The Stacks project

Comments 1 to 20 out of 8650 in reverse chronological order.

\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*}

On left comment #9308 on Theorem 35.4.22 in Descent

OK, the typo is that a subscript was missing on the in the sentence, right? I fixed that. But I think it immediately follows from that sentence that is is an isomorphism. I did add another reference to the first sentence of the last paragraph to clarify. Changes here.


On Fiasco left comment #9307 on Lemma 59.82.2 in Étale Cohomology

In the end of proof of lemma, why does Zariski locally come from a section of ? And why are these local sections compatible such that they can glue altogether?

For the first question, let's be more careful(with the same notations). Let be an open subscheme which factors through for some . So defines a section of named , we want to show its restriction is equal to . Note that has an etale cover , So we only need to check each piece . We look at the stalks and choose a geometric point .

On the one hand, the stalk of restriction of is just the stalk of at pt, which is considered as a geometric point of .

On the other hand, the stalk of is just the stalk of at pt, which is considered as a geometric point of .

Now the key point is we only know pt factors through . But by definition and coincide at some scheme which is etale over such that and both factor through . So can be strictly "smaller" than .


On left comment #9306 on Lemma 31.28.5 in Divisors

Good catch! OK, I rewrote the proof to make it work in the non-Noetherian case. More interesting was the case of Lemma 31.28.4. Finally, Lemma 31.31.3 was unfixable and I needed to assume the Noetherian assumption. See these changes.


On left comment #9305 on Lemma 5.26.7 in Topology

Thanks and fixed here.


On left comment #9304 on Section 5.1 in Topology

Fixed.


On left comment #9303 on Remark 63.9.5 in More Étale Cohomology

Dear Cop 223, I understand what you are saying, but in the setup as in this chapter, I do not think this "helps".


On left comment #9302 on Lemma 33.44.12 in Varieties

Yes. Going to leave as is.


On left comment #9301 on Lemma 48.15.1 in Duality for Schemes

Indeed. Weakend assumptions here.


On left comment #9300 on Section 63.1 in More Étale Cohomology

this.


On left comment #9299 on Lemma 43.17.2 in Intersection Theory

Good catch! Fixed here.


On left comment #9298 on Section 17.26 in Sheaves of Modules

Thanks and fixed here.


On left comment #9297 on Lemma 12.6.3 in Homological Algebra

Thanks for the references. Going to leave as is.


On left comment #9296 on Lemma 10.157.3 in Commutative Algebra

Thanks and fixed here.


On left comment #9295 on Lemma 13.27.7 in Derived Categories

Thanks for the typo. I added a description of the composition law on Ext groups. See this commit.


On left comment #9294 on Section 59.33 in Étale Cohomology

Oops. Fixed here.


On left comment #9293 on Section 10.63 in Commutative Algebra

No: the annihilator of in is not a prime ideal.


On left comment #9292 on Lemma 76.14.3 in More on Morphisms of Spaces

Thanks and fixed here.


On left comment #9291 on Section 28.16 in Properties of Schemes

Thanks and fixed here.


On left comment #9290 on Section 42.44 in Chow Homology and Chern Classes

See Definition 42.7.6.


On left comment #9289 on Lemma 15.83.8 in More on Algebra

OK, this is such a technical lemma that a slogan does not work well, I think.

I like your and Yuchen Wu's suggestion of the additional lemma. I have added it here.