The Stacks project

Comments 721 to 740 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 #8992 on Lemma 34.9.8 in Topologies on Schemes

Thanks and fixed here.

On ZL left comment #8991 on Section 20.28 in Cohomology of Sheaves

A small typo: In the first graph, the second point beginning with "for any additive functor ..." the derived categry and the homotopy category should switch place.

On left comment #8990 on Lemma 10.134.4 in Commutative Algebra

@#8988: the edit about the Tor-2 has already been made as you can see when you click on the link I gave. I will correspondingly update the website when all the comments have been addressed, see also this page

On Nico left comment #8989 on Lemma 92.5.8 in The Cotangent Complex

Tiny typo. The short exact sequence "0\to\Omega_{B/A}\otimes_BP_n\to\Omega_{P_n/A}\to\Omega_{P_n/B}" needs a at the end.

On Zheng Yang left comment #8988 on Lemma 10.134.4 in Commutative Algebra

As noted above, is there supposed to be a condition on ?

Assuming this, we can use the Tor spectral sequence (Tag 061Z) for tensored by to conclude the result in the last sentence.

On left comment #8987 on Lemma 10.115.3 in Commutative Algebra

Maybe nitpicking, but shouldn't the case be handled separately? Just saying something like "for , is a finitely-generated -vector space; thus is finite over ."

On Laurent Moret-Bailly left comment #8986 on Lemma 4.14.10 in Categories

Even more precisely: assume that exists for all . Then the resulting functor has a colimit if and only if does, and then the colimits coincide. (And if this is the case, it does not follow that the colimits with fixed exist).

On Devang Agarwal left comment #8985 on Lemma 20.25.4 in Cohomology of Sheaves

Possible small typo: the equality occurs in

On left comment #8984 on Section 110.19 in Examples

I think the initially stated generators for are incorrect. The correct definition is as given during the proof, but then in the defining equations of the give for example and so , not .

So I think .

On left comment #8983 on Lemma 94.8.2 in Algebraic Stacks

THanks and fixed here.

On left comment #8982 on Section 12.5 in Homological Algebra

Thanks and fixed here.

On left comment #8981 on Lemma 13.14.3 in Derived Categories

Going to leave this as is for now.

On left comment #8980 on Lemma 47.15.3 in Dualizing Complexes

Directly from the definition, if has finite injective dimension and if is bounded below (resp above), then is bounded above (resp below). Namely, you can represent by a bounded below (resp above) complex and by a bounded complex of injectives, and then the is computed by the hom complex which has the requisite boundedness.

Maybe this should be added to Section 15.69?

On left comment #8979 on Definition 20.49.1 in Cohomology of Sheaves

Thanks and fixed here.

On left comment #8978 on Lemma 13.11.6 in Derived Categories

Thanks for the typo which is fixed here. I think that looking at Lemma 13.6.11 whose statement refers to Lemma 13.6.10 we do see that the kernel of each localization functor is the set of acyclic complexes. I think the reader should at this point be sufficiently familiar with computations in localizations of categories to be able to see why such an exists, but yes I agree that this can be done using Lemma 4.27.6.

On left comment #8977 on Lemma 13.10.2 in Derived Categories

Thanks for the typo. Fixed here.

On left comment #8976 on Section 39.13 in Groupoid Schemes

This would have to be done somewhere in the chapter on simplicial methods I think and then this section could refer back to that. Going to leave as is, but I would welcome a small set of changes implementing this.

On left comment #8975 on Lemma 13.10.2 in Derived Categories

Well, I think you are explaining in more detail exactly what the reader has to do when verifying the proof as given in the text. Although I think your comments are helpful for the reader who chances upon this web location, I am going to leave the text as is for now.

On left comment #8974 on Lemma 12.14.10 in Homological Algebra

OK, I think that this is the sort of "easy" case. Even if we cannot choose splittings compatible with the maps, then there is a relation between the two deltas. In order words, we should upgrade Lemma 12.14.12 to the case of a map between termwise split short exact sequences (not compatible with splittings) whose conclusion is some homotopy between two maps involving deltas. Maybe this is what commenter R in #5086 had in mind?

On left comment #8973 on Lemma 20.23.6 in Cohomology of Sheaves

Good catch! Thanks. I checked with the script and I have changed the formula for a description of the permutation which is better for humans than a formula. See changes.


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