The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

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


On 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 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 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 instead of .


On 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 is missing in the left side of


On Name* left comment #4059 on Section 36.26 in More on Morphisms

Who is Y in the proof of lema o55j?


On 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 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 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 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 is a scheme, then the fibre product is an affine scheme.


On 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 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 be a morphism of schemes. Assume integral and surjective, noetherian, and affine. Then is affine.

Chevalley's theorem is the case where is finite. We reduce to this case by writing as the colimit of its finitely generated (hence finite) -algebras . By the lemma, is affine for some . Since it is finite over , we may apply EGA II.


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

Line 3 of proof: should be (twice).


On Kęstutis Česnavičius left comment #4044 on Lemma 16.5.2 in Smoothing Ring Maps

The assumption seems to be missing from the statement. It is used in the proof.


On 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 BAH left comment #4036 on Section 23.4 in Divided Power Algebra

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


On BAH left comment #4035 on Section 23.4 in Divided Power Algebra


On 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 , sign "-" instead of sign "=".


On Dario Weißmann left comment #4033 on Section 28.8 in Morphisms of Schemes

*elements


On 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 should be distinct element of .


On 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.