The Stacks project

Comments 1121 to 1140 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 Hayama Kazuma left comment #8552 on Lemma 37.62.1 in More on Morphisms

typo: “in a neighbourhood of ”, here, is ?

On Long Liu left comment #8551 on Remark 13.36.7 in Derived Categories

In Remark 0ATH (1), 'D(A)' should be 'D'.

On left comment #8550 on Proposition 10.138.8 in Commutative Algebra

The second to last sentence of the proof is incomplete.

On left comment #8549 on Lemma 10.140.9 in Commutative Algebra

In the second paragraph of the proof, instead of Lemma 00TG, it seems more relevant to cite Lemma 00TF.

On Bernd Siebert left comment #8548 on Section 34.7 in Topologies on Schemes

Typo in Lemma 021X: , ,

On Max L. left comment #8547 on Section 12.28 in Homological Algebra

Hi! In definition 013F, the "J" should be a "P", right?

On Shizhang left comment #8546 on Lemma 26.2.2 in Schemes

The last bit of the proof: ``is a local ring \emph{isomorphism}''.

On DU Changjiang left comment #8545 on Definition 38.34.11 in More on Flatness

Maybe and should be and ?

On ZL left comment #8544 on Lemma 78.19.6 in Groupoids in Algebraic Spaces

Typo: "Let be a pre-relation" should be "Let be a pre-equivalence relation"

On Michael Barz left comment #8542 on Lemma 37.64.2 in More on Morphisms

I'm a little confused by the proof of lemma 094Q -- how does lemma 092C imply that condition (2) implies condition (1)? It seems like we want to prove that is flat as a -module, so we should take in lemma 092C, but then I'm not sure if lemma 092C actually applies.

On Jiwan Jung left comment #8540 on Section 10.90 in Commutative Algebra

What is the exact sequence 0→Ker(φ)→E′→E→0 in the proof of Lemma 10.90.3? Maybe the first term is Im(φ) not Ker(φ).

On DU Changjiang left comment #8539 on Section 61.5 in Pro-étale Cohomology

It seems that there is a typo in the discussion before the formula \ref{https://stacks.math.columbia.edu/tag/0972}: it should be instead of .

On Minki Lee left comment #8538 on Section 43.7 in Intersection Theory

I think I see a typo in the definition; the pullback of should be , not , right?

On Mark left comment #8537 on Lemma 60.21.2 in Crystalline Cohomology

should be for all and

On left comment #8536 on Lemma 29.54.5 in Morphisms of Schemes

Yes, you are right that when we use the term birational in the proof here and in 29.54.6 we need to restrict to quasi-compact opens. I fixed it here by replacing "birational" by "locally birational". I think "locally birational" is sufficiently clear, but if multiple people disagree I will change it.

On left comment #8535 on Lemma 29.54.5 in Morphisms of Schemes

@#8530 Thank you for the explanation. What got me confused with respect to the birationality issue is that in the proof, third paragraph, it is said "as is birational..." Nonetheless, none of or need to have a finite amount of irreducible components (so cannot be birational under the current definition of birationality, 29.50.1).

The same abuse of terminology happens in the proof of 29.54.6, when it is said " is integral and birational."

On left comment #8533 on Lemma 5.12.12 in Topology

Agreed. Going to leave as is until more people chime in.

On left comment #8532 on Lemma 10.22.2 in Commutative Algebra

Hmm, I think it is fine. I mean that ideal really does cut out the connected component of by the previous sentence, Lemma 10.21.3, the fact that , and for an idempotent . So I think the proof is fine as is.

I guess an alternative proof would be to first say that the connected component of is as in Lemma 10.22.1 and then explain which idempotents are in . Do people prefer this?

On left comment #8531 on Lemma 28.7.9 in Properties of Schemes

Very good. I fixed it as you suggested here.

On left comment #8530 on Lemma 29.54.5 in Morphisms of Schemes

Did you read the footnote? It explains why the formulation in (3) is as given; your alternative formulation, although OK in spirit, does not work with the current definition of birational.

Uniqueness in (4) guarantees that the morphisms constructed will glue. Namely, if and are the morphisms constructed for open, then by uniqueness . This type of argument is used in a number of places in the Stacks project without further explanation.


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