The Stacks project

Comments 1941 to 1960 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 #7627 on Section 26.20 in Schemes

Thanks. See this.


On left comment #7626 on Section 48.19 in Duality for Schemes

Thanks. See this.


On left comment #7625 on Situation 48.20.1 in Duality for Schemes

No reason.


On left comment #7624 on Lemma 4.18.3 in Categories

Indeed! Hmm... not worth changing to me but if others chime in... also this occurs in more places than just here.


On left comment #7623 on Lemma 7.8.5 in Sites and Sheaves

Thanks. Changed it here.


On left comment #7622 on Section 25.10 in Hypercoverings

Thanks very much. I made precise the category we are using and I fixed the mistake you pointed out here.


On left comment #7621 on Section 4.4 in Categories

Dear Fawzy N. Hegab, this is indeed just a tad pedantic! Since the word "final" occurs in the text and not in a statement or proof of a mathematical result, I am not going to change it.


On left comment #7620 on Lemma 6.33.1 in Sheaves on Spaces

OK, I am going to leave this as is.


On left comment #7619 on Section 33.5 in Varieties

Dear Jinying An, I think this is how tensor products work. For your last question, you would need to assume that is a closed point.


On left comment #7618 on Lemma 47.16.11 in Dualizing Complexes

OK, I carefully checked the proof and I cannot fault it. @everyone: if you find a mistake in a proof, please in your comment point explicitly to the first step in the proof that does not work. Thanks very much!


On left comment #7617 on Example 9.26.8 in Fields

Thanks. See this.


On left comment #7616 on Section 12.6 in Homological Algebra

OK, I changed the order here. I guess it is a good exercise to show that you get the same thing if you do it the other way around.


On left comment #7615 on Lemma 94.10.10 in Algebraic Stacks

@#7464: I think the references you gave are not really relevant for the proof (this may not have been the point you were trying to make of course). For example, we cannot use Lemma 4.35.9 because we do not know is faithful or fully faithful at the start of the proof (this is the whole point). We cannot use Lemma 100.8.4 because it would be a forward reference.


On left comment #7614 on Lemma 101.16.1 in Morphisms of Algebraic Stacks

Very good! OK, I have decided to go with your second option and added this as possible abuse of language here. But I would be interested in somebody carefully going through all the possible cases and add "2-" everywhere and then removing this additional abuse. What do people think?


On left comment #7613 on Section 4.2 in Categories

For the Stacks project our default kind of category has a set of objects and for every two objects a set of morphisms. But we also study some categories, such as the category of sets, where this is not the case; ideally these should all be listed in Remark 4.2.2. In particular, we make the meta-mathematical claim that the categories we use in the Stacks project whose objects and/or morphisms do not form a set can be listed. We appreciate help with this. Thanks!


On left comment #7612 on Lemma 4.24.9 in Categories

No.


On left comment #7611 on Lemma 59.69.3 in Étale Cohomology

THanks. Tried to improve here.


On left comment #7610 on Section 10.132 in Commutative Algebra

Tried to improve this a little bit. See here.


On left comment #7609 on Lemma 4.41.1 in Categories

Thanks. Fixed here.


On left comment #7608 on Lemma 10.157.6 in Commutative Algebra

We don't have to: the module has no embedded and no associated primes so (1) holds.