Stacks project -- Comments

#8096 on tag 0C30 by Rubén Muñoz--Bertrand

Rubén Muñoz--Bertrand — Sat, 04 Feb 2023 01:56:45 GMT

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.

#8093 on tag 00ZN by Andrea Panontin

Andrea Panontin — Wed, 01 Feb 2023 12:01:37 GMT

In Lemma 00ZO I think that $\varphi(id_V)$ should be $\varphi(\alpha)$ .

#8092 on tag 00ZN by Andrea Panontin

Andrea Panontin — Tue, 31 Jan 2023 11:25:39 GMT

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.

#8091 on tag 08N8 by Stacks Project

Stacks Project — Thu, 26 Jan 2023 05:15:58 GMT

Thanks and fixed here.

#8090 on tag 02LV by Stacks Project

Stacks Project — Thu, 26 Jan 2023 05:10:27 GMT

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?

#8089 on tag 05CW by Steven Sam

Steven Sam — Tue, 24 Jan 2023 04:26:36 GMT

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."

#8088 on tag 0C0D by Laurent Moret-Bailly

Laurent Moret-Bailly — Tue, 24 Jan 2023 12:12:14 GMT

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).

#8087 on tag 0ASX by Laurent Moret-Bailly

Laurent Moret-Bailly — Tue, 24 Jan 2023 11:46:36 GMT

It follows from Comment 8086 that the content ideal is principal.

#8086 on tag 0AS6 by Laurent Moret-Bailly

Laurent Moret-Bailly — Tue, 24 Jan 2023 11:41:03 GMT

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.

#8085 on tag 01IZ by Yassin Mousa

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

Maybe one should mention oi Lemma 26.12.7, that the factorisation is unique. This is given by Lemma 26.4.6.