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#4658 on tag 05DW by Johan

Johan — Mon, 11 Nov 2019 03:41:19 GMT

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 $A \otimes_k \overline{k}$ to be irreducible with $\overline{k}$ equal to either the algebraic closure or the separable algebraic closure. I will add this the next time I go through the comments.

#4657 on tag 0AV3 by Remy

Remy — Mon, 11 Nov 2019 12:09:20 GMT

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.)

#4656 on tag 05DW by Hao

Hao — Mon, 11 Nov 2019 09:40:30 GMT

A possibly helpful equivalent definition: If $A$ is a $k$ algebra, then $A$ is geometrically integral iff $A\otimes_k \overline{k}$ is a domain.

#4655 on tag 0BQ8 by Johan

Johan — Sun, 10 Nov 2019 03:47:44 GMT

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.

#4654 on tag 08FB by Dylan Spence

Dylan Spence — Sun, 10 Nov 2019 01:03:46 GMT

In the statement of the lemma, should $\mathcal{O}_S$ actually be $\mathcal{O}_Y$ ?

#4653 on tag 0BQ8 by Rex

Rex — Sun, 10 Nov 2019 12:56:26 GMT

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?

#4652 on tag 0BQ8 by Johan

Johan — Sun, 10 Nov 2019 10:02:38 GMT

A new comment by Johan on tag 0BQ8.

#4651 on tag 0BQ8 by Rex

Rex — Sun, 10 Nov 2019 03:23:39 GMT

Shouldn't the source category of the fibre functor to be, not finite etale coverings, but finite etale coverings with basepoint?

#4650 on tag 0FLT by Johan

Johan — Fri, 08 Nov 2019 08:15:54 GMT

First, of all, we should add a variant of this lemma where $X$ and $Y$ are flat over $S$ in which case this lemma is a lot easier to prove. Secondly, in the formula $R\Gamma (X, \sigma _{\geq n}\mathcal{F}^\bullet ) \otimes _ A^\mathbf {L} R\Gamma (Y, \mathcal{G}^\bullet )$ there is a typo and $n$ should be $N$ . Thirdly, the statement that this lives in degrees $\geq N + b$ needs to be justified by proving a bound on the tor dimension of $R\Gamma (Y, \mathcal{G}^\bullet )$ which follows from the results in the next paragraph.

#4649 on tag 0F32 by Noah Olander

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

A couple typos:

You should say that $(V,v) \to (Y,y)$ is an elementary étale neighborhood since you assume Y is unibranch, not geometrically unibranch.

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