Stacks project -- Comments https://stacks.math.columbia.edu/recent-comments.xml Stacks project, see https://stacks.math.columbia.edu en stacks.project@gmail.com (The Stacks project) pieterbelmans@gmail.com (Pieter Belmans) https://stacks.math.columbia.edu/static/stacks.png Stacks project -- Comments https://stacks.math.columbia.edu/recent-comments.rss #8096 on tag 0C30 by Rubén Muñoz--Bertrand https://stacks.math.columbia.edu/tag/0C30#comment-8096 A new comment by Rubén Muñoz--Bertrand on tag 0C30. Okay, I feel this proof can be slightly upgraded. First, lemma 10.44.1 only works in characteristic $p$. Of course, the characteristic zero case is obvious, but why not just use 10.43.6 instead?

Now for the tricky part. The term universal homeomorphism is not yet defined, it only appears in definition 29.45.1 and in the context of schemes, despite its use in the title of section 10.46 (maybe that's a bit confusing, but this is not the main issue here).

It seems that throughout the Stacks Project, a great care is taken to not use the term universal homeomorphism for morphisms of rings. Instead, it is written in terms of morphisms on spectra (for instance in the proof of lemma 58.14.2), or with a sentence such as "induces a universal homeomorphism on spectra" (see for instance lemma 29.46.11).

So if we want to keep this consistency, I propose to first say that by lemma 10.45.4 (or by definition) $k^{perf}/k$ is purely inseparable, so we can apply lemma 10.46.10 to find that $k^{perf}\otimes_{k}K$ has a unique prime ideal.

If this is not nitpicking for you, then I would suggest to also edit the very few remaining cases of universal homeomorphisms of rings in the Stacks Project: tags 29.47.6, 37.24.7, 109.39 (twice) and maybe 29.46.8.

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Rubén Muñoz--Bertrand Sat, 04 Feb 2023 01:56:45 GMT
#8093 on tag 00ZN by Andrea Panontin https://stacks.math.columbia.edu/tag/00ZN#comment-8093 A new comment by Andrea Panontin on tag 00ZN. In Lemma 00ZO I think that $\varphi(id_V)$ should be $\varphi(\alpha)$.

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Andrea Panontin Wed, 01 Feb 2023 12:01:37 GMT
#8092 on tag 00ZN by Andrea Panontin https://stacks.math.columbia.edu/tag/00ZN#comment-8092 A new comment by Andrea Panontin on tag 00ZN. In the proof of theorem 00ZP you write "the sheafification functor is the right adjoint of the inclusion functor", it should be the left adjoint.

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Andrea Panontin Tue, 31 Jan 2023 11:25:39 GMT
#8091 on tag 08N8 by Stacks Project https://stacks.math.columbia.edu/tag/08N8#comment-8091 A new comment by Stacks Project on tag 08N8. Thanks and fixed here.

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Stacks Project Thu, 26 Jan 2023 05:15:58 GMT
#8090 on tag 02LV by Stacks Project https://stacks.math.columbia.edu/tag/02LV#comment-8090 A new comment by Stacks Project on tag 02LV. Dear Miles, what about $(x + y)/(x + 2y)$? This corresponds to the function which is $1$ when $y = 0$ and $2$ when $x = 0$, so it corresponds to $(1, 2)$ in $k(x) \times k(y)$. OK?

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Stacks Project Thu, 26 Jan 2023 05:10:27 GMT
#8089 on tag 05CW by Steven Sam https://stacks.math.columbia.edu/tag/05CW#comment-8089 A new comment by Steven Sam on tag 05CW. 4th paragraph of proof says: "Let \overline{x_i} \in coker(\phi)$." Should say something like "Let$\overline{x_1},\dots,\overline{x_n} \in \coker(\phi)$be a finite set of generators." ]]> Steven Sam Tue, 24 Jan 2023 04:26:36 GMT #8088 on tag 0C0D by Laurent Moret-Bailly https://stacks.math.columbia.edu/tag/0C0D#comment-8088 A new comment by Laurent Moret-Bailly on tag 0C0D. Not sure this is useful, but the lemma works for any valuation ring $R$: instead of Krull's intersection theorem, use the fact that $a$ has a content ideal (Lemma 0ASX) which is principal by Comment 8087. (I do realize that Lemma 0ASX comes a bit later). ]]> Laurent Moret-Bailly Tue, 24 Jan 2023 12:12:14 GMT #8087 on tag 0ASX by Laurent Moret-Bailly https://stacks.math.columbia.edu/tag/0ASX#comment-8087 A new comment by Laurent Moret-Bailly on tag 0ASX. It follows from Comment 8086 that the content ideal is principal. ]]> Laurent Moret-Bailly Tue, 24 Jan 2023 11:46:36 GMT #8086 on tag 0AS6 by Laurent Moret-Bailly https://stacks.math.columbia.edu/tag/0AS6#comment-8086 A new comment by Laurent Moret-Bailly on tag 0AS6. It may be worth pointing out that $F'$ is of finite type. Namely, there is $F''\subset F'$ of finite type such that $x\in F''\otimes_R M$, and $F''=F'$ since$F' is minimal.

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Laurent Moret-Bailly Tue, 24 Jan 2023 11:41:03 GMT
#8085 on tag 01IZ by Yassin Mousa https://stacks.math.columbia.edu/tag/01IZ#comment-8085 A new comment by Yassin Mousa on tag 01IZ. Maybe one should mention oi Lemma 26.12.7, that the factorisation is unique. This is given by Lemma 26.4.6.

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Yassin Mousa Tue, 24 Jan 2023 10:13:55 GMT