The Stacks project

Comments 481 to 500 out of 9050 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 #9239 on Section 80.6 in Bootstrap

Thanks and fixed here.

On left comment #9238 on Lemma 10.106.2 in Commutative Algebra

This doesn't work because there is no ring map . A posteriori we see that there is a map of multiplicative monoids -- but this isn't true in general for Noetherian local rings.

On left comment #9237 on Lemma 61.8.7 in Pro-étale Cohomology

Thanks and fixed here.

On left comment #9236 on Section 13.3 in Derived Categories

Going to leave as is.

On left comment #9235 on Section 61.10 in Pro-étale Cohomology

The sentence is meant to say that if "h" or if "ph", then the pro- topology (as informally introduced in this section) is the V topology. Does that help?

On left comment #9234 on Lemma 10.131.6 in Commutative Algebra

Dear Et, I think that if in the last sentence of the first proof then in the direct sum so we're done by fiat. I did remove a superfluous sentence here.

On left comment #9233 on Section 10.131 in Commutative Algebra

Going to leave as is.

On left comment #9232 on Lemma 31.4.5 in Divisors

Thanks and fixed here.

On left comment #9231 on Section 27.8 in Constructions of Schemes

Yes.

On left comment #9230 on Lemma 92.24.2 in The Cotangent Complex

Thanks and fixed here.

On left comment #9229 on Section 4.13 in Categories

Thanks for the typo which is fixed here. Yes, strictly speaking we need to tell the reader the maps.

On left comment #9228 on Lemma 10.37.9 in Commutative Algebra

This was already fixed here.

On left comment #9227 on Section 15.11 in More on Algebra

@#8770: OK, thanks. I added this here.

@#8861: Yes, this is more or less a historical artifact. (I think there is a typo in your comment because Lemma 85.23.3 does not seem related). There is also Lemma 15.93.10. Since these lemmas aren't wrong, I think I am going to leave them as they are now. But we could move some of them to the obsolete chapter, I guess. Let's see if others chime in.

On left comment #9226 on Lemma 13.4.16 in Derived Categories

Going to leave as is.

On left comment #9225 on Section 32.5 in Limits of Schemes

THanks and fixed here.

On left comment #9224 on Lemma 12.8.2 in Homological Algebra

That's not a comment on this lemma however, so I am going to leave this lemma as is.

On left comment #9223 on Lemma 12.4.2 in Homological Algebra

One could add another equivalent property:

(4) Every idempotent endomorphism is of the form , where and is a section of .

Proof. (3)(4). Clear.

(4)(3). Suppose that is a retraction of . Then is an idempotent endomorphism and thus so is . Hence, , where and is a section of . We already know that , , , so by Remark 12.3.6 it suffices to see , . Note , thus by monicity of . Analogously, .

On left comment #9222 on Lemma 12.8.2 in Homological Algebra

@#9219 I am aware about the SP convention that everything is a small category (unless otherwise stated). My point is that inside the SP itself there may be an application of this result to some non-small category, like, say, the derived category of some non-small abelian category (like -modules or some category of modules over a scheme). It is not far-fetched that something like this might already appear (although I don't know any instance throughout the SP now).

On left comment #9221 on Lemma 61.19.8 in Pro-étale Cohomology

Haha! Fixed here.

On left comment #9220 on Remark 13.10.4 in Derived Categories

Going to leave as is.


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