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Comments 1 to 20 out of 8248 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 #8863 on Lemma 13.4.16 in Derived Categories

In the statement, second sentence, one can weaken " is an additive full subcategory of " to just " is a non-empty preadditive full subcategory of " (where a preadditive subcategory of a preadditive category is a subcategory with a preadditive structure such that the inclusion functor is additive). Condition (2) implies additivity for :

Pick any object in . By (TR1), the triangle is distinguished. Thus, (2) implies that has a zero object. On the other hand, by Lemma 13.4.11(3) and (TR2), the triangle is distinguished. Now (2) implies that has the direct sum for and . This shows that is additive.


On Xiaolong Liu left comment #8862 on Lemma 10.37.9 in Commutative Algebra

We need replace "" into "".


On Manuel Hoff left comment #8861 on Section 15.11 in More on Algebra

It seems to me that Lemma 0ALJ and Lemma 0CT7 both are special cases of Lemma 0DC7. Also the proofs of all three lemmas are quite similar, they all reduce to Lemma 0ALI.


On Francisco Gallardo left comment #8860 on Section 4.13 in Categories

Proof of Lemma 08LR, last part of the proof. It says ''commutes and if and such that..." It should say . Also, maybe I'm not seeing it, but saying that isn't enough? Should it be ? Thanks in advance.


On Daniel McCormick left comment #8859 on Lemma 92.24.2 in The Cotangent Complex

It seems that and have been reversed in the statement of the lemma. It should say and .


On Sean left comment #8858 on Section 27.8 in Constructions of Schemes

My previous comment was incorrect. The point is whenever vanishes for all suffciently large, we get .


On Noah Olander left comment #8857 on Lemma 31.4.5 in Divisors

It might be nice to add the rephrasing of (2):

(3) contains every associated point of .


On Fiasco left comment #8856 on Section 10.131 in Commutative Algebra

Sorry, I'm caring about lemma 031G. Since if all are invariant, then we have by universal property and Yoneda lemma, where the canonical map comes from lemma 00RS. Briefly, given a projective system of schemes over a base, if we consider(via pull back) all those differential modules on the limit scheme, then we have a correspondence. But if we consider(via push forward) all those differential modules on the base scheme, do we still have a correspondence?


On Et left comment #8855 on Lemma 10.131.6 in Commutative Algebra

At the end of the first proof, it would be helpful to say that we are done in the case thanks to the Leibnitz rule.


On Wataru left comment #8854 on Section 61.10 in Pro-étale Cohomology

The following sentence (right before Lemma 0EVP) is not well parsed:

"The following lemma tells us that the pro-h topology is equal to the pro-ph topology is equal to the V topology."

Or is it intentional?


On left comment #8853 on Section 13.3 in Derived Categories

Since now is only assumed to be an autoequivalence, in sequence 13.3.2.1 not all four consecutive terms are honest triangles, right? (For instance is not an actual triangle, since might not be strictly equal to .)


On left comment #8852 on Section 13.3 in Derived Categories

In case it's worth to point this out, actually TR1+TR2+TR4 implies TR3. See Lemma 2.2 in <cite authors="May, J.P.">May, J.P., The additivity of traces in triangulated categories, Adv. Math. 163, No.1, 34-73 (2001). ZBL1007.18012.</cite>


On left comment #8851 on Proposition 13.5.6 in Derived Categories

Under the risk of sounding nitpicking, I want to ask: when in the statement we say "and such that the localization functor is exact", shouldn't we speficy the natural isomorphism ? (Which just some automorphism of , granted that .) Under each choice of , I think we do get actually different triangulated structures on by imposing exactness of (a unique one for each , yeah, but still different.)

I know that most of the time in these sections when we say that some functor is triangulated we do not specify the natural transformation (that is part of the data of a triangulated functor), although maybe in this case it's worth to do so.


On Wataru left comment #8850 on Lemma 61.8.7 in Pro-étale Cohomology

There's a typo in the following part of the proof of Lemma 097X, where should be with a prime ' (there, the ring has not been defined yet):

"the map is surjective too."


On Wataru left comment #8849 on Section 61.8 in Pro-étale Cohomology

There's a typo in the proof of Lemma 097U, where the last prime ' should be deleted:

"By Algebra, Lemma 04D1 we can write for some étale ring map "

There are other B' later in the proof, which are fine.


On Et left comment #8848 on Lemma 10.106.2 in Commutative Algebra

Proposed simpler proof: by lemma 00IP, hence the map is an inclusion. By lemma 00NO is an integral domain, and hence so is .


On Simon Vortkamp left comment #8847 on Section 80.6 in Bootstrap

Typo in Theorem 03Y3 assumption (b): It should be 'representable by algebraic spaces' instead of 'representable by algebraic paces'.


On Sean left comment #8846 on Section 27.8 in Constructions of Schemes

In 01M9, I believe that is nonzero but the ring of global sections is zero.


On Ignazio Longhi left comment #8845 on Section 10.3 in Commutative Algebra

I am afraid there is an ambiguity in (46) and (47). Listing them as two different items suggests that they denote different notions. However, in Definition 0518 they are explained to have the same meaning.


On Z. He left comment #8844 on Lemma 13.4.22 in Derived Categories

I think should be additive, since Definition 12.12.1 requires the cohomological -functors to be additive.