Stacks project -- Comments

#9763 on tag 0GVI by Branislav Sobot

Branislav Sobot — Wed, 11 Sep 2024 06:34:10 GMT

It seems to me that you also need to choose $t\in K$ so that it is $\kappa(\mathfrak{m})$ -linearly idependent from $s+m$ , and then also choose $\varphi$ to map $s+m$ to $\mathfrak{m}$ . Otherwise, I don't see why the obtained locus doesn't contain $\mathfrak{m}$ .

#9762 on tag 00N4 by Branislav Sobot

Branislav Sobot — Wed, 11 Sep 2024 06:29:39 GMT

Sorry, but I don't see why the case $d=0$ can't occur. Simply take $M=A/\mathfrak{q}$ for some ideal of definition $\mathfrak{q}$ . I think that this lemma is simply wrong in this case, while the case $d=0$ can't occur in the next lemma.

#9761 on tag 0FBI by Sasha

Sasha — Wed, 11 Sep 2024 01:52:39 GMT

"gysin" is a name, so it should be capitalized everywhere.

#9760 on tag 08ZW by Patrick Rabau

Patrick Rabau — Wed, 11 Sep 2024 12:33:02 GMT

The final paragraph of the proof of Lemma 08ZY refers to Lemma 08ZN with the phrase "the intersection of all open and closed subsets of $X$ containing $x$ ". That may be ambiguously read to mean the intersection of the collection of all open sets and all closed sets containing $x$ , which is not the desired meaning. Using the usual term "clopen set" would make things shorter and clearer.

#9758 on tag 01RN by Laurent Moret-Bailly

Laurent Moret-Bailly — Tue, 10 Sep 2024 02:06:34 GMT

In addition to #9757, the expression birational schemes should be defined.

#9757 on tag 01RN by Fanjun Meng

Fanjun Meng — Mon, 09 Sep 2024 04:37:35 GMT

In the paragraph after Definition 29.50.1, it should be "the existence of a birational morphism".

#9756 on tag 04VX by Félix

Félix — Mon, 09 Sep 2024 03:34:26 GMT

What is K in this context?

#9755 on tag 00J4 by Stacks project

Stacks project — Mon, 09 Sep 2024 01:45:04 GMT

Maybe we should add the lemma that any product of local rings is the product of the localizations at its maximal ideals.

#9754 on tag 04EA by Litonlaser

Litonlaser — Sat, 07 Sep 2024 01:50:50 GMT

Excellent, I saw other comments about the formula, which is excellent! As a beauty machine manufacturer, Litonlaser is very appreciative.

#9753 on tag 04EA by Fiasco

Fiasco — Fri, 06 Sep 2024 12:49:04 GMT

Some possible typos in the proof of lemma10.149.1:

line -4: $\beta'(g)=\beta(g)+\sum_{i}\delta_{i}\cdot \beta(\frac{\partial g}{\partial x_{i}})$

line -3: $d: J'/J^{2} \rightarrow \Omega_{P/R}\otimes_{P} S, g\mapsto \sum_{i} \pi(\frac{\partial g}{\partial x_{i}})\cdot dx_{i}$ , where $\pi: P\rightarrow S$ is the natural quotient map.