The Stacks project

Comments 741 to 760 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 #8972 on Definition 10.12.6 in Commutative Algebra

Thanks and fixed here.

On left comment #8971 on Section 10.12 in Commutative Algebra

Thanks and fixed here.

On left comment #8970 on Section 10.10 in Commutative Algebra

Yes! Thanks and fixed here.

On left comment #8969 on Lemma 65.16.4 in Algebraic Spaces

I think the lemma is OK as it stands. The point of the lemma is that in checking that the restriction of to is an algebraic space, it suffices to prove the functor on the category of schemes over is an algebraic space. Note that it scarcely makes sense to consider the functor on the category of schemes over .

On left comment #8968 on Section 59.89 in Étale Cohomology

Thanks to Niels and Alexander. Fixed here.

On left comment #8967 on Section 21.2 in Cohomology on Sites

Dear Leonard, all of this should be discussed (and mostly is discussed, sometimes implicitly) elsewhere and not in this section. Some small comments on your questions: it is best to think of a global section (of a sheaf or of a presheaf) as a compatible family of sections over all objects of the site. Then it is immediately clear that the set of global sections of a presheaf of abelian groups is an abelian group. I think this also answers questions 1 and 2: the category A is the category of abelian groups and homs in abelian groups are abelian groups.

On left comment #8966 on Lemma 10.50.5 in Commutative Algebra

Thanks and changes here.

On left comment #8965 on Lemma 10.97.3 in Commutative Algebra

Much better. Thanks! Changes are here.

On left comment #8964 on Lemma 28.26.4 in Properties of Schemes

Sure. Change is here.

On left comment #8963 on Lemma 48.2.5 in Duality for Schemes

Thanks and fixed here.

On left comment #8962 on Lemma 7.38.2 in Sites and Sheaves

Thanks for catching this silly error! Fixed here.

On left comment #8961 on Lemma 20.11.7 in Cohomology of Sheaves

This is a reasonable concern to have, but since the proof is fine I am going to leave this as is.

On left comment #8960 on Lemma 12.13.12 in Homological Algebra

Thanks and fixed here.

On left comment #8959 on Lemma 47.25.2 in Dualizing Complexes

Thanks for this remark. At the moment we have only defined relatively perfect complexes in the flat case, see Remark 36.35.14 for why. Note that with option (B) in that remark what you say would be correct. So I have simply restated this lemma in a way avoiding the notion of "relatively perfect" in the non-flat case. See here.

On left comment #8958 on Lemma 13.37.3 in Derived Categories

Well, the proof as written now works if , so we can just leave it as it is now.

On left comment #8957 on Lemma 10.51.2 in Commutative Algebra

Yes. Going to leave as is.

On left comment #8956 on Lemma 13.5.9 in Derived Categories

Thanks and fixed here.

On left comment #8955 on Lemma 7.29.6 in Sites and Sheaves

Thanks and fixed here.

On left comment #8954 on Lemma 13.5.8 in Derived Categories

Going to leave as is.

On left comment #8953 on Proposition 13.5.6 in Derived Categories

@#8851: OK, this does sound extremely nitpicky. In the statement of the proposition we say so clearly that is the isomorphism we'll be using? The Stacks project is written for human consumption and I think it is common practice in mathematics to assume that a piece of data that has already been given is the one that is being used? (In other words, one should point out when the "obvious" choice isn't being used!)


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