The Stacks project

Lemma 29.6.5. Let $f : X \to Y$ be a quasi-compact morphism. Let $Z$ be the scheme theoretic image of $f$. Let $z \in Z$1. There exists a valuation ring $A$ with fraction field $K$ and a commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[rr] \ar[d] & & X \ar[d] \ar[ld] \\ \mathop{\mathrm{Spec}}(A) \ar[r] & Z \ar[r] & Y } \]

such that the closed point of $\mathop{\mathrm{Spec}}(A)$ maps to $z$. In particular any point of $Z$ is the specialization of a point of $f(X)$.

Proof. Let $z \in \mathop{\mathrm{Spec}}(R) = V \subset Y$ be an affine open neighbourhood of $z$. By Lemma 29.6.3 the intersection $Z \cap V$ is the scheme theoretic image of $f^{-1}(V) \to V$. Hence we may replace $Y$ by $V$ and assume $Y = \mathop{\mathrm{Spec}}(R)$ is affine. In this case $X$ is quasi-compact as $f$ is quasi-compact. Say $X = U_1 \cup \ldots \cup U_ n$ is a finite affine open covering. Write $U_ i = \mathop{\mathrm{Spec}}(A_ i)$. Let $I = \mathop{\mathrm{Ker}}(R \to A_1 \times \ldots \times A_ n)$. By Lemma 29.6.3 again we see that $Z$ corresponds to the closed subscheme $\mathop{\mathrm{Spec}}(R/I)$ of $Y$. If $\mathfrak p \subset R$ is the prime corresponding to $z$, then we see that $I_{\mathfrak p} \subset R_{\mathfrak p}$ is not an equality. Hence (as localization is exact, see Algebra, Proposition 10.9.12) we see that $R_{\mathfrak p} \to (A_1)_{\mathfrak p} \times \ldots \times (A_ n)_{\mathfrak p}$ is not zero. Hence one of the rings $(A_ i)_{\mathfrak p}$ is not zero. Hence there exists an $i$ and a prime $\mathfrak q_ i \subset A_ i$ lying over a prime $\mathfrak p_ i \subset \mathfrak p$. By Algebra, Lemma 10.50.2 we can choose a valuation ring $A \subset K = \kappa (\mathfrak q_ i)$ dominating the local ring $R_{\mathfrak p}/\mathfrak p_ iR_{\mathfrak p} \subset \kappa (\mathfrak q_ i)$. This gives the desired diagram. Some details omitted. $\square$

[1] By Lemma 29.6.3 set-theoretically $Z$ agrees with the closure of $f(X)$ in $Y$.

Comments (5)

Comment #2762 by BCnrd on

The notation to denote "fraction field" (twice) near the end of the proof seems kind of awful (not only because in the argument there is a map called ). Is this really the standard notation in this document? Why not instead?

Comment #2765 by Dario WeiƟmann on

Why not use the notation introduced in Example 10.9.8 (3)? Or - as in this case we are talking about the residue field at a prime - use the notation introduced in 25.2.1 (2)?

Comment #2766 by BCnrd on

Dario's suggestion of for the fraction field of seems fine since such notation has been standard from EGA, and I think the meaning of is also immediately recognized by anyone, but I am not a fan of to denote the fraction field of a domain since it is not sufficiently universally known and so potentially a bit obscure (though admittedly anyone who has actually understood the proof up to there can infer what it must mean). In a work this massive, which essentially nobody reads linearly, it is best not to use slightly non-standard notation unless there is an easily-identified and systematically maintained list of all such notation, but alas there seems to be no such list; the Notation section in the "Preliminaries" Part is a bit brief!)

Comment #2767 by sdf on

There is a fairly standard notation of or or for field of fractions at least in my part of the world. But having a full-stop as part of the notation is aesthetically less than ideal, this has been pointed out before I think.

Comment #2875 by on

OK already! I have now changed this by not having any notation for the fraction field of a domain. It turns out it isn't that frequently used and often it is used (as in the current situation) when we are taking the residue field at a prime. See here to see why this kind of change is painful.

There are also:

  • 9 comment(s) on Section 29.6: Scheme theoretic image

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 02JQ. Beware of the difference between the letter 'O' and the digit '0'.