The Stacks project

Comments 1 to 20 out of 8837 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 Peilin Lee left comment #9516 on Section 29.39 in Morphisms of Schemes

Proof of lemma 29.39.1, in the explaination of condition (1), should be a quasi-coherent module instead of a module?

On Branislav Sobot left comment #9515 on Lemma 10.30.1 in Commutative Algebra

I guess is not the ring generated by these elements, but rather the -subalgebra

On left comment #9514 on Lemma 19.12.4 in Injectives

In the proof, in “then is a functor” one could write '' instead to make it agree with the style displayed in the statement.

When the proof says “the cokernel of is isomorphic to the direct sum of the cokernels of the maps hence acyclic” we are using that is AB5. Specifically, we are using firstly that AB5AB4 (as it was pointed out in #9401) and secondly that in an AB4 category a direct sum of acyclic complexes is acyclic (actually, this is equivalent to the AB4 condition, see Proposition 1 here).

On left comment #9513 on Lemma 19.12.3 in Injectives

Correction to #9512: it suffices to consider some cardinal greater than and such that there is some other cardinal strictly in-between.

On left comment #9512 on Lemma 19.12.3 in Injectives

Logic issues in the construction of the transfinite sequence of subcomplexes . Cases (1) and (3) are independent of previous steps, but it seems (2) is not: it requires to make a choice since is not unique. Instead of transfinite recursion on , maybe what one actually needs is the generalized axiom of dependent choices? Let be an aleph. This axiom says:

[1, Sect. 8.1]. Let be a nonempty set and let be a binary relation such that for every and every -sequence of elements of there exists such that . Then there is a function such that for every .

(One has [1, Theorem 8.1].)

The axioms alone do not provide a transfinite sequence indexed on the class of all ordinals, but rather a transfinite sequence of length equal to an aleph. However, for our proof it suffices to consider some cardinal greater than (as it is hinted in #9511, 4).

A disclaimer: All set theory I know I have learnt it in the last month, from scratch. I hadn't ever been into any kind of foundations of math, logical or set-theoretic ones.

(Lest this comment not parse well in your device, you can consult the original code here.)


  1. T. Jech, The Axiom of Choice

On left comment #9511 on Lemma 19.12.3 in Injectives

  1. In the statement, to match the style used in Lemma 19.12.6 and in Derived Categories, Section 13.31, one could display “K-injective” instead of “-injective.”

  2. In the construction of the subcomplexes , in case (3), is a limit ordinal, we are inadvertently using that has AB5 to get that the filtered colimit of subobjects is again a subobject (maybe it's worth mentioning AB5?).

  3. When the proof says “note that the transition maps in the system are surjective” how is this fact being used? Lemma 12.31.8 does not require the transition maps being surjective as a hypothesis.

  4. In “it is clear that for a suitably large ordinal ” one could add “(take an ordinal greater than ).”

  5. For-posterity details for those who care: The reason we get exact is because is exact (this is equivalent to being injective for all ), so a s.e.s. of cochain complexes is sent to another s.e.s. of cochain complexes . Application of Homology, Lemma 12.13.12 to gives exactness of \eqref{seq} (Homology, Equation

On Branislav Sobot left comment #9510 on Lemma 6.16.3 in Sheaves on Spaces

Isn't it a little bit weird to talk about sheafification in this lemma, since you introduce it only after this subsection?

On left comment #9509 on Lemma 12.31.8 in Homological Algebra

Typos in the proof, second paragraph: in “let ” it should be in the limit, and in “we will find a ”, instead of it should be .

On left comment #9508 on Section 19.11 in Injectives

The previous comments #9501 and #9502 should be inside Tag 19.11.7. I thought I was commenting inside the latter result, my bad.

On left comment #9507 on Lemma 19.12.2 in Injectives

In the proof, I got a little bit confused because means different things in “let and be a morphism” and “ for some .” Maybe in the former case one could write “let and be a morphism” instead? If I got it right, the base case of the downward induction is and one fixes some .

On nkym left comment #9506 on Lemma 59.71.8 in Étale Cohomology

In the proof of (1), the definition of should be replaced by its complement.

On nkym left comment #9505 on Lemma 59.64.4 in Étale Cohomology

Oh you mean the local values in 03RU by "the values"

On nkym left comment #9504 on Lemma 59.64.4 in Étale Cohomology

I thought (2) did not imply (1). For example, let be a geometric point, and .

On Shizhang left comment #9503 on Lemma 76.9.5 in More on Morphisms of Spaces

Proof for finite order thickenings, second paragraph, line 2: should be ``the map is surjective''.

On left comment #9502 on Section 19.11 in Injectives

(Since the previous doesn't currently compile well on my browser, I assume it won't either in other people's browsers, so you may look at the comment's plain text here.)

On left comment #9501 on Section 19.11 in Injectives

Typo: In “note that is a functor” I think it should be instead. This was also pointed out in #7169.

In the proof, after “pick any ordinal whose cofinality is greater than ,” I think we later use that is actually a limit ordinal when we say “then we see that factors through for some by Proposition 19.11.5.” Maybe one should mention some of this? ( will always be a limit ordinal since and by Sets, Comment #9498),

Also, in the transfinite recursion, we are not only defining for each but also a natural transformation for each , right? And such that is injective for all and for . In other words, we are getting a functor , where is the totally ordered class of ordinal numbers and is the wide subcategory of where the morphisms are the natural transformations whose components are all injective maps. After thinking for a while, the following is what I came up with.

For each ordinal number , denote to the set of ordinal numbers (this is , the successor of ). For each ordinal number , we want to define a functor such that for all . The desired functor will be obtained by taking the union of the functors .

We do so by transfinite recursion in .

  • Base step. For , define as .

  • case. Suppose such a functor is defined. For an ordinal , define as where is the natural transformation whose component at is the morphism in the cocartesian square that defines , and in the second-to-last equality we are using the whiskering notation from Categories, Section 4.28. Since is injective, so is the morphism in the second-to-last equality; hence also the morphism in the last equality is injective.

  • Limit case. Suppose is a limit ordinal and that is defined for all . We define , on the one hand, by setting to be the union of the functors for , and on other hand, by For , the map is defined to be the leg at of the limiting cocone \eqref{colim}. Since is AB5, by the Lemma in Comment #9497, is injective.

On left comment #9500 on Section 10.5 in Commutative Algebra


On Jack Gallahan left comment #9499 on Section 10.5 in Commutative Algebra

Is there an implied 0 of the left in the exact sequence that appears in the definition of a finitely presented module?

On left comment #9498 on Proposition 3.7.2 in Set Theory

In case it's any interesting, one could add to the statement “moreover, if , then such an ordinal is necessarily a limit ordinal.” Indeed, for an ordinal , is a sucessor ordinal if and only if .

On left comment #9497 on Proposition 19.11.5 in Injectives

I think the phrase “so suppose to the contrary that all of the were proper subobjects of ” may be deleted. The argument does not do a proof by contradiction.

For the slow-thinkers out there (like me myself) the justification of the sentence “we only need to show that the map ( is a surjection” is because is left-exact (Homology, Lemma 12.5.8) and because of the following

Lemma. Suppose is an AB5 abelian category. If is a direct system in such that is injective for all , then is injective.

Proof. We may assume that is initial in (replace by the directed subset , which is cofinal, and apply Categories, Lemma 4.17.2). The components of the morphism of direct systems are all injective. Hence is injective.