The Stacks project

Comments 1 to 20 out of 10041 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 Alejandro González Nevado left comment #10992 on Lemma 8.11.2 in Stacks

SS: Category equivalences preserve being a gerbe.

On Alejandro González Nevado left comment #10991 on Lemma 8.7.1 in Stacks

SS: Inertias preserve kind of stacks.

On Peng Du left comment #10990 on Lemma 10.45.4 in Commutative Algebra

The sentence ``... consider the ring map . This is injective and satisfies (a) and (b)'' does not seem make sense.

On Alejandro González Nevado left comment #10989 on Lemma 8.6.4 in Stacks

SS: Category equivalences preserve being a stack in setoids.

On Alejandro González Nevado left comment #10988 on Lemma 8.6.3 in Stacks

SS: Stacks in setoids correspond to stacks in sets via the respective unique equivalent categories fibred in sets.

On Alejandro González Nevado left comment #10987 on Lemma 8.6.2 in Stacks

SS: The stacks in sets are the sheaves.

On Alejandro González Nevado left comment #10986 on Lemma 8.5.4 in Stacks

SS: Category equivalences preserve being a stack in groupoids.

On Alejandro González Nevado left comment #10985 on Remark 7.48.4 in Sites and Sheaves

I think that in conditions (0') - (3') you do not mean to use but instead, right?

On Alejandro González Nevado left comment #10984 on Remark 7.48.4 in Sites and Sheaves

I think that in conditions (0') - (3') you do not mean to use but instead, right?

On Anonymous left comment #10983 on Lemma 37.22.5 in More on Morphisms

Apologies if this is already somewhere in the Stacks project, but the cited Lemma 10.163.3 also shows the following.

Let be a Cohen–Macaulay morphism of locally Noetherian schemes. If is such that is Cohen–Macaulay,

then is also Cohen–Macaulay.

And sorry for this, but: I wonder if it's better to use "Cohen–Macaulay" instead of "Cohen-Macaulay", i.e. an n-dash instead of a hyphen.

--

The weird formatting above is because the comment box resued to let me typeset the two local rings on the same line...

On ani left comment #10982 on Section 77.13 in Flatness on Algebraic Spaces

What does "bis" in the title mean? Is this a technical term? - I have not seen this before in the context of algebraic geometry.

On Alejandro González Nevado left comment #10981 on Section 7.32 in Sites and Sheaves

@S. Nivodski For references and examples about sites or topoi without points, one can check \ref{here}{https://mathoverflow.net/questions/306483/locales-as-spaces-of-ideal-imaginary-points/}.

On Junyan Xu left comment #10980 on Lemma 10.30.8 in Commutative Algebra

By you probably mean .

On Alejandro González Nevado left comment #10979 on Lemma 7.29.6 in Sites and Sheaves

SS: A morphism of topoi factors via a special cocontinuos functor and a continuous morphism of sites that commutes with fibre products.

On Alejandro González Nevado left comment #10978 on Lemma 7.26.6 in Sites and Sheaves

SS: Sheaves canonically reconstruct via (are) their absolute glueing data.

On left comment #10977 on Lemma 33.24.1 in Varieties

Little typo in the proof: It should be instead of .

On Alejandro González Nevado left comment #10976 on Lemma 7.17.7 in Sites and Sheaves

SS: On a site, the canonical map of colimits: (1) inherits the injectivity of all the transition maps, (2) transforms quasi-compactness of objects into its own injectivty, (3) transforms the combination of the two previous conditions into bijectivity, (4) transforms the existence of a finitely indexed cofinal system of coverings with quasi-compact fibered products into bijectivity.

On Alejandro González Nevado left comment #10975 on Lemma 7.17.3 in Sites and Sheaves

SS: "On a site, quasi-compactness coincides with the existence of finitely indexed subsurjections of sheaves."

On Alejandro González Nevado left comment #10974 on Lemma 7.17.2 in Sites and Sheaves

Wouldn't it be better to put separate names to these properties and introduce them separately as a weaker notions of quasi-compactness on objects of a site? (2) is quasi-compactness by finite refinements and (3) quasi-compactness by finite subcoverings. Thus we can also establish the slogan: "On sites, quasi-compactness implies quasi-compactness by finite refinements, which implies quasi-compactness by finite subcoverings. The reverse implications are not true".

On Alejandro González Nevado left comment #10973 on Lemma 7.15.2 in Sites and Sheaves

SS: A morphism of sites canonically induces a morphism of topoi.


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