The Stacks project

In the second proof, shouldn't it be $\mu_i\otimes 1=\psi\circ\lambda_i$ instead of $\lambda_i=\psi\circ (\mu_i\otimes 1)$?

Yes, it is in the SP, see for instance Lemma 01WM.

@#7377: A universal homeomorphism is integral (EGA IV, 18.12.10). So perhaps this should be (resp. already is) in the Stacks Project.

It may be useful to include the statement that the same holds for any $A \rightarrow B$ that is a universal homeomorphism on spectra (sorry if this is already in the Stacks Project but I missed it!). I think this follows from the characterization of Henselian pairs in terms of lifting idempotents.

Hello!

In the proof of (2), showing the surjection of the canonical map to completion, the sequence of equations must have an "f" in the coefficient of x_1. That is, " ... = x-x_0+f e_1 = x-x_0 -f x_1 + f^2 e_2 = ..."

Thanks

A rather trivial observation: The flatness hypothesis on $f$ seems to be superfluous except for (1) because of openness of flatness. So I think, one could slightly strengthen the result by supposing only that $f$ is locally of finite presentation and add in (1) the condition that $f$ is flat in $x$ in the description of $W$.

Sorry, I just realized that the flatness is of course also needed for (3). So please forget my comment.

In tag 00S1, "φ:P→P is a morphism of presentations from α to α′" should be "φ:P→P' is a morphism of presentations from α to α′".

I think it would be better to indicate where we use the assumption that $f$ is representable by algebraic spaces. I guess this is used to ensure that $X\times_Y T$ is a quasi-compact algebraic space, hence admits an \' etale cover by an affine scheme $U$; the composite $U\to Y$ then factors through some $Y_{\mu}$, and the same is true for $X\times_Y T$ by the sheaf property.

A rather trivial observation: The flatness hypothesis on $f$ seems to be superfluous except for (1) because of openness of flatness. So I think, one could slightly strengthen the result by supposing only that $f$ is locally of finite presentation and add in (1) the condition that $f$ is flat in $x$ in the description of $W$.

OK. I see. Even though $a\in \mathcal{O}_{X,x}$ can not be lifted to an element in $\Gamma(X,\mathcal{O}_X)$ in general, it can be lifted to an element in $\Gamma(X,\mathcal{K}'_X)$, which makes $ad-cf$ well-defined.

I checked a bit that the notation $\mathrm{trdeg}_R(S)$ probably has never defined for a ring, only for fields. Maybe it is a bit clearer to mention what it means.

I think in lemma 31.24.2, in order to define $ad-cf$ as a section in $\Gamma(X,\mathcal{K}_X)$, one should first lift $a\in A_{\mathfrac{p}}$ to a global section $a\in A$. But $A\to A_{\mathfrac{p}}$ is not surjective, seems that $ad-cf$ is not well-defined.

In definition 37.1.11, there is a slight typo : it says "soure" instead of "source."

Maybe the vertical arrows should all go downward?

Hi Zongzhu Lin, I think 005K might be the reference you are looking for? Alternatively, I think it's not so hard to see the claim from the definition of constructibility.

The page has been archived by the Wayback Machine. At least some parts of the book are still available through it: https://web.archive.org/web/20110707004531/http://www.math.uzh.ch/index.php?pr_vo_det&key1=1287&key2=580&no_cache=1

Typo in the proof of lemma 28.22.11: the first sentence should be 'A_1，A_2 ⊂A'

In (2), it should in fact suffice to assume that $Y$ is locally of finite type over $S$. (This is also consistent with Lemma 02FW, to which the proof refers).

@7361: That is a limitation of how things are being displayed. There is a work-around possible, the question is whether there are enough cases of this causing confusion to put in the effort. I'll put it on the possible features list though, thanks for noticing!

Strangely, in "tags" mode, parts (1) and (2) are also converted to tags in the proof (but not in the statement).