The Stacks project

Comments 1001 to 1020 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 #8691 on Lemma 31.23.8 in Divisors

Typo in (2): it should be “let be an invertible -module.”

Part (1) can be stated more generally as:

Let be a morphism of locally ringed spaces. Assume pullbacks of meromorphic functions are defined for . Let , and suppose we have a morphism of -modules. Then there is a canonical morphism of -modules. (In particular, setting , the map to be the unit of and taking global sections gives the pullback map pullback map .)

The construction is the composite of canonical maps:

I regarded this more general form of part (1) useful for instance if one is working with the sheaf of meromorphic differentials on a -variety , i.e., . Pullbacks of meromorphic functions are defined for the normalization . Hence one obtains a morphism . When is a curve, this last morphism is used by Serre in Algebraic Groups and Class Fields to define the “sheaf of regular meromorphic differentials.”


On left comment #8690 on Lemma 36.13.6 in Derived Categories of Schemes

The distinguished triangle is backwards (wrong truncations). So we need to rewrite the proof.


On I. Vanni left comment #8689 on Lemma 15.22.11 in More on Algebra

I think some 's should be 's.


On Sveta M left comment #8688 on Lemma 18.34.3 in Modules on Sites

Typo in the second equation in the proof: omitted closing bracket in .


On Micheal left comment #8687 on Section 35.8 in Descent

A small typo in remark 35.8.4, after "Hence, given S′/S we let", f_{sites, *}O(S'/S) should equal to O(T\times_{S}S'/T)


On Anonymous left comment #8686 on Lemma 96.17.1 in Sheaves on Algebraic Stacks

Does part (2) of this lemma need a flatness hypothesis as in Tag 20.11.11?


On Anne left comment #8685 on Theorem 15.90.16 in More on Algebra

It doesn’t seem to be clear that suffices to prove that is fully faithful. I thought about using Tag 05ER? But in general, \varphi is not flat.


On Xiaolong Liu left comment #8684 on Definition 20.47.1 in Cohomology of Sheaves

In (1) we may replace ' is strictly perfect' into ' is strictly perfect'.


On Figo left comment #8683 on Proposition 35.3.9 in Descent

Oh I am sorry. "Apply to x_inx_i$ and use the same argument.


On Figo left comment #8682 on Proposition 35.3.9 in Descent

I am sorry that I cannot get from the fact . I know that from the conclusion by (1). And as , I only can conclude that .


On Haohao Liu left comment #8681 on Lemma 3.12.1 in Set Theory

To Comment #8193: Then for a topological space , the category of abelian sheaves on is an abelian subcategory of the category of abelian presheaves on in the comment's sense. In general, the inclusion functor is not exact, so is not an abelian subcategory of in the sense of p.7 of "Introduction to Homological Algebra" by Weibel.


On Xueping Huang left comment #8680 on Section 7.5 in Sites and Sheaves

In the first paragraph: ``Note that by the previous section this functor commutes with all limits. " Does commute with colimits as well?


On Peter Fleischmann left comment #8679 on Section 6.29 in Sheaves on Spaces

Proof of Lemma 009F (2), end of paragraph: typo, we need phi_{i'i''}(s') two times (instead of phi_{i'i''}(s))


On Gabriel left comment #8676 on Section 21.4 in Cohomology on Sites

On lemma 03AJ we have a bijection between two abelian groups. It would be nice to say that this is indeed an isomorphism. (Said otherwise, it would be nice to explain that the contracted product of torsors "is" the addition on H^1.)


On Life_is_hard left comment #8675 on Section 7.32 in Sites and Sheaves

Do we have any explicit example of a point of the site ? I thought it should be the skyscraper sheaf, but the skyscraper sheaf is a contravariant functor instead of a covariant functor.


On ZL left comment #8674 on Lemma 21.34.6 in Cohomology on Sites

There is a typo "" should be ""


On Figo left comment #8673 on Lemma 7.12.4 in Sites and Sheaves

Is here ?


On Matthew left comment #8672 on Section 12.3 in Homological Algebra

@#8671 No, simple misread on my part.


On Matthew left comment #8671 on Section 12.3 in Homological Algebra

In the proof of Lemma 12.3.4: "It follows that is the identity since..."

I think that this should read "It follows that is the identity since..." Otherwise the map in question is not well-defined. The rest of the argument works fine.


On Sveta M left comment #8670 on Lemma 50.24.2 in de Rham Cohomology

Typo: "In Situation 0G8G the psuhforward" should be "... pushforward".