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Comments 1 to 20 out of 8812 in reverse chronological order.

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On left comment #9489 on Section 12.25 in Homological Algebra

After Lemma 12.25.1, in “these spectral sequences define two filtrations on . We will denote these and ,” perhaps one could spell out , for . That is, we are plugging into Definition 12.24.5.


On left comment #9488 on Lemma 12.24.6 in Homological Algebra

Analogously as in Comment #9487:

  1. In the statement, instead of “the limit exists” shouldn't we specify “the limit of the spectral sequence of Lemma 12.24.2 exists”?
  2. In the proof, second sentence, instead of , , shouldn't one write , ? (We are working over an abelian category that might not satisfy AB3.)

On left comment #9487 on Lemma 12.23.5 in Homological Algebra

  1. In the statement, instead of “the limit exists” shouldn't we specify “the limit of the spectral sequence of Lemma 12.23.2 exists”?
  2. In the proof, second sentence, instead of , , shouldn't one write , ? (We are working over an abelian category that might not satisfy AB3.)

On left comment #9486 on Lemma 12.23.2 in Homological Algebra

Typo: In the statement, I think the very last equality should be .


On Ziqiang Cui left comment #9485 on Section 26.14 in Schemes

Can you please explain why in the proof of lemma 26.14.1 the commutative diagram imply the isomorphism of sheaves . What I think is all three morphisms in the commutative diagram are isomorphisms and hence induce a sequence of isomorphisms . But I could be wrong.


On Ziyu Lü left comment #9484 on Section 6.33 in Sheaves on Spaces

Maybe a easy question but why in Lemma 6.33.2 is isomorphism. Let be open for fixed . Let then for any other index , is uniquely determined by the image of . In detail, we construct and , where in the second map if , otherwise set . And they are mutually inverses.


On Huang left comment #9483 on Section 17.22 in Sheaves of Modules

Should Lemma 17.22.2.(1) be correct only for being coherent(or finite presentation), then it follows from the stalk isomorphism of intern Hom, while the general case the proof would not go through since surjective of sheaves only implies surjective on the stalk level?


On left comment #9482 on Remark 12.21.5 in Homological Algebra

The diagram should compile like this. I don't know why Mathjax doesn't compile it well here in the Stacks Project (same happened to me in #6829).


On left comment #9481 on Remark 12.21.5 in Homological Algebra

How one can describe exactly how is induced? What I thought is to define as the connecting homomorphism arisen from the snake lemma applied to the commutative diagram with exact rows Note that this also gives a description of in Lemma 12.21.2 if . I don't know if there is another way, though. The footnote seems to be using something else, instead of the snake lemma.


On left comment #9480 on Section 12.22 in Homological Algebra

Maybe in the list of properties (1) – (3) to be found before Definition 12.22.5 one could mention Lemma 12.21.4? It seems it is what one needs.


On left comment #9479 on Section 12.21 in Homological Algebra

(See also [1, Sect. 2.6, Proposition 6.1].)


On left comment #9478 on Section 12.21 in Homological Algebra

I disagree with the “obviously” before Remark 12.21.5 in “note that in the situation of the lemma we obviously have .” The non-obvious thing to me is the fact that taking inverse of subobjects commutes with unions (more generally, it will commute with subobject sums) and that taking image of subobjects commutes with intersections.

It is not a difficult matter, though: Given an abelian category and an object of , denote to the class of isomorphism classes of subobjects of . Then is partially ordered. Given a morphism in , one can define maps which are taking image of a subobject and taking inverse image of a subobject along , see [1, p. 40]. Moreover, the former is left adjoint to the latter [1, Sect. 2.6, Proposition 6.8]. Hence, taking direct images commutes with arbitrary colimits and taking inverse images with arbitrary limits.

References

  1. N. Popescu, Abelian categories with applications to rings and modules

On Lukas Krinner left comment #9477 on Lemma 61.23.1 in Pro-étale Cohomology

I have two small comments on this lemma:

In order to apply Lemma 03Q6 at the end of the proof of (1) we need to be quasi-compact and quasi-separated. \

The last equation in the proof of (1) says: By Lemma 099S we have

To be precise this should be where is the canonical map. Moreover, the used lemma is not 099S but Lemma 0GLZ.


On left comment #9476 on Lemma 10.97.3 in Commutative Algebra

Alternative suggested proof: By Lemma 10.97.2 it is flat. Thus, if is some non-zero -module, to see that it suffices to see that , for finitely generated submodules . By Lemma 10.97.1, . Since , the map is injective by Krull's intersection theorem (Lemma 10.51.4), so implies .


On Olivier Benoist left comment #9475 on Lemma 37.43.4 in More on Morphisms

Typo : in the proof of Lemma 0F2N, should be replaced by .


On Olivier Benoist left comment #9474 on Lemma 37.43.4 in More on Morphisms

Typo : in the proof of Lemma 0F2N, should be replaced by .


On left comment #9473 on Lemma 13.23.3 in Derived Categories

Thank you! I think I just confused myself. I'm curious: why do you say this section is "a bit obsolete"?


On left comment #9472 on Lemma 13.23.6 in Derived Categories

Regarding the last sentence of the proof: I think there are currently no “remarks following Definition 13.23.2.” One could instead substitute this last sentence by “This is true by the commutative square in proof of Lemma 13.23.3. In this square, if is a quasi-isomorphism, then is a quasi-isomorphism too, hence so is ; by Proposition 13.23.1, is an iso in .”

In the statement, one could sharpen:

  1. “then for a unique exact functor...,” since this is what we actually get from the proof (say, from Lemma 13.5.7).

  2. “which is quasi-inverse in the -category of triangulated categories to the canonical functor...” (I wrote the proof below).

Finally, at the end of the proof, shouldn't we say something like “we leave to the reader to check that is quasi-inverse to ” at least? Here's the proof anyway:

First we note that the components assemble into a natural transformation , where is the inclusion. This is witnessed by the commutative square in the proof of Lemma 13.23.3. Now, on the one hand, is isomorphic to the composite since becomes a natural isomorphism when restricted to , i.e., is an isomorphism if (use Proposition 13.23.1). By Comment #9471, this natural isomorphism is in the -category of triangulated categories. On the other hand, leveraging again naturality of it is not difficult to see that is isomorphic to the composite . Again, Comment #9471 implies this natural isomorphism lives in the -category of triangulated categories.


On left comment #9471 on Lemma 13.23.5 in Derived Categories

One could add to the statement something like “Moreover, the morphisms assemble into a -morphism in the -category of triangulated categories (see explanation before Definition 13.3.4).” Indeed, the proof already gives for free.


On Ryo Suzuki left comment #9470 on Lemma 15.91.1 in More on Algebra

In the proof a spectral sequence 15.67.0.1 is used. But why we can use it? There is an assumption that is bounded below in Section 15.67...