The Stacks project

Comments 681 to 700 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 Liam left comment #9035 on Lemma 10.43.6 in Commutative Algebra

In Lemma 030U. Why we can choose x1,…,xr+1∈K as in Lemma 030Q. Does separable + finitely generated property impliy separably generated ?

On left comment #9034 on Lemma 10.63.16 in Commutative Algebra

Regarding "the displayed inclusion and equality in the Noetherian case follows from Lemma 10.63.15": I think the displayed inclusion is independent of the invoked lemma (one observes that the -annihilator of in is the -annihilator of in if ), and it is only the converse inclusion what requires the mentioned lemma.

On left comment #9033 on Lemma 10.62.2 in Commutative Algebra

Just to save thinking time, here are the details: by induction in . For there's only prime , and we have , whence , by Lemma 10.40.5. Supposing the result true for , we apply Lemma 10.40.9 and obtain . By the induction hypothesis, ; on the other hand, .

On left comment #9032 on Lemma 10.63.4 in Commutative Algebra

I think one needs to treat the case separately. Something like "the case is the observation that a non-zero element of has annihilator equal to ."

On DU Changjiang left comment #9028 on Lemma 59.73.13 in Étale Cohomology

there is a typo in the proof: in the third line, a right parenthesis is missing. The right one should be

On João Candeias left comment #9027 on Lemma 9.8.9 in Fields

I think this is still not quite correct, as a countable product of countables is not countable (already a countable product of finite sets isn't countable), so in particular if we take , we see that a countable product of copies of has bigger cadinality than that of (which is equal to ). Hence this is not a valid argument.

However, we are only interested in a subset of , which can be obtained as a countable union (all possible degrees ) of finite products (all possible coefficients of the polynomial) of , and this is indeed bounded by .

On James left comment #9026 on Lemma 92.10.2 in The Cotangent Complex

"zero" is repeated twice in the 3rd line of the proof.

On Liam left comment #9025 on Section 26.21 in Schemes

In the proof of Lemma 26.21.7 (b), it's not obvious that the collection of U V forms an open cover, however, we can check this easily by checking that every k-point is in one of this collection for every field.

On Zheng Yang left comment #9024 on Section 10.134 in Commutative Algebra

There is a typo below Defn. 07BN in the description of the cotangent complex. The sentence should be corrected to read: "... the chain complex associated to the simplicial module..."

On jack left comment #9023 on Remark 10.78.4 in Commutative Algebra

Here the formula defining R is recursive, but it should just be C^\inft(\mathbb{R}) of course. 052H has the same problem since it is a copy of this.

On Zhenhua Wu left comment #9022 on Lemma 9.16.3 in Fields

In the last line, the function should be .

On left comment #9021 on Section 12.6 in Homological Algebra

@#8412 Thanks and fixed here.

On left comment #9020 on Lemma 10.153.3 in Commutative Algebra

Thanks to the comments 8410, 8561, 8796, 8797. Fixed here.

On left comment #9019 on Section 10.36 in Commutative Algebra

Thanks and fixed here.

On left comment #9018 on Proposition 13.16.8 in Derived Categories

Thanks, this is indeed better. Fixed here.

On left comment #9017 on Theorem 58.6.2 in Fundamental Groups of Schemes

Hahaha! Yes, of course. The notation is that is the map between groups and that the functor between the categories of sets with group actions is supposed to be induced by . See Lemma 58.3.11. Going to leave as is for now.

On Zhenhua Wu left comment #9016 on Lemma 9.16.6 in Fields

The part ''It follows from Lemma 09HQ that is normal over and that it is the smallest normal subextension of containing needs clarification. To be specific: 1)why is an field; 2)how do we use lemma 09HQ to show is normal; 3) how do we show that it is the smallest normal subextension of containing .

On James left comment #9015 on Section 105.7 in Introducing Algebraic Stacks

Another mistake (mine of course): the dimension calculation above is 6-5. Apologies for the spam.

On left comment #9014 on Lemma 13.16.7 in Derived Categories

Thanks and fixed here.

On left comment #9013 on Lemma 13.16.5 in Derived Categories

Thanks and fixed here.


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