spaces are the ...
Elías Guisado Villalgordo's user avatar
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Characterization of isomorphisms of ringed spaces

Let $ (X,\mathscr O_X) $ and $ (Y,\mathscr O_Y) $ be ringed spaces over the same unspecified commutative ring. My book defines a morphism between $ (X,\mathscr O_X) $ and $ (Y,\mathscr O_Y) $ as a ...
GeometriaDifferenziale's user avatar
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Spectrum is ringed space

Hello I just started studying theory about affine schemes and I am now studying ringed spaces. I see everywhere the Spec(R) is a ringed space but I can’t find a proof. I think it’s obvious since the ...
Mathematician's user avatar
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Subvarieties of an abstract affine variety

I'm really confused on how to transfer constructions from "concrete" affine varieties (i.e. zeroes of polynomial equations in an affine space) and (abstract) affine varieties (i.e. ringed ...
Kandinskij's user avatar
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Is there a morphism of ringed spaces between smooth (resp., complex) manifolds that is not local?

$\def\bbC{\mathbb{C}} \def\sO{\mathcal{O}} \def\hom{\operatorname{Hom}} \def\rs{\mathsf{RS}} \def\lrs{\mathsf{LRS}} \def\swf{\mathsf{SWF}} \def\k{\operatorname{K}} \def\ent{\mathrm{ent}} \def\spec{\...
Elías Guisado Villalgordo's user avatar
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Do restrictions preserve ring structure of ringed space?

Let $(X, \mathcal{O})$ be a ringed space, i.e. $X$ is a topological space and $\mathcal{O}$ is a sheaf of rings on the open subsets of $X$. I would like to show that for two global sections $a, b\in \...
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"Punctured stalk" of a locally ringed space (at a closed point) is the fraction field of the stalk?

Let $(X,\mathcal{O})$ be a locally ringed space. For $x \in X$ a closed point (i.e. $\{x\}$ is closed in $X$), let $\mathcal{O}_x$ denote the stalk of $\mathcal{O}$ at $x$; and define the "...
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On a ringed space: If a section has zero germ at $x$, must it be zero on some neighborhood of $x$?

Let $F$ be a sheaf of commutative rings or Abelian groups on a topological space $X$, let $x \in X$ be a point, let $U$ be an open neighborhood of $x$ in $X$. Let $f \in F(U)$, and suppose the germ $...
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Refinement to previous question: is the conjecture true or false in the affine case?

This question is a follow-up to this other question. There I asked to prove or disprove the conjecture stated at the end. What I want to ask here now is a refinement of this conjecture. For this ...
Elías Guisado Villalgordo's user avatar
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Locally, every coherent sheaf is isomorphic to the cokernel of a homomorphism $\phi: \mathcal{A}^q \to \mathcal{A}^p$

Say, we have a topological space $X$ and a sheaf $\mathcal{F}$ over a sheaf of rings $\mathcal{A}$ on $X$. The sheaf $\mathcal{F}$ is said to be coherent, if the following two conditions are satisfied:...
Paul Joh's user avatar
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The radical presheaf is not a sheaf

$\def\sO{\mathcal{O}} \def\sI{\mathcal{I}} $Let $(X,\sO_X)$ be a ringed space. Let $\sI\subset\sO_X$ be an ideal sheaf. We define the radical presheaf of $\sI$, denoted $\sqrt[p]{\sI}$, as the ...
Elías Guisado Villalgordo's user avatar
1 vote
1 answer
187 views

Is a morphism from a quasi-affine variety to a quasi-projective variety given by globally defined regular maps?

$\def\bbA{\mathbb{A}} \def\bbP{\mathbb{P}} \def\sO_{\mathcal{O}}$The following discussion is strictly classical. Throughout this question, I will use the notions of (i) sheaf of $k$-algebras, (ii) $k$-...
Elías Guisado Villalgordo's user avatar
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2 answers
128 views

How can I think about a morphism of locally ringed spaces?

A locally ringed space is a pair $(X,\mathcal{O}_X)$ of a topological space $X$ and a sheaf of rings $\mathcal{O}_X$. Then we say that $(f,f^b):(X,\mathcal{O}_X)\rightarrow (Y,\mathcal{O}_Y)$ is a ...
user1294729's user avatar
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1 vote
1 answer
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Is the module sum presheaf a sheaf?

$\def\O{\mathcal{O}} \def\M{\mathcal{M}} \def\N{\mathcal{N}} \def\P{\mathcal{P}} $Given a ringed space $(X,\O{_X})$, an $\O_X$-module $\P$ and $\mathcal{O}_X$-submodules $\M,\N\subset\P$ we define the ...
Elías Guisado Villalgordo's user avatar
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function on structure sheaf and its values at points

I have some trouble understanding the concept of a function in Vakil's lecture notes. First, let $X = Spec(A)$ for some ring $A$. A function $f$ is a section in $\mathcal{O}_X(X)$, thus an element of ...
Paul Joh's user avatar
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1 vote
1 answer
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Existence of morphism from a locally ringed space $X \to Spec(\mathbb{F}_p)$

Let $X$ be a topological space, such that $(X, \mathcal{O}_X)$ is locally ringed. Let $A$ be a ring. We showed in the lecture that there is a natural bijection of $Hom((X, \mathcal{O}_X),(Spec(A), \...
Paul Joh's user avatar
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Surjectivity of $\mathscr G\to f_*f^{-1}\mathscr G$

Let $f:X\to Y$ be a continuous map, and let $\mathscr G$ be a sheaf on $Y$. There is a canonical morphism $\varphi:\mathscr G\to f_*f^{-1}\mathscr G$, hence a map $\varphi_y:\mathscr G_y\to f_*f^{-1}\...
Brad Bitta's user avatar
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1 answer
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Function vanishing on ringed spaces

In Vakil's notes on locally ringed spaces, he claimed that "we can't even make sense of the phrase of 'function vanishing' on ringed spaces in general. " Could someone explain what this ...
Nancium's user avatar
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Morphisms of algebraic varieties are regular?

I want to understand a proof that establishes the fact that every map between abstract algebraic varieties (ie, a ringed space on k-algebras which is locally isomorphic to a Zariski closed on the ...
math3341's user avatar
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1 answer
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For a locally ringed space $(X,\mathcal{O})$, is there a sheaf of ideals whose stalks are the max ideals of the stalks of $\mathcal{O}$? [closed]

For $X$ a locally ringed space with structure sheaf $\mathcal{O}$, for each $x \in X$ let $\mathcal{M}_x$ denote the max ideal of the stalk $\mathcal{O}_x$ of $\mathcal{O}$ at $x$. In general, does ...
Indraneel Tambe 2's user avatar
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1 answer
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Is the forgetful functor from locally ringed spaces to topological spaces a full functor? Faithful? What about when restricted to schemes?

In detail, given locally ringed spaces $X,Y$ with underlying topological spaces $X_0,Y_0$, can every continuous map $f_0 : X_0 \rightarrow Y_0$ lift to a morphism of ringed spaces $f : X\rightarrow Y$...
I.A.S. Tambe's user avatar
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2 votes
1 answer
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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 ...
I.A.S. Tambe's user avatar
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1 answer
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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 $k$...
I.A.S. Tambe's user avatar
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3 votes
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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 \...
I.A.S. Tambe's user avatar
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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^{...
Plantation's user avatar
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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 ...
gf.c's user avatar
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3 votes
0 answers
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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 $$ ...
Elías Guisado Villalgordo's user avatar
1 vote
1 answer
186 views

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}...
Elías Guisado Villalgordo's user avatar
1 vote
1 answer
135 views

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 ...
GoogleME's user avatar
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1 vote
2 answers
176 views

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 ...
Elías Guisado Villalgordo's user avatar
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0 answers
100 views

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}]\...
Stabilo's user avatar
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2 votes
1 answer
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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 ...
Pont's user avatar
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5 votes
1 answer
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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 $\mathfrak X$-...
Suzet's user avatar
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2 votes
1 answer
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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 ...
Elías Guisado Villalgordo's user avatar
1 vote
0 answers
23 views

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 ...
Psi's user avatar
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