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Suppose that $X$ is a scheme and $x\in X$ is a point. The stalk of $X$ at $x$ is a (local) ring and we can form its spectrum $Y_x=\rm{Spec}(\mathcal{O}_{X,x})$.

There is a canonical map $Y_x\to X$. We can define it by fixing an affine neighborhood $x\in U\cong \rm{Spec}(R)$, making $x$ as a prime ideal in $R$ and $\mathcal{O}_{X,x}\cong R_x$ a localization. The quotient $R\to R_x$ then induces the map of schemes $Y_x\to U \subseteq X$.

My question is this: is there a name for this construction? Are there familiar methods or theorems where it arises?

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Topologically the scheme $\rm{Spec}(\mathcal{O}_{X,x})$ is exactly the intersection of all neighbourhoods of $x$ and algebraically it contains every infinitesimal neighbourhood of $X$.
Although technically it is not the germ of$X$ at $x$, it seems to me that it contains so much information about that germ that it could be considered a materialization of that germ .
Also: technically it is not a subscheme of $X$ (since it is not locally closed) but I can imagine a world where a broader notion of subscheme would allow the monomorphism $j: S=\rm{Spec}(\mathcal{O}_{X,x})\hookrightarrow X$ to be called a subscheme, an almost open subscheme if you will.
One argument for that broader point of view is that the induced canonical morphism $j^{-1}\mathcal O_X=\mathcal O_X\mid S \stackrel {\cong}{\to} \mathcal O_S$ of sheaves over $S$ is an isomorphism, just as if $S\subset X$ were open.

In particular, given a morphism of rings $A\to B$ , the corresponding morphism of affine schemes $\phi:\rm{Spec} (B)\to \rm{Spec} (A)$ and a prime ideal $\mathfrak p\subset A$, the morphism $\rm{Spec} (B_{\mathfrak p}) \to \rm{Spec}(A_{\mathfrak p} )$ is a pleasant thickening of the genuine fiber $\phi^{-1}(\mathfrak p)=\rm{Spec}(B\otimes_A \kappa(\mathfrak p))$ of $\mathfrak p$.
Considerations of such almost germs $\rm{Spec}(A_{\mathfrak p} )$ of $\rm{Spec}(A)$ at $\mathfrak p$ may be of help when following reasonings in commutative algebra.

My first contact with one of these almost germs (or almost open subschemes) was in Mumford's Red Book, Chapter Two, §1, Example F, where he describes an example as "... a startling way to make a scheme out of the non closed points in the plane" and draws one of his celebrated pictures to illustrate the notion.

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In EGA1, 2.4 this is called the local scheme of $X$ at $x$.

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Ah, so the official terminology is reminiscent of germ of $X$ at $x$. – Georges Elencwajg Jul 25 '13 at 20:21

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