The Stacks project

66.19 Monomorphisms

Here is another case where monomorphisms are representable. Please see More on Morphisms of Spaces, Section 74.4 for more information.

Lemma 66.19.1. Let $S$ be a scheme. Let $Y$ be a disjoint union of spectra of zero dimensional local rings over $S$. Let $f : X \to Y$ be a monomorphism of algebraic spaces over $S$. Then $f$ is representable, i.e., $X$ is a scheme.

Proof. This immediately reduces to the case $Y = \mathop{\mathrm{Spec}}(A)$ where $A$ is a zero dimensional local ring, i.e., $\mathop{\mathrm{Spec}}(A) = \{ \mathfrak m_ A\} $ is a singleton. If $X = \emptyset $, then there is nothing to prove. If not, choose a nonempty affine scheme $U = \mathop{\mathrm{Spec}}(B)$ and an étale morphism $U \to X$. As $|X|$ is a singleton (as a subset of $|Y|$, see Morphisms of Spaces, Lemma 65.10.9) we see that $U \to X$ is surjective. Note that $U \times _ X U = U \times _ Y U = \mathop{\mathrm{Spec}}(B \otimes _ A B)$. Thus we see that the ring maps $B \to B \otimes _ A B$ are étale. Since

\[ (B \otimes _ A B)/\mathfrak m_ A(B \otimes _ A B) = (B/\mathfrak m_ AB) \otimes _{A/\mathfrak m_ A} (B/\mathfrak m_ AB) \]

we see that $B/\mathfrak m_ AB \to (B \otimes _ A B)/\mathfrak m_ A(B \otimes _ A B)$ is flat and in fact free of rank equal to the dimension of $B/\mathfrak m_ AB$ as a $A/\mathfrak m_ A$-vector space. Since $B \to B \otimes _ A B$ is étale, this can only happen if this dimension is finite (see for example Morphisms, Lemmas 29.55.9 and 29.55.10). Every prime of $B$ lies over $\mathfrak m_ A$ (the unique prime of $A$). Hence $\mathop{\mathrm{Spec}}(B) = \mathop{\mathrm{Spec}}(B/\mathfrak m_ A)$ as a topological space, and this space is a finite discrete set as $B/\mathfrak m_ A B$ is an Artinian ring, see Algebra, Lemmas 10.52.2 and 10.52.6. Hence all prime ideals of $B$ are maximal and $B = B_1 \times \ldots \times B_ n$ is a product of finitely many local rings of dimension zero, see Algebra, Lemma 10.52.5. Thus $B \to B \otimes _ A B$ is finite étale as all the local rings $B_ i$ are henselian by Algebra, Lemma 10.152.10. Thus $X$ is an affine scheme by Groupoids, Proposition 39.23.9. $\square$

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Your email address will not be published.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 06RY. Beware of the difference between the letter 'O' and the digit '0'.