## Tag `01J9`

Chapter 25: Schemes > Section 25.13: Points of schemes

Lemma 25.13.3. Let $X$ be a scheme. Points of $X$ correspond bijectively to equivalence classes of morphisms from spectra of fields into $X$. Moreover, each equivalence class contains a (unique up to unique isomorphism) smallest element $\mathop{\mathrm{Spec}}(\kappa(x)) \to X$.

Proof.Follows from the discussion above. $\square$

The code snippet corresponding to this tag is a part of the file `schemes.tex` and is located in lines 2309–2316 (see updates for more information).

```
\begin{lemma}
\label{lemma-characterize-points}
Let $X$ be a scheme. Points of $X$ correspond bijectively
to equivalence classes of morphisms from spectra of
fields into $X$. Moreover, each equivalence class contains
a (unique up to unique isomorphism) smallest element
$\Spec(\kappa(x)) \to X$.
\end{lemma}
\begin{proof}
Follows from the discussion above.
\end{proof}
```

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