The Stacks project

Comments 1 to 20 out of 8202 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 Jakob Werner left comment #8817 on Lemma 18.40.11 in Modules on Sites

The first sentence of the lemma should read Let f be a morphism.


On Hsueh-Yung Lin left comment #8816 on Section 28.7 in Properties of Schemes

Small typo: In the proof of Lemma 28.7.9, I think we should cite Lemma 28.7.4 instead of Lemma 28.7.2.


On Laurent Moret-Bailly left comment #8815 on Lemma 110.12.5 in Examples

First displayed formula in proof: should be (twice).


On Subhranil Deb left comment #8814 on Lemma 15.55.4 in More on Algebra

I believe the map should be instead of .


On amnon yekutieli left comment #8813 on Section 110.12 in Examples

trying again to cite:

Yekutieli, A. Flatness and Completion Revisited. Algebr Represent Theor 21, 717–736 (2018). https://doi.org/10.1007/s10468-017-9735-7


On amnon yekutieli left comment #8812 on Section 110.12 in Examples

In Theorem 7.2 of my paper Flatness and Completion Revisited (DOI \ref{https://doi.org/10.1007/s10468-017-9735-7} there is an example of a ring homomorphism A \to B that's not flat, but it is adically flat (i.e. completely flat), and also A \to \hat{B} is flat.


On Thomas Manopulo left comment #8811 on Section 53.2 in Algebraic Curves

I think in Lemma 0BXY it should be mentioned that the point x lies in U, no?


On Samuel Tiersma left comment #8810 on Section 111.28 in Exercises

Exercise 02EM, part (1): the linear form should be coprime to capital F, not little f.


On left comment #8809 on Lemma 10.69.5 in Commutative Algebra

Typo: in the iso after "is the quotient of", there's a missing in the denominator of the LHS.


On Chris left comment #8808 on Section 29.11 in Morphisms of Schemes

What does the notation mean? I can't find it anywhere else in the stacks project.


On Branislav Sobot left comment #8807 on Lemma 10.66.13 in Commutative Algebra

In the second paragraph it should be istead of in two places


On ZW left comment #8806 on Lemma 58.5.5 in Fundamental Groups of Schemes

Is the definition of "finite étale" here just finite and étale? Since finite flat is not the same as finite locally free, could you explain why, in the last paragraph, "since Y → Z is finite étale and hence finite locally free ..." without any noetherian assumption? Thanks.


On Yuto Masamura left comment #8805 on Remark 50.12.2 in de Rham Cohomology

We have an extra ")" in the first equation.


On Rudolf Tange left comment #8804 on Section 49.15 in Discriminants and Differents

I am rather puzzled by the identity for a finite flat A module B. I also notice that for finite and flat you claim , but this is not in accordance with the case that is a smooth variety over an algebraically closed field of characteristic and the absolute Frobenius morphism where we have , see Sect 1.3, p21 in the book on Frobenius splittings by Brion and Kumar. Note that in that case .


On Rudolf Tange left comment #8803 on Section 49.15 in Discriminants and Differents

I am rather puzzled by the identity for a finite flat A module B. I also notice that for finite and flat you claim , but this is not in accordance with the case that is a smooth variety over an algebraically closed field of characteristic and the absolute Frobenius morphism where we have , see Sect 1.3, p21 in the book on Frobenius splittings by Brion and Kumar. Note that in that case .


On Maxime CAILLEUX left comment #8802 on Lemma 5.8.17 in Topology

The case also works as would be empty hence connected.


On JJ left comment #8801 on Lemma 38.10.9 in More on Flatness

Shouldn't this work just as well for a morphisim locally of finite type?

If is ring map of finite type, is a finitely generated -module, and is a prime of such that is flat over , then take a finite polynomial ring surjecting onto . We still have that is a finite -module, and if is the preimage of in , then is still flat over . So proposition 05I5 guarantees that is finitely presented over . But then is also finitely presented over .


On ZL left comment #8800 on Proposition 96.14.3 in Sheaves on Algebraic Stacks

Typo: the th line from the below "Note that if is a special ..." should be "Note that if is a special ..."


On Maxime CAILLEUX left comment #8799 on Section 5.21 in Topology

For 5.21.2 a proof could be :

Naturally it suffices to show it for the union of two nowhere dense sets as the result will follow by induction. So, let be nowhere dense subsets. As for every pair of subsets of , so if is an open subset of , then, as but as is nowhere dense and , we have that and since is nowhere dense, .

For 5.21.5 a proof could be :

As is closed in , and as is an homeomorphism, is closed in and since it contains , so which then gives . As is nowhere dense, so as .


On Runchi left comment #8798 on Section 6.17 in Sheaves on Spaces

Lemma 007Z. I'm confused by last two lines of proof. And I wonder if means ? Also I noticed the textbook by Gortz, saying colimit of is precisely .