The Stacks project

Comments 1 to 20 out of 9245 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 #10094 on Lemma 10.38.3 in Commutative Algebra

should be used instead of as in the previous lemma and it should be mentioned that .


On left comment #10093 on Lemma 10.38.3 in Commutative Algebra

should be used instead of as in the previous lemma and it should be mentioned that .


On Chris left comment #10092 on Lemma 27.10.7 in Constructions of Schemes

Just to be clear, for Lemma 0B5I, this simple tensor is a priori an elment of ?


On Chris left comment #10091 on Lemma 27.10.7 in Constructions of Schemes

Just to be clear, for Lemma 0B5I, this simple tensor is a priori an elment of F(D_+(f))\otimes O_X(-nd)?


On Ryan Rueger left comment #10090 on Remark 8.2.4 in Stacks

The term "on the nose" is used here for the first time (according to the search function) and is used later on as well a few times. From context I gather it roughly means actual equality (instead of up to isomorphism)? I think it would be helpful for this notion to be defined somewhere concretely (or defined on-the-nose so to speak)


On left comment #10089 on Remark 8.2.4 in Stacks

The term "on the nose" is used here for the first time (according to the search function) and is used later on as well a few times. From context I gather it roughly means actual equality (instead of up to isomorphism)? I think it would be helpful for this notion to be defined somewhere concretely (or defined on-the-nose so to speak)


On Haodong Yao left comment #10088 on Section 26.21 in Schemes

In Lemma 01KO, (2) why do we bother saying that " is a finite union of affine opens in ", instead of saying that " is quasi-compact"? Is this to compare with Lemma 01KP, (1)(a) where we require to be affine?

Also personally I think it is more clear and parallel to rewrite Lemma 01KP in the form of Lemma 01KO, i.e. 3 things are equivelent : being separated is equivalent to for any pair of affine opens such that ... , and is equivalent to there is an affine open covering such that ... This is because in this form Lemma 01KO and Lemma 01KP would easily imply that being separated and quasi-separated is local on the target (which is also missing in this tag)

In Remark 0816, "Moreover, if and are morphisms ", do you want to say "are morphisms in "?


On Jhan-Cyuan Syu left comment #10087 on Lemma 21.48.1 in Cohomology on Sites

In the statement, should be a ringed site.


On student left comment #10086 on Lemma 35.35.1 in Descent

In line with the proof of Lemma 02VX, which references Lemmas 01S1 and 01U9, a stronger statement holds: A surjective and flat morphism is a universal epimorphism in the category of schemes.


On Alex Scheffelin left comment #10085 on Remark 47.7.9 in Dualizing Complexes

should be


On Wenqi Li left comment #10084 on Section 51.22 in Local Cohomology

In the paragraph before Lemma 51.22.1, "we denote " all the should probably be .


On Otmar Venjakob left comment #10083 on Section 95.16 in Examples of Stacks

The composition of with is probably given by i.e., the order of and should be changed


On ZL left comment #10082 on Lemma 37.33.1 in More on Morphisms

Typos : the third paragraph is miswritten as . The last paragraph, third line from below "cup product" (missing space).


On left comment #10081 on Item 50

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On René Bruin left comment #10080 on Lemma 10.112.7 in Commutative Algebra

Am I right that there is a typo in the sentence And then we may choose lying over ?

If I understand it correctly, then it should be , since by the going down property, one defines a chain of prime ideals lying over the chain of prime ideals in the sense that the preimage of under the map equals .


On Junyan Xu left comment #10079 on Lemma 10.30.5 in Commutative Algebra

Let me present a proof that does not rely on localizations or quotients (so it applies to semirings as well). This lemma is a special case (where is injective and ) of the following more general going-up result:

If is a homomorphism of commutative semirings and is an ideal in , then for any prime ideal , there exists a prime ideal such that . If is minimal over , then of course .

This result directly follows from the following fact: if is a submonoid of a commutative semiring and is an ideal disjoint from it, then there exists a maximal such ideal containing , and any such ideal is prime. A formalized proof is available at https://leanprover-community.github.io/mathlib4_docs/Mathlib/RingTheory/Ideal/Maximal.html#Ideal.isPrime_of_maximally_disjoint. To obtain the result, just notice that the submonoid is disjoint from an ideal (or subset) iff .

The implication (1) ⇒ (2) in https://stacks.math.columbia.edu/tag/00FL also follows from this general result by taking , so is the kernel of . (1) says that the kernel is contained in the nilradical of , but every prime contains the nilradical, so a minimal prime over the nilradical is the same as a minimal prime of .

In https://stacks.math.columbia.edu/tag/00HY, one can directly show the image is equal to the zero locus of . Every prime ideal in clearly contains , and on the other hand, if a prime ideal contains then there is a prime such that by the general result, so because and is closed under specialization.


On left comment #10078 on Section 5.5 in Topology

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On left comment #10077 on Remark 13.19.5 in Derived Categories

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On left comment #10076 on Remark 13.19.5 in Derived Categories

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On left comment #10075 on Remark 13.19.5 in Derived Categories

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