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 #3645 on tag 0DYE by Brian Conrad https://stacks.math.columbia.edu/tag/0DYE#comment-3645 A new comment by Brian Conrad on tag 0DYE. I should have also mentioned that currently this Lemma is only mentioned in two proofs, neither of which is impacted by imposing the requirement $V(J)=V(IB)$ (though the second proof which claims to invoke this Lemma doesn't really clearly indicate where it is used -- that proof says this Lemma is used, but never explicitly invokes it with a cross-reference at any step).

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Brian Conrad Sat, 20 Oct 2018 11:35:49 GMT
#3644 on tag 0DYE by Brian Conrad https://stacks.math.columbia.edu/tag/0DYE#comment-3644 A new comment by Brian Conrad on tag 0DYE. At the start of the proof, the wrong lemma is cited: should cite Lemma 09XK. In the second sentence, the invocation of the universal property doesn't make any sense unless $J$ lands in $I^h(A^h \otimes_A B) = I(A^h \otimes_A B)$, which would hold if $J=IB$. But there is the flexibility to replace $J$ with any bigger ideal of $B$ without affecting the henselian hypothesis on $(B, J)$, so you should really assume $V(J)=V(IB)$: this would ensure $(B,IB)$ is also henselian by Lemma 09XJ and would ensure that when $A$ is local with maximal ideal $I$ we can take $J$ to be the Jacobson radical of $B$ (rather than demanding $J=IB$, which would be an unpleasant requirement for such cases).

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Brian Conrad Sat, 20 Oct 2018 11:30:36 GMT
#3643 on tag 0A03 by Brian Conrad https://stacks.math.columbia.edu/tag/0A03#comment-3643 A new comment by Brian Conrad on tag 0A03. In the second proof, which at the end invokes a lemma that has locality hypotheses, you need to first indicate why $A^h$ is local. One way based on general principles beyond the local setting is to recall that for any pair $(B,J)$ the ideal $J^h$ of $B^h$ is contained in the Jacobson radical (by the henselian property for the pair $(B^h, J^h)$) and by construction $J^h = JB^h$ yet always $B^h/JA^h = B/J$ (so when $B$ is local with maximal ideal $J$, the ideal $J^h$ is maximal and so is the unique maximal ideal of $J^h$, moreover with the same residue field as $B$).

Note that in this argument, we are using two facts currently recorded in Lemma 0AGU, which currently appears just after the present Lemma in section 0EM7 (no circularity, since Lemma 0AGU doesn't use the present lemma, so their order should be swapped).

Also, this shows that before swapping the order of the two lemmas, the "first proof" isn't entirely satisfactory since it isn't giving that $\mathfrak{m}^h$ is also the maximal ideal (which would however be "known" if Lemma 0AGU were put before the present lemma, as recommended above).

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Brian Conrad Sat, 20 Oct 2018 11:15:45 GMT
#3642 on tag 0CAI by Brian Conrad https://stacks.math.columbia.edu/tag/0CAI#comment-3642 A new comment by Brian Conrad on tag 0CAI. Near the start of the proof, replace "source is a coprod $\mathcal{F}_0$" with "source $\mathcal{F}_0$ is a coproduct", 2nd line after the displayed expression replace "coprod" with "coproduct", and on 3rd to last line replace $U_b$ with $V_b$.

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Brian Conrad Sat, 20 Oct 2018 10:49:01 GMT
#3641 on tag 009F by Brian Conrad https://stacks.math.columbia.edu/tag/009F#comment-3641 A new comment by Brian Conrad on tag 009F. In the proof of (4), it should be mentioned that the hypothesis on $U$ forces it to be quasi-compact, so injectivity holds by (2).

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Brian Conrad Sat, 20 Oct 2018 10:48:40 GMT
#3640 on tag 0CAM by Brian Conrad https://stacks.math.columbia.edu/tag/0CAM#comment-3640 A new comment by Brian Conrad on tag 0CAM. The reference to Lemma 0902 in the proof is a typo; you meant to refer to Lemma 0903 (and more specifically to part (1) of that Lemma).

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Brian Conrad Sat, 20 Oct 2018 10:48:00 GMT
#3639 on tag 09Z0 by Brian Conrad https://stacks.math.columbia.edu/tag/09Z0#comment-3639 A new comment by Brian Conrad on tag 09Z0. In the parenthetical early in the 3rd paragraph of the proof, replace "quasi-compact etale" with "quasi-compact separated etale" (it is implicit in the Lemma invoked there, and also necessary as well), so earlier when $U$ is made affine it should have been noted (with cross-reference) that $U \to X$ is thereby separated. In the 2nd to last line of this same paragraph, don't call the neighborhood of $z$ in $Z$ by the name $V$, since the notation $V$ already has an entirely different meaning in the overall proof.

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Brian Conrad Sat, 20 Oct 2018 10:47:20 GMT
#3638 on tag 09ZF by Brian Conrad https://stacks.math.columbia.edu/tag/09ZF#comment-3638 A new comment by Brian Conrad on tag 09ZF. In the final paragraph of the proof, replace "every closed subscheme $T \subset X$" with "every non-empty closed subscheme $T \subset X$".

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Brian Conrad Sat, 20 Oct 2018 01:31:41 GMT
#3637 on tag 0EM6 by Brian Conrad https://stacks.math.columbia.edu/tag/0EM6#comment-3637 A new comment by Brian Conrad on tag 0EM6. In the proof, replace "radical" with "Jacobson radical".

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Brian Conrad Sat, 20 Oct 2018 12:02:09 GMT
#3636 on tag 0DYD by Brian Conrad https://stacks.math.columbia.edu/tag/0DYD#comment-3636 A new comment by Brian Conrad on tag 0DYD. The proof that (1) implies (2) seems to be incomplete as written since to do the second factorization lifting you'd need to know that the lifted monic factorization in $(A/I)[T]$ satisfies the "generate the unit ideal" condition, which is not addressed. But anyway it seems easier to just argue by the same method as in the proof of the converse: using notation as set up there, in the composition $B \to B/IB \to B/JB$ the first and second maps induce bijections on sets of idempotents bu the assumption (1), so the composite is also such a bijection and hence (2) holds (by one of the equivalent characterizations of "henselian pair").

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Brian Conrad Fri, 19 Oct 2018 11:59:27 GMT