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 #4658 on tag 05DW by Johan https://stacks.math.columbia.edu/tag/05DW#comment-4658 A new comment by Johan on tag 05DW. Although you are right that it is equivalent, it is not completely trivial to see the equivalence. The corresponding result for "geometrically reduced" is Lemma 10.43.3 and the corresponding result for "geometrically irreducible" is Lemma 10.46.3 except that unfortunately it is missing the part where we say it is enough to look for the spectrum of to be irreducible with equal to either the algebraic closure or the separable algebraic closure. I will add this the next time I go through the comments.

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Johan Mon, 11 Nov 2019 03:41:19 GMT
#4657 on tag 0AV3 by Remy https://stacks.math.columbia.edu/tag/0AV3#comment-4657 A new comment by Remy on tag 0AV3. After the second sentence, couldn't you just conclude directly from Tag 15.23.5? (In fact, you don't need to take the image, because exactness on the right would not be needed.)

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Remy Mon, 11 Nov 2019 12:09:20 GMT
#4656 on tag 05DW by Hao https://stacks.math.columbia.edu/tag/05DW#comment-4656 A new comment by Hao on tag 05DW. A possibly helpful equivalent definition: If is a algebra, then is geometrically integral iff is a domain.

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Hao Mon, 11 Nov 2019 09:40:30 GMT
#4655 on tag 0BQ8 by Johan https://stacks.math.columbia.edu/tag/0BQ8#comment-4655 A new comment by Johan on tag 0BQ8. The answer to your last question is no. Everything else you say is fine (except I cannot comment on your state of mind). Suggest reading the text by Lenstra (especially section 3) to clarify things, see reference in Section 55.1.

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Johan Sun, 10 Nov 2019 03:47:44 GMT
#4654 on tag 08FB by Dylan Spence https://stacks.math.columbia.edu/tag/08FB#comment-4654 A new comment by Dylan Spence on tag 08FB. In the statement of the lemma, should actually be ?

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Dylan Spence Sun, 10 Nov 2019 01:03:46 GMT
#4653 on tag 0BQ8 by Rex https://stacks.math.columbia.edu/tag/0BQ8#comment-4653 A new comment by Rex on tag 0BQ8. I'm confused. If you leave out basepoints, then objects in the source category will have nontrivial automorphisms. A natural automorphism of the fibre functor will be required to intertwine with these automorphisms. Does this not force the etale fundamental group as defined above to consist only of the central elements in what should be the fundamental group?

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Rex Sun, 10 Nov 2019 12:56:26 GMT
#4652 on tag 0BQ8 by Johan https://stacks.math.columbia.edu/tag/0BQ8#comment-4652 A new comment by Johan on tag 0BQ8. No.

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Johan Sun, 10 Nov 2019 10:02:38 GMT
#4651 on tag 0BQ8 by Rex https://stacks.math.columbia.edu/tag/0BQ8#comment-4651 A new comment by Rex on tag 0BQ8. Shouldn't the source category of the fibre functor to be, not finite etale coverings, but finite etale coverings with basepoint?

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Rex Sun, 10 Nov 2019 03:23:39 GMT
#4650 on tag 0FLT by Johan https://stacks.math.columbia.edu/tag/0FLT#comment-4650 A new comment by Johan on tag 0FLT. First, of all, we should add a variant of this lemma where and are flat over in which case this lemma is a lot easier to prove. Secondly, in the formula there is a typo and should be . Thirdly, the statement that this lives in degrees needs to be justified by proving a bound on the tor dimension of which follows from the results in the next paragraph.

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Johan Fri, 08 Nov 2019 08:15:54 GMT
#4649 on tag 0F32 by Noah Olander https://stacks.math.columbia.edu/tag/0F32#comment-4649 A new comment by Noah Olander on tag 0F32. A couple typos:

You should say that is an elementary ├ętale neighborhood since you assume Y is unibranch, not geometrically unibranch.

Right after that, you should say is the unique point of mapping to not vice versa.

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Noah Olander Thu, 07 Nov 2019 07:37:18 GMT