The Stacks project

Comments 2041 to 2060 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 #7521 on Lemma 29.43.4 in Morphisms of Schemes

Yes, that's true, but we already have this result in Lemma 29.43.16.

On Firmaprim left comment #7520 on Lemma 29.43.4 in Morphisms of Schemes

Hello,

We are also proving that a projective morphism over an affine scheme is H-projective here.

I think it would be useful to make another lemma before the lemma 01WB with this statement and then make the lemma 01WB as a corollary of the new lemma.

On Hao Peng left comment #7519 on Lemma 10.52.4 in Commutative Algebra

in fact the condition that M is a finite module is not needed here.

On Hao Peng left comment #7518 on Section 43.17 in Intersection Theory

There is previous defined intersection product in 42.62, what does that relate to the notion defiend here?

On Hao Peng left comment #7517 on Lemma 13.28.2 in Derived Categories

I think in the proof we should use the usual truncation instead of the stupid truncation.

On Laurent Moret-Bailly left comment #7516 on Lemma 33.43.2 in Varieties

Here is a more elementary (and, I think, more natural) proof: since is irreducible, is not closed in . In particular, the immersion is not proper. By Lemma 29.41.7 (and because is separated), is not proper either.

On Andrea Panontin left comment #7515 on Section 59.31 in Étale Cohomology

Small typo: after Definition 04FS the sheaf is denoted first by and then by (mathbf first and mathcal later).

On David Holmes left comment #7514 on Lemma 33.43.2 in Varieties

Dear WhatJiaranEatsTonight,

I quite agree with your counterexample. Moreover, Lemma 26.22.1 refered to in the proof uses separatedness. Though happily this tag is only applied in once place, and in that place it is applied to a separated scheme (over a field).

Best wishes, David

On David Holmes left comment #7513 on Section 1.1 in Introduction

You can use \ref; for example removing the spaces from \ref { 0001 } yields 1.1. For more, see the 'Markdown' link in the text above the comments box.

On David Holmes left comment #7512 on Lemma 59.32.5 in Étale Cohomology

Dear Haohao Liu, Henselian rings are local by definition in the stacks project, see 03QF. One could argue that the grammar is ambiguous (the def only tells us when a local ring is henselian, and could be interpreted as giving no information on when a non-local ring is henselian). But I think there it is quite standard to intrepret a definition written in this way as meaning that every local ring is henselian. Best wishes, David

On Hao Peng left comment #7511 on Lemma 45.11.2 in Weil Cohomology Theories

maybe the last assertion shoule be placed prior to condition 3 to avoid confusion that X is equidimebsional or not.

On Hao Peng left comment #7510 on Lemma 45.7.6 in Weil Cohomology Theories

This one is jot trivial though. You need to use Poincare duality to deduce it from tag 0FGX.

On Matthieu Romagny left comment #7509 on Section 27.17 in Constructions of Schemes

Missing word in 5th sentence of this section: We first describe the value of this sheaf on schemes T mapping into the relative Proj.

On left comment #7508 on Lemma 29.36.7 in Morphisms of Schemes

No.

On Peng Du left comment #7507 on Lemma 29.36.7 in Morphisms of Schemes

Don't we need the disjoint union to be finite (in both statement and proof)?

On Haohao Liu left comment #7506 on Lemma 59.32.5 in Étale Cohomology

Is the ring local?

On Xiaolong Liu left comment #7505 on Lemma 20.15.1 in Cohomology of Sheaves

At the above of the diagram, it should be instead of .

On left comment #7504 on Lemma 50.24.3 in de Rham Cohomology

See introduction to Section 48.15.

On left comment #7503 on Lemma 37.17.3 in More on Morphisms

This lemma needs to be moved earlier because the result is used silently in Lemmas 45.14.3, 45.14.4, and 45.14.15. I supposed we could move it and Lemma 37.17.1 to their own section in some section later than Section 37.16. Then in the proof we cannot use the computations from Section 50.16, so these would have to be replicated in a local remark or something (don't change Section 50.16).

On Hao Peng left comment #7502 on Lemma 50.24.3 in de Rham Cohomology

Shouldn't it be the case that the trace map is canonical if induced from the canonial isomorphism and canonical trace \?

I'm thinking about this because there should be a perfect pairing that is compatible with base change.


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