
Comments 1 to 20 out of 3390 in reverse chronological order.

On Daniel Levine left comment #3510 on Section 54.79 in Étale Cohomology

Typo: I believe the the second $\mathbb{P}^1_T$ in the morphism $g':\mathbb{P}^1_T \to \mathbb{P}^1_T$ in Theorem 095T should be $\mathbb{P}^1_S$.

On Dmitrii left comment #3509 on Section 10.120 in Commutative Algebra

In the proof of Lemma 02MJ it says "In this case the left hand side of the formula"; I think this should be replaced by the "right hand side of the formula".

On Yicheng Zhou left comment #3508 on Lemma 27.26.4 in Properties of Schemes

By using direction (2) to (1) of Lemma 28.11.3, one can give a more schematic proof as follows. By hypothesis we have locally $\mathcal{L}\cong\mathcal{O}_X$, therefore the open immersion $\iota: X_s \hookrightarrow X$ is locally given by a principal open set (in an affine neighborhood). By the lemma cited, $\iota$ is affine, therfore $U \cap X_ s = \iota^{-1}(U)$ is affine whenever $U$ is affine.

On Jonas Ehrhard left comment #3507 on Lemma 10.38.12 in Commutative Algebra

By the claimed exactness of the diagram we have $0 = \mathrm{Ker}(M \otimes K \rightarrow M^{(I)})$, and

• $M'' \otimes N = \mathrm{Coker}(M''\otimes K \rightarrow (M'')^{(I)})$
• $M' \otimes N = \mathrm{Coker}(M'\otimes K \rightarrow (M')^{(I)})$
• $M \otimes N = \mathrm{Coker}(M \otimes K \rightarrow ((M)^{(I)})$

Then the snake connects $0 = \mathrm{Ker}(M \otimes K \rightarrow M^{(I)}) \rightarrow \mathrm{Coker}(M''\otimes K \rightarrow (M'')^{(I)}) \rightarrow \mathrm{Coker}(M'\otimes K \rightarrow (M')^{(I)}) \rightarrow \mathrm{Coker}(M \otimes K \rightarrow ((M)^{(I)})\rightarrow 0$.

On Manuel Hoff left comment #3506 on Lemma 10.38.12 in Commutative Algebra

Hi, I think it's better to say "diagram chasing" here than "snake lemma". Seriously though, can somebody explain where the snake is?

On Jonas Ehrhard left comment #3505 on Proposition 10.58.5 in Commutative Algebra

Lemma 00JD (10.54.1) needs $S$ to be Artinian. I don't see why this should be the case, as the ideals

seem to give a possibly infinite descending sequence. $S = J_0 \supset J_1 \supset J_2 \supset \cdots$. This sequence stabilises iff $I^d = I^{d+1}$ for $d \gg 0$, which must not be true. For example consider the localisation $k[x]_{(x)}$ of the polynomial ring at the maximal ideal $(x)$ and $I = (x)$.

On left comment #3504 on Lemma 15.70.7 in More on Algebra

Dear Ravi, fair question. It is the 5th condition in Lemma 15.70.2. I have edited the proof to clarify. See here.

On left comment #3503 on Lemma 45.24.1 in Dualizing Complexes

OK, yes thanks. Fixed here.

On left comment #3502 on Lemma 55.3.2 in Crystalline Cohomology

Thanks very much! Fixed here.

On left comment #3501 on Remark 55.8.7 in Crystalline Cohomology

Thanks and fixed here.

On left comment #3500 on Remark 55.8.6 in Crystalline Cohomology

Yes, this is a mistake. Thanks very much for pointing this out. I have fixed this here.

On left comment #3499 on Lemma 55.6.6 in Crystalline Cohomology

OK, I have checked this proof and it seems OK to me.

But I think there should be another proof of this lemma as well. Namely, we should be able to show directly that given a $D$-module $M$ an $A$-derivation $\vartheta : B \to M$ is the same thing as a divided power $A$-derivation $\theta : D \to M$ using the universal property of $D$. To do this consider the square zero thickening $D \oplus M$ of $D$. There is a divided power structure on $\overline{J} \oplus M$ if we set the higher divided power operations zero on $M$. Consider the $A$-algebra map $B \to D \oplus M$ whose first component is the given map $B \to M$ and second component is $\vartheta$. By the universal property we get a corresponding map $D \to D \oplus M$ whose second component should be the map $\theta$ corresponding to $\vartheta$.

I didn't check this completely, but this should work and we should replace the proof given in the Stacks project by this argument.

On left comment #3498 on Lemma 45.5.7 in Dualizing Complexes

Thanks for this. See change here.

On left comment #3497 on Section 45.3 in Dualizing Complexes

On left comment #3496 on Section 45.2 in Dualizing Complexes

Thanks for the fix. See changes here.

Lemma 45.2.3 is stated as used later, so I think it is fine as is.

On left comment #3495 on Lemma 27.2.2 in Properties of Schemes

This is good. Thanks. Change is here.

On left comment #3494 on Lemma 49.11.3 in Algebraic and Formal Geometry

OK, I have made this change in this case. However, when writing these sections I tried to have some consistency in the numbering of the conditions listed in the lemmas. So I want to be careful in changing this in other lemmas, etc. So unless somebody is going to review the chapter as a whole and look more carefully to see if things can be simplified and/or strengthened (as may very well be the case), I would prefer to leave things as is for now.

On left comment #3493 on Lemma 7.46.11 in Sites and Sheaves

Thanks and fixed here.

On left comment #3492 on Section 104.3 in A Guide to the Literature

Yes, of course. It is impossible to keep lists up to date; the only solution seems to be to not have lists of things. For this kind of thing, I strongly encourse people to just edit the latex file and email me. In this case I minimally added this here. Thanks.

On left comment #3491 on Lemma 10.134.6 in Commutative Algebra

OK, I added the conclusion from Nakyama's lemma. But I kept the other statement as well because it is how I think about it. See change here. Thanks very much.