# Questions tagged [ringed-spaces]

For questions on ringed spaces or locally ringed spaces

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### Group actions on ringed topological spaces

Letting $(X,\mathcal{O}_{X})$ be a ringed topological space, and $G$ a group of automorphisms on $X$, I'm confused on how an element $g\in G$ is supposed to act on an abelian group $\mathcal{O}_{X}(U)$...
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• 319
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### Direct image of locally ringed space

Let $(X,\mathcal{O}_X)$ be a ringed space and $Y$ be topological space and $f: X \to Y$ a continuous map. Then $(Y,f_*\mathcal{O}_X)$ is obviously a ringed space, where $f_*\mathcal{O}_X$ is the ...
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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-...
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• 3,431
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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 ...
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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. ...
• 404
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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 ...
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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 ...
1 vote
134 views

### 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 ...
• 3,536
1 vote
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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 "...
• 2,461
1 vote
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• 559
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• 1,538
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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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### 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 ...