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The Stacks project

Lemma 10.41.5. Let R \to S be a ring map. Let T \subset \mathop{\mathrm{Spec}}(R) be the image of \mathop{\mathrm{Spec}}(S). If T is stable under specialization, then T is closed.

Proof. We give two proofs.

First proof. Let \mathfrak p \subset R be a prime ideal such that the corresponding point of \mathop{\mathrm{Spec}}(R) is in the closure of T. This means that for every f \in R, f \not\in \mathfrak p we have D(f) \cap T \not= \emptyset . Note that D(f) \cap T is the image of \mathop{\mathrm{Spec}}(S_ f) in \mathop{\mathrm{Spec}}(R). Hence we conclude that S_ f \not= 0. In other words, 1 \not= 0 in the ring S_ f. Since S_{\mathfrak p} is the directed colimit of the rings S_ f we conclude that 1 \not= 0 in S_{\mathfrak p}. In other words, S_{\mathfrak p} \not= 0 and considering the image of \mathop{\mathrm{Spec}}(S_{\mathfrak p}) \to \mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R) we see there exists a \mathfrak p' \in T with \mathfrak p' \subset \mathfrak p. As we assumed T closed under specialization we conclude \mathfrak p is a point of T as desired.

Second proof. Let I = \mathop{\mathrm{Ker}}(R \to S). We may replace R by R/I. In this case the ring map R \to S is injective. By Lemma 10.30.5 all the minimal primes of R are contained in the image T. Hence if T is stable under specialization then it contains all primes. \square


Comments (6)

Comment #3001 by David Holmes on

"Since S_p is the directed limit of the rings S_f..." Here the "limit" should be "colimit" I think.

Comment #7873 by Ryo Suzuki on

It can be proved considering constructible topology.

First, is quasi-compact in constructible topology by Lemma 5.23.2. Hence is also quasi-compact in constructible topology. Now is Hausdorff in constructible topology, again by Lemma 5.23.2. So is closed in constructible topology. By Lemma 5.23.6, is closed in usual topology.

Comment #10038 by on

I think that the argument by Ryo Suzuki \ref{https://stacks.math.columbia.edu/tag/00HY#comment-7873} is better since it shows that this is a purely topological statement: the image of a spectral map between spectral spaces is closed if and only if it is stable under specialization.

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  • 7 comment(s) on Section 10.41: Going up and going down

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