The Stacks project

Comments 1 to 20 out of 10385 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 nkym left comment #11363 on Lemma 15.50.5 in More on Algebra

In the statement, probably it is not meant that the indices for x and y are equal

On left comment #11362 on Section 31.29 in Divisors

What is the definition of a rank 1 coherent sheaf on an integral scheme? It seems to me that rank is defined only for locally free sheaves in https://stacks.math.columbia.edu/tag/0B37

On 乳法专家 left comment #11361 on Lemma 59.75.4 in Étale Cohomology

The f's should be g.

On Rankeya Datta left comment #11360 on Lemma 10.154.3 in Commutative Algebra

I think this Lemma and the next one hold when we replace étale by smooth. Is it possible to add the smooth case to the statements?

On Joe Bloggs left comment #11359 on Definition 4.21.1 in Categories

While these definitions from The Stacks project are technically precise, I wonder if distinguishing between a directed set and a directed partially ordered set is always necessary in practice. Sometimes mathematicians overcomplicate definitions that could be simplified, much like how one might use a text repeater to expand simple strings into complex patterns without adding real clarity. Is there a specific case where the antisymmetry requirement fundamentally changes the outcome?

On 乳法专家 left comment #11358 on Lemma 59.55.5 in Étale Cohomology

In the first paragraph of the proof, in the definition of E_1, there is a missing subindex p

On 乳法专家 left comment #11357 on Lemma 59.55.5 in Étale Cohomology

The exact sequence is on Y_{étale} not on X_{étale}

On 乳法专家 left comment #11356 on Lemma 59.46.1 in Étale Cohomology

In the statement of the lemma, one should also mention that W is étale over X

On 乳法专家 left comment #11355 on Lemma 59.46.3 in Étale Cohomology

In the first paragraph of the proof, one should probably say U=X-Z.

On shubhankar left comment #11353 on Lemma 110.72.4 in Examples

The 'unwinding definitions' part can likely be made more explicit if needed by noting that the map on is the induced map on isomorphism classes.

On left comment #11349 on Lemma 4.33.3 in Categories

A converse holds: If is strongly -cartesian and is strongly -cartesian, then is strongly -cartesian.

On shubhankar left comment #11348 on Definition 29.54.1 in Morphisms of Schemes

It would be better if this referred to whereever relative normalization is defined.

On nkym left comment #11347 on Theorem 58.29.3 in Fundamental Groups of Schemes

Maybe excellent schemes have not been defined.

On left comment #11346 on Lemma 59.63.5 in Étale Cohomology

Thanks and fixed here.

On fherzig left comment #11345 on Section 7.29 in Sites and Sheaves

For the SGA4 reference at the top, it's Exp. III, not IV.

On fherzig left comment #11344 on Lemma 7.29.1 in Sites and Sheaves

I think it would be worth stating a special case of this lemma, which is useful in practice (cf. [SGA4, Exp. III, Thm. 4.1]). Namely, this lemma applies in the situation where is fully faithful such that moreover (a) is a cover in iff is a cover in , and (b) any object of has a cover of the form with .

This is analogous to some results in section 6.30, and likewise it might be nice to state them also e.g. for sheaves of modules.

I see that in practice this lemma is used in the Stacks Project for fully faithful morphisms that satisfy (b) and a slight weakening of (a). If I'm not mistaken, the slight weakening of (a) that is needed is precisely that the topology (rather than the site) of is induced from that of , as in the SGA4 reference.

On Sanskar left comment #11343 on Section 33.41 in Varieties

In Lemma 0C1R, the notation x has two different roles: a fixed closed point in the set-up and the singular points of X in, for example part (2).

On left comment #11342 on Lemma 10.119.7 in Commutative Algebra

Piotr Achinger send an example: If is a nontrivial extension of fields with algebraically closed in , then the ring is a local normal domain of dimension but not a valuation ring.

On nkym left comment #11341 on Lemma 15.39.1 in More on Algebra

has not been defined in the statement.

On nkym left comment #11340 on Lemma 15.37.5 in More on Algebra

"along the right column of the diagram above" should be "along the bottom row of the diagram above"


feed