The Stacks project

Comments 1181 to 1200 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 Et left comment #8480 on Lemma 22.5.4 in Differential Graded Algebra

This proposition isn't hard to prove directly, but I wonder where you use : "and because ∏ is an exact functor on the category of families of abelian groups."?


On Minotaurus left comment #8479 on Section 20.25 in Cohomology of Sheaves

This article is not well written. Namely, the Cech hypercohomology is NOT a limit, but a colimit or a direct limit. Why haven't you given the formula to show what you mean by a limit? You are confusing the reader. You should correct everywhere the deceptive reference to a limit.


On Et left comment #8476 on Lemma 10.99.11 in Commutative Algebra

Instead of saying that there is a module map , it would be better to say that the map factors through the surjective map


On Et left comment #8475 on Lemma 10.134.12 in Commutative Algebra

Is the map a splitting? If so it would be clearer to just deduce injectivity from that.


On Et left comment #8474 on Lemma 10.134.4 in Commutative Algebra

Suggestion: since this proposition uses multiple rings, maybe add a subscript for the tensor products indicating over what they are taken


On Et left comment #8473 on Lemma 10.99.9 in Commutative Algebra

Where is the induction here specifically done? Wouldn't it be enough just to replace R by R/I^k+1, M by M/I^k+1 and I by I/I^k+1 and apply the precceding lemma?


On Et left comment #8472 on Lemma 10.99.8 in Commutative Algebra

Maybe note in the almost last paragraph that you are using the snake lemma? Took me a moment to figure out what you were trying to do


On Paul left comment #8471 on Lemma 6.29.1 in Sheaves on Spaces

I guess in the proof of (2) and (4) the notation was heavily relaxed, since why should hold, actually it holds for some restriction of s and s'. Also it should be or better of some . I recommend to mention it or rework completly the proof.


On Laurent Moret-Bailly left comment #8470 on Lemma 10.5.4 in Commutative Algebra

In the proof, the minimality of is not needed, so "a finite set of generators" would be more to the point. Also, it would be worth pointing out that the 's are finite modules, and perhaps that the last quotient is finitely presented if is.


On Ryo Suzuki left comment #8469 on Lemma 34.9.2 in Topologies on Schemes

I think Lemma 03GI is used to deduce is quasi-compact. It might worth to note it explicitly.


On left comment #8468 on Lemma 26.21.9 in Schemes

For the sake of having some reference that is actually instructive, here's a neat proof of the cartesianity of the square https://mathoverflow.net/a/80812/101848


On left comment #8467 on Lemma 26.21.7 in Schemes

Minor typo: instead of in "we see that can be identified with ," one should write .


On left comment #8466 on Lemma 26.18.2 in Schemes

I think it could be nice to reference https://stacks.math.columbia.edu/tag/001U#comment-3413 to make explicit the formal nature of the argument (I believe such little hints are quite useful for those recently initiated to algebraic geometry—and often to category theory too—reading this webpage, like the version of myself one year ago).

An alternative reformulation of the proof would be by placing this lemma after remark https://stacks.math.columbia.edu/tag/01JW#comment-8464 and then just invoking the remark plus 26.17.6.


On left comment #8465 on Lemma 26.17.6 in Schemes

In the last sentence, I would give the hint "the third is a combination of the first two, plus the pasting law for pullbacks, https://stacks.math.columbia.edu/tag/001U#comment-3413 ."


On left comment #8464 on Section 26.18 in Schemes

I think the following could be an interesting remark to add after 26.18.3: Let be a property of morphisms of schemes. Then is preserved under arbitrary base change as in 26.18.3 (1) if and only if it is preserved under arbitrary base change as in 26.18.3 (2). To see (), use the diagram of the proof of 26.18.2, where the top square is cartesian. To see (), use again the same diagram but with , so .


On Abhishek Jha left comment #8463 on Section 9.8 in Fields

In the proof of lemma 9.8.5, it should say that the elements are linearly dependent over K.


On left comment #8462 on Section 26.10 in Schemes

I think it would be informative if after «we usually just say “let be a locally closed subscheme of ” since we may recover from the morphism ,» one added «namely,


On M left comment #8461 on Section 10.113 in Commutative Algebra

At the start of the proof of Lemma 02IJ, "Suppose that R⊂S′⊂S is a finitely generated R-subalgebra of S." is slightly odd in that it gives the initial impression that the string of inclusions is a finitely generated R-subalgebra of S (although it easy to decipher the meaning a moment later). I propose "Suppose that R⊂S′⊂S, where S' is a finitely generated R-subalgebra of S."


On ZL left comment #8459 on Lemma 66.27.3 in Properties of Algebraic Spaces

A typo: in the fourth line, "" should be .

Here is identified with the localization morphism in 66.18.11


On ZL left comment #8458 on Section 7.28 in Sites and Sheaves

I would like to ask how strict the equalities "" in the section are?

To elaborate my question, first suppose that we have morphisms of topoi , and . We say if and . However, if we only know , then by adjunction we know that there exists a canonical isomorphism . Hence we get a canoical -isomorphism .

In this regard, for example in lemma 7.28.4, we have . Hence we have . However, we only have canonical isomorphisms , for and both involving taking inductive limits. Hence what seems to me is that we have a canonical isomorphism .

In the same manner, in lemma 7.28.1, we verified the equlity to conclude that . But again by adjunction of functors, we a priori only knows that canonically.

Is there a remedy for this problem?