The Stacks project

Comments 1501 to 1520 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 Adrian left comment #8131 on Section 33.36 in Varieties

Frobenii are divided into "frobenius" (e.g., Definition 03SM) and "Frobenius" (e.g., in Definition 0CC9 and Remark 0CCG). I guess these are just misprints and "frobenius" should always be "Frobenius".

On Et left comment #8130 on Lemma 10.134.11 in Commutative Algebra

It may be worth saying a word on why the two complexes are equal. The equality in degree 0 is clear, but in degree 1 it relies on the fact we are quotienting by I^2 and the surjectivity of the presentation.

On left comment #8128 on Definition 10.120.14 in Commutative Algebra

Here the empty product (the case ) gives the ideal .

On quasicompact left comment #8127 on Section 29.38 in Morphisms of Schemes

You can search latex on this site, for example search for "r_{\mathcal{L},\psi}" (including the quotation marks).

On Jinyong An left comment #8126 on Section 29.38 in Morphisms of Schemes

What is the assoicated map in the Lemma 29.38.7 ? Where can I find the definition? And in the proof of the Lemma 29.38.7, what does the sentence "This map is adjoint to a map " exactly means? Where can I find associated reference?

On Anonymous left comment #8125 on Definition 94.19.3 in Algebraic Stacks

Typo: I think "is the algebraic space" should read "is the algebraic stack"

On anon left comment #8124 on Lemma 10.53.5 in Commutative Algebra

More generally, if a ring R is a product of local rings, then the factors in this product can be obtained by localizing at R's maximal ideals.

On anon left comment #8123 on Lemma 10.53.5 in Commutative Algebra

The claim that the factors in the product are localizations is not justified. See here for a justification: https://math.stackexchange.com/questions/4643653/in-lemma-10-53-5-of-stacks-about-commutative-artinian-rings-how-did-they-use/4643742#4643742

On quasicompact left comment #8122 on Lemma 12.5.20 in Homological Algebra

Slogan: five lemma

On left comment #8118 on Definition 39.9.1 in Groupoid Schemes

Explanation is in the first couple of sentences here.

On Zhenhua Wu left comment #8117 on Definition 39.9.1 in Groupoid Schemes

Cool! But have you added this remark on the site? I don't see it in this section.

On Zhenhua Wu left comment #8116 on Section 31.23 in Divisors

I suggest we add the example that if is a complex manifold where is the sheaf of holomorphic functions on . Then is indeed the sheaf of meromorphic functions on in the classical sense. That's where the concept comes from.

On Reginald Anderson left comment #8115 on Definition 10.6.1 in Commutative Algebra

The way this page is currently written reads is "a ring map R -> S is if finite type/presentation..." "...if certain properties of R and S hold, independent of the map given"

On Elías Guisado left comment #8113 on Lemma 29.21.7 in Morphisms of Schemes

The additional equivalence with “ is a finitely presented -module” follows easily from the statement (and maybe it is worth adding it?). On the one hand, if is of finite presentation, it follows that is of finite type by 17.11.3. For the converse, consider the exact sequence If is of finite type, then given , we can find a surjection in some open neighborhood of , and, hence, we can replace by in the last sequence (after restricting) while maintaining exactness. Thus, is of finite presentation.

On Owen left comment #8110 on Lemma 10.127.15 in Commutative Algebra

it should say (3) Each is finite over , not 'of finite.'

On Laurent Moret-Bailly left comment #8109 on Example 29.43.2 in Morphisms of Schemes

This is repeated later as Lemma 31.30.5. But the condition on can be relaxed, at least if is quasicompact: by 10.56.2 and 27.11.8 it is enough to assume that is finitely generated over and is finite over . This is useful in practice, but I could not find this statement in SP (or in EGA!).

On Et left comment #8108 on Lemma 12.6.4 in Homological Algebra

I think it's worth giving a sentence on how to obtain exactness at the end. It seems to me there is something non trivial to show there (my proof used the snake lemma in a necessary way) and I wasn't able to find references on this either. (of course, there are plenty of references that construct the sequence with the other defintion of Ext)

On Laurent Moret-Bailly left comment #8107 on Lemma 27.10.4 in Constructions of Schemes

Line 6 of proof: should contain . Line 9: should be .

On Laurent Moret-Bailly left comment #8106 on Lemma 30.19.3 in Cohomology of Schemes

The gradings on and are completely irrelevant (and not even mentioned in the proof), so why not forget about them? The same comment applies to 69.20.4.

On Et left comment #8105 on Lemma 12.5.8 in Homological Algebra

Seems to be a typo on the second exact sequence


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