The Stacks project

Lemma 37.52.2. Let $A \to B$ be a local homomorphism of local rings, and $g \in \mathfrak m_ B$. Assume

  1. $A$ and $B$ are domains and $A \subset B$,

  2. $B$ is essentially of finite type over $A$,

  3. $g$ is not contained in any minimal prime over $\mathfrak m_ AB$, and

  4. $\dim (B/\mathfrak m_ AB) + \text{trdeg}_{\kappa (\mathfrak m_ A)}(\kappa (\mathfrak m_ B)) = \text{trdeg}_ A(B)$.

Then $A \subset B/gB$, i.e., the generic point of $\mathop{\mathrm{Spec}}(A)$ is in the image of the morphism $\mathop{\mathrm{Spec}}(B/gB) \to \mathop{\mathrm{Spec}}(A)$.

Proof. Note that the two assertions are equivalent by Algebra, Lemma 10.30.6. To start the proof let $C$ be an $A$-algebra of finite type and $\mathfrak q$ a prime of $C$ such that $B = C_{\mathfrak q}$. Of course we may assume that $C$ is a domain and that $g \in C$. After replacing $C$ by a localization we see that $\dim (C/\mathfrak m_ AC) = \dim (B/\mathfrak m_ AB) + \text{trdeg}_{\kappa (\mathfrak m_ A)}(\kappa (\mathfrak m_ B))$, see Morphisms, Lemma 29.28.1. Setting $K$ equal to the fraction field of $A$ we see by the same reference that $\dim (C \otimes _ A K) = \text{trdeg}_ A(B)$. Hence assumption (4) means that the generic and closed fibres of the morphism $\mathop{\mathrm{Spec}}(C) \to \mathop{\mathrm{Spec}}(A)$ have the same dimension.

Suppose that the lemma is false. Then $(B/gB) \otimes _ A K = 0$. This means that $g \otimes 1$ is invertible in $B \otimes _ A K = C_{\mathfrak q} \otimes _ A K$. As $C_{\mathfrak q}$ is a limit of principal localizations we conclude that $g \otimes 1$ is invertible in $C_ h \otimes _ A K$ for some $h \in C$, $h \not\in \mathfrak q$. Thus after replacing $C$ by $C_ h$ we may assume that $(C/gC) \otimes _ A K = 0$. We do one more replacement of $C$ to make sure that the minimal primes of $C/\mathfrak m_ AC$ correspond one-to-one with the minimal primes of $B/\mathfrak m_ AB$. At this point we apply Lemma 37.52.1 to $X = \mathop{\mathrm{Spec}}(C) \to \mathop{\mathrm{Spec}}(A) = S$ and the locally closed subscheme $Z = \mathop{\mathrm{Spec}}(C/gC)$. Since $Z_ K = \emptyset $ we see that $Z \otimes \kappa (\mathfrak m_ A)$ has to contain an irreducible component of $X \otimes \kappa (\mathfrak m_ A) = \mathop{\mathrm{Spec}}(C/\mathfrak m_ AC)$. But this contradicts the assumption that $g$ is not contained in any prime minimal over $\mathfrak m_ AB$. The lemma follows. $\square$


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 053S. Beware of the difference between the letter 'O' and the digit '0'.