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 3390 in reverse chronological order.


On Daniel Levine left comment #3510 on Section 54.79 in √Čtale Cohomology

Typo: I believe the the second in the morphism in Theorem 095T should be .


On Dmitrii left comment #3509 on Section 10.120 in Commutative Algebra

In the proof of Lemma 02MJ it says "In this case the left hand side of the formula"; I think this should be replaced by the "right hand side of the formula".


On Yicheng Zhou left comment #3508 on Lemma 27.26.4 in Properties of Schemes

By using direction (2) to (1) of Lemma 28.11.3, one can give a more schematic proof as follows. By hypothesis we have locally , therefore the open immersion is locally given by a principal open set (in an affine neighborhood). By the lemma cited, is affine, therfore is affine whenever is affine.


On Jonas Ehrhard left comment #3507 on Lemma 10.38.12 in Commutative Algebra

By the claimed exactness of the diagram we have , and

Then the snake connects .


On Manuel Hoff left comment #3506 on Lemma 10.38.12 in Commutative Algebra

Hi, I think it's better to say "diagram chasing" here than "snake lemma". Seriously though, can somebody explain where the snake is?


On Jonas Ehrhard left comment #3505 on Proposition 10.58.5 in Commutative Algebra

Lemma 00JD (10.54.1) needs to be Artinian. I don't see why this should be the case, as the ideals

seem to give a possibly infinite descending sequence. . This sequence stabilises iff for , which must not be true. For example consider the localisation of the polynomial ring at the maximal ideal and .


On left comment #3504 on Lemma 15.70.7 in More on Algebra

Dear Ravi, fair question. It is the 5th condition in Lemma 15.70.2. I have edited the proof to clarify. See here.


On left comment #3503 on Lemma 45.24.1 in Dualizing Complexes

OK, yes thanks. Fixed here.


On left comment #3502 on Lemma 55.3.2 in Crystalline Cohomology

Thanks very much! Fixed here.


On left comment #3501 on Remark 55.8.7 in Crystalline Cohomology

Thanks and fixed here.


On left comment #3500 on Remark 55.8.6 in Crystalline Cohomology

Yes, this is a mistake. Thanks very much for pointing this out. I have fixed this here.


On left comment #3499 on Lemma 55.6.6 in Crystalline Cohomology

OK, I have checked this proof and it seems OK to me.

But I think there should be another proof of this lemma as well. Namely, we should be able to show directly that given a -module an -derivation is the same thing as a divided power -derivation using the universal property of . To do this consider the square zero thickening of . There is a divided power structure on if we set the higher divided power operations zero on . Consider the -algebra map whose first component is the given map and second component is . By the universal property we get a corresponding map whose second component should be the map corresponding to .

I didn't check this completely, but this should work and we should replace the proof given in the Stacks project by this argument.


On left comment #3498 on Lemma 45.5.7 in Dualizing Complexes

Thanks for this. See change here.


On left comment #3497 on Section 45.3 in Dualizing Complexes

Thanks very much. Added your version here.


On left comment #3496 on Section 45.2 in Dualizing Complexes

Thanks for the fix. See changes here.

Lemma 45.2.3 is stated as used later, so I think it is fine as is.


On left comment #3495 on Lemma 27.2.2 in Properties of Schemes

This is good. Thanks. Change is here.


On left comment #3494 on Lemma 49.11.3 in Algebraic and Formal Geometry

OK, I have made this change in this case. However, when writing these sections I tried to have some consistency in the numbering of the conditions listed in the lemmas. So I want to be careful in changing this in other lemmas, etc. So unless somebody is going to review the chapter as a whole and look more carefully to see if things can be simplified and/or strengthened (as may very well be the case), I would prefer to leave things as is for now.


On left comment #3493 on Lemma 7.46.11 in Sites and Sheaves

Thanks and fixed here.


On left comment #3492 on Section 104.3 in A Guide to the Literature

Yes, of course. It is impossible to keep lists up to date; the only solution seems to be to not have lists of things. For this kind of thing, I strongly encourse people to just edit the latex file and email me. In this case I minimally added this here. Thanks.


On left comment #3491 on Lemma 10.134.6 in Commutative Algebra

OK, I added the conclusion from Nakyama's lemma. But I kept the other statement as well because it is how I think about it. See change here. Thanks very much.