The Stacks project

29.58 Miscellany

Results which do not fit elsewhere.

Lemma 29.58.1. Let $f : Y \to X$ be a morphism of schemes. Let $x \in X$ be a point. Assume that $Y$ is reduced and $f(Y)$ is set-theoretically contained in $\{ x\} $. Then $f$ factors through the canonical morphism $x = \mathop{\mathrm{Spec}}(\kappa (x)) \to X$.

Proof. Omitted. Hints: working affine locally one reduces to a commutative algebra lemma. Given a ring map $A \to B$ with $B$ reduced such that there exists a unique prime ideal $\mathfrak p \subset A$ in the image of $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$, then $A \to B$ factors through $\kappa (\mathfrak p)$. This is a nice exercise. $\square$

Lemma 29.58.2. Let $f : Y \to X$ be a morphism of schemes. Let $E \subset X$. Assume $X$ is locally Noetherian, there are no nontrivial specializations among the elements of $E$, $Y$ is reduced, and $f(Y) \subset E$. Then $f$ factors through $\coprod _{x \in E} x \to X$.

Proof. When $E$ is a singleton this follows from Lemma 29.58.1. If $E$ is finite, then $E$ (with the induced topology of $X$) is a finite discrete space by our assumption on specializations. Hence this case reduces to the singleton case. In general, there is a reduction to the case where $X$ and $Y$ are affine schemes. Say $f : Y \to X$ corresponds to the ring map $\varphi : A \to B$. Denote $A' \subset B$ the image of $\varphi $. Let $E' \subset \mathop{\mathrm{Spec}}(A') \subset \mathop{\mathrm{Spec}}(A)$ be the set of minimal primes of $A'$. By Algebra, Lemma 10.30.5 the set $E'$ is contained in the image of $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A') \subset \mathop{\mathrm{Spec}}(A)$. We conclude that $E' \subset E$. Since $A'$ is Noetherian we have $E'$ is finite by Algebra, Lemma 10.31.6. Since any other point in the image of $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ is a specialization of an element of $E'$ and in $E$, we conclude that the image is contained in $E'$ (by our assumption on specializations between points of $E$). Thus we reduce to the case where $E$ is finite which we dealt with above. $\square$


Comments (0)


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Your email address will not be published.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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0H1L. Beware of the difference between the letter 'O' and the digit '0'.