Comments 1 to 20 out of 3989 in reverse chronological order.


On Dario Weißmann left comment #4223 on Remark 55.6.10 in Crystalline Cohomology

Concerning the definition of the maps $\nabla:M\otimes \Omega^i \to M\otimes \Omega^{i+1}$:

Several $\otimes$ should be $\wedge$, i.e.,

On Dario Weißmann left comment #4222 on Lemma 55.6.3 in Crystalline Cohomology

Concerning the second paragraph of the proof. I suggest replacing "We will show this by induction ...." (till the end of the paragraph) by the shorter (and easier) direct calculation: The claim is clear for $n=1$. Assume $n>1$. Note that $\delta_i(z-y)$ lies in $(K\cap J(1))^{[2]}$ for $i>1$. Calculating modulo $K^2 + (K\cap J(1))^{[2]}$ we have The claim follows.

On left comment #4221 on Lemma 54.70.8 in Étale Cohomology

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On Dario Weißmann left comment #4219 on Remark 55.13.2 in Crystalline Cohomology

Typo in the definition of the multiplication: $f \omega_2'+f'\omega_2'$ has too many $'$s

There is also an issue how the multiplication formula is displayed. It overlaps with the sidebar. Works fine in the pdf version though.

On left comment #4217 on Lemma 54.78.7 in Étale Cohomology

(And I gave a short proof of the fact that I don't know how to count...)

On left comment #4216 on Lemma 54.78.7 in Étale Cohomology

4th sentence of the proof: sch -> such.

On Frank left comment #4215 on Section 22.23 in Differential Graded Algebra

It seems better to clarify the second property of graded tensor product: the $p,q$ in $x\in M^p,y\in N^q$ are different from those in the direct sum $\bigoplus_{p+q=n}$. In other words, here $p+q$ is only assumed to be no greater than $n$. It seems better to choose different names for them.

On David Holmes left comment #4214 on Lemma 60.4.6 in Properties of Algebraic Spaces

I came here to make the same comment as Wessel, but then I saw Wessel's comment, so I won't (maybe I just did...). So maybe count this as another vote for changing it??

On 羽山籍真 left comment #4213 on Lemma 62.9.1 in Decent Algebraic Spaces

For the last sentence of the proof, $V$ has been used to denote an open of $Y$, so it's not a subspace of $Z$ indeed, maybe replace it by $V\prime$?

On Sean Cotner left comment #4212 on Lemma 34.20.16 in Descent

Small point: after replacing S by f(X), you still use the assumption that S is affine. I think this can be fixed by instead base changing to an arbitrary affine open contained in f(X), after which the rest of the proof goes through.

On Aaron left comment #4211 on Lemma 10.29.5 in Commutative Algebra

I think it should be: $\text{Spec}(S)\to\text{Spec}(R)$ hits all the minimal primes.

On Che Shen left comment #4210 on Section 30.14 in Divisors

0B3P follows immediately from 01R3 (1). Maybe we can use this as proof instead of "omitted"?

On Zhiyu Zhang left comment #4209 on Section 48.16 in Local Cohomology

Kunz theorem is pretty good, will other properties about Frobenius action (and singularity) be added in the future? For example, Remark 13.6. in Ofer Gabber. Notes on some t-structures. In Geometric aspects of Dwork theory. Vol. I, II, pages 711–734. Walter de Gruyter GmbH & Co. KG, Berlin, 2004. shows that F-finiteness on a noetherian ring $A$ with $p=0$ will imply $A$ is quotient by a regular ring hence the existence of dualizing complex. The proof is very short and elegant.

On 羽山籍真 left comment #4208 on Lemma 32.41.3 in Varieties

The second last paragraph: "we obtain a specialization \eta' to t_j with \eta' \in X^0". Here \eta' is in T^0 (not X^0), right?

On HayamaKazuma(羽山籍真) left comment #4207 on Lemma 32.41.1 in Varieties

（2）Is it be f|_{U_i}: U_i \rightarrow Y ?

On slogan_bot left comment #4206 on Lemma 15.50.7 in More on Algebra

Suggested slogan: "Henselization of a ring inherits good properties of formal fibers"

On Pierre left comment #4205 on Section 10.130 in Commutative Algebra

In lemma 00RW, at some point the conclusion is that $\Omega _{S \otimes _ R S/S} = \Omega _{S/R} \otimes _ S (S \otimes _ R S)$ which is said to follow from lemma 00RV, but in lemma 00RV the tensor product is over the base change $R\to S$, and in the 00RW the tensor product is not over the base change but over the induced map $S\to S\otimes_R S$. Is that a problem or am I missing something?

On Nicolas Müller left comment #4204 on Lemma 54.80.8 in Étale Cohomology

I have a hard time understanding how Lemma 0959 is used here to prove that $j_!f_* f^{-1}\mathcal{G} = \overline{f}_* j'_!f^{-1}\mathcal{G}$. Instead, wouldn't it be better to combine Propositions 03QP and 03S5 to show that both sides agree on the stalks? Then one also doesn't need to argue that the diagram is cartesian. (Also, at least in the preview, the LaTeX doesn't render properly in this comment.)

On left comment #4203 on Lemma 10.95.1 in Commutative Algebra

In Lemma 81.4.5 in Section 81.4 the topology on the submodule $K$ is the topology inherited from $M$ (it is given by the submodules $K \cap I^nM$). But in the current Section 10.95 there is no mention whatsoever of topologies or completion with respect to any topology. We are just considering $I$-adic completion straight up. Maybe we should be a little bi more careful in the statement of Lemma 81.4.5. Thanks for the comment.

On left comment #4202 on Section 10.96 in Commutative Algebra

@#4200 Well, it may be that we can shorten the proof at the end there, but flat surjections of rings aren't necessarily isomorphisms.