Recent comments—The Stacks project

Comments 1 to 20 out of 3855 in reverse chronological order.

On Mar 19, 2019 Dan Dore left comment #4066 on Section 55.10 in Crystalline Cohomology

Maybe it's a good idea to remind the reader here of Definition 7.45.1, defining global sections for a site without a terminal object, then spelling out what this means for the cristalline site?

On Mar 19, 2019 Frid left comment #4065 on Section 7.19 in Sites and Sheaves

In the last line of the proof of Lemma 7.19.3., it should say "part (4)" and "Part (5)" instead of "part (2)" and "Part (3)".

On Mar 18, 2019 Kestutis Cesnavicius left comment #4064 on Lemma 16.9.3 in Smoothing Ring Maps

One could insert references to this lemma: Lemma 12.2 in Swan "Néron-Popescu desingularization" or Lemma 2 in Popescu "General Néron desingularization."

Also, I think it would be better to write $A \otimes_R R_{\mathfrak{p}}$ instead of $A_{\mathfrak{p}}$ .

On Mar 16, 2019 Yicheng Zhou left comment #4063 on Section 32.34 in Varieties

Just a small typo: in the paragraph between Definition 08AD and Lemma 08AE, a $d$ is missing in the left side of $H^0(\mathbf{P}^n_k,\mathcal{O}_{\mathbf{P}^n_k})=k[T_0,\ldots,T_n]_d$

On Mar 13, 2019 Name* left comment #4059 on Section 36.26 in More on Morphisms

Who is Y in the proof of lema o55j?

On Mar 12, 2019 Laurent Moret-Bailly left comment #4057 on Section 29.13 in Cohomology of Schemes

At the beginning, it would be better to refer to 05YU than 01ZT (the former being more general).

On Mar 12, 2019 Laurent Moret-Bailly left comment #4056 on Lemma 31.11.1 in Limits of Schemes

Typo in third paragraph of proof: "as a directed limit of schemes..."

On Mar 11, 2019 Johan left comment #4055 on Lemma 31.4.13 in Limits of Schemes

OK, this theorem is Proposition 31.11.2 where we can even remove the assumption of Noetherianness.

On Mar 11, 2019 Matthew Emerton left comment #4054 on Lemma 79.14.7 in Formal Algebraic Spaces

Lemma 0AIG has a locally quasi-finite hypothesis, which doesn't seem to hold in the generality of the present lemma (unless I'm missing something). But it seems that the second argument just works anyway to show that if $T$ is a scheme, then the fibre product $X\times_Y T$ is an affine scheme.

On Mar 11, 2019 Matthieu Romagny left comment #4053 on Lemma 15.11.6 in More on Algebra

In Condition (5) of the statement of the Lemma: has a root

On Mar 11, 2019 Laurent Moret-Bailly left comment #4052 on Lemma 31.4.13 in Limits of Schemes

As an interesting application of Lemma 01Z6, we get the following generalization of Chevalley's theorem (EGA II, (6.7.1)). (I could not fint it in the text, but maybe I did not look in the right place.)

Theorem. Let $f:X\to Y$ be a morphism of schemes. Assume $f$ integral and surjective, $Y$ noetherian, and $X$ affine. Then $Y$ is affine.

Chevalley's theorem is the case where $f$ is finite. We reduce to this case by writing $\mathscr{A}:=f_*\mathscr{O}_X$ as the colimit of its finitely generated (hence finite) $\mathscr{O}_Y$ -algebras $\mathscr{A}_i$ . By the lemma, $\underline{\mathrm{Spec}}_Y(\mathscr{A}_i)$ is affine for some $i$ . Since it is finite over $Y$ , we may apply EGA II.

On Mar 11, 2019 Laurent Moret-Bailly left comment #4051 on Lemma 31.4.13 in Limits of Schemes

Line 3 of proof: $f_{0i}$ should be $f_{i0}$ (twice).

On Mar 08, 2019 Kęstutis Česnavičius left comment #4044 on Lemma 16.5.2 in Smoothing Ring Maps

The assumption $\varphi(J) = 0$ seems to be missing from the statement. It is used in the proof.

On Mar 07, 2019 Kęstutis Česnavičius left comment #4040 on Proposition 16.5.3 in Smoothing Ring Maps

At the end of the statement 'colimit' should be 'filtered colimit.'

On Mar 05, 2019 BAH left comment #4036 on Section 23.4 in Divided Power Algebra

Hi, forget that confused comment (if you can't delete it).

On Mar 05, 2019 BAH left comment #4035 on Section 23.4 in Divided Power Algebra

On Mar 05, 2019 BAH left comment #4034 on Section 55.2 in Crystalline Cohomology

Hi. Small typo: in the proof of lemma 55.2.4, second line after introducing $\mathcal{R}$ , sign "-" instead of sign "=".

On Mar 04, 2019 Dario Weißmann left comment #4033 on Section 28.8 in Morphisms of Schemes

*elements

On Mar 04, 2019 Dario Weißmann left comment #4032 on Section 28.8 in Morphisms of Schemes

In the example given directly after the defintion of a dominant morphism the $\lambda_i$ should be distinct element of $k$ .

On Mar 02, 2019 Laurent Moret-Bailly left comment #4027 on Section 25.10 in Schemes

@#4025, #2519, and Johan: The terminology is a bit confusing, as the comments demonstrate. The point is that in 25.10.2, "closed subspace" is not meant in the topological sense but in the ringed space sense of Definition 01HN (aka 25.4.4). The text should be more explicit about this (first of all by referring to that definition!). The same goes for open subchemes. The fact that the two notions (open/closed) are defined separately for locally ringed spaces (and there is no definition of "immersion" or "subspace" for these, as far as I could see) makes it a bit hard to get a clear picture.