Lemma 26.13.1. Let $X$ be a scheme. Let $R$ be a local ring. The construction above gives a bijective correspondence between morphisms $\mathop{\mathrm{Spec}}(R) \to X$ and pairs $(x, \varphi )$ consisting of a point $x \in X$ and a local homomorphism of local rings $\varphi : \mathcal{O}_{X, x} \to R$.

**Proof.**
Let $A$ be a ring. For any ring homomorphism $\psi : A \to R$ there exists a unique prime ideal $\mathfrak p \subset A$ and a factorization $A \to A_{\mathfrak p} \to R$ where the last map is a local homomorphism of local rings. Namely, $\mathfrak p = \psi ^{-1}(\mathfrak m)$. Via Lemma 26.6.4 this proves that the lemma holds if $X$ is an affine scheme.

Let $X$ be a general scheme. Any $x \in X$ is contained in an open affine $U \subset X$. By the affine case we conclude that every pair $(x, \varphi )$ occurs as the end product of the construction above the lemma.

To finish the proof it suffices to show that any morphism $f : \mathop{\mathrm{Spec}}(R) \to X$ has image contained in any affine open containing the image $x$ of the closed point of $\mathop{\mathrm{Spec}}(R)$. In fact, let $x \in V \subset X$ be any open neighbourhood containing $x$. Then $f^{-1}(V) \subset \mathop{\mathrm{Spec}}(R)$ is an open containing the unique closed point and hence equal to $\mathop{\mathrm{Spec}}(R)$. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (5)

Comment #5879 by Zhenhua Wu on

Comment #5880 by Zhenhua Wu on

Comment #5882 by Johan on

Comment #5888 by Zhenhua Wu on

Comment #6089 by Johan on

There are also: