Recent comments—The Stacks project

Comments 1361 to 1380 out of 9050 in reverse chronological order.

On Mar 24, 2023 Xu left comment #8279 on Lemma 15.114.4 in More on Algebra

In the last paragraph of the proof, "is obtained by adjoining $\pi_B/\theta$ " should be " $\theta/\pi_B$ ".

On Mar 23, 2023 Et left comment #8278 on Lemma 10.47.14 in Commutative Algebra

Why does every prime in $\overline{k} \otimes k'$ being maximal imply that every residue field is isomorphic to $\overline{k}$ ?

On Mar 22, 2023 anon left comment #8277 on Section 35.13 in Descent

In the proof of 0EUD I think that the target of $h_i$ should be $T$ instead of $Y$ .

On Mar 21, 2023 Anon left comment #8276 on Definition 10.6.1 in Commutative Algebra

In both (1) and (2) the condition is that something is true as $R$ -algebras. The $R$ -algebra structure on $S$ that this refers to comes from the given map $R \to S$ . So the conditions do refer to the given ring map.

On Mar 21, 2023 Luis Turcio left comment #8275 on Definition 10.6.1 in Commutative Algebra

Maybe what causes confution is that the surjection $R[x_1,\ldots,x_n]\to S$ in (1) and the isomorphism $R[x_1,\ldots,x_n]/(f_1,\ldots,f_m)\cong S$ in (2) both extends the given ring map $R\to S$ . Am I right? If the above is the case I agree with Stacks Project when it says " this is indeed a common practice in mathematics".

On Mar 21, 2023 Laurent Moret-Bailly left comment #8274 on Lemma 15.23.12 in More on Algebra

In the proof, "is annihilated by $\mathfrak{p}R_\mathfrak{p}$ " should be "whose annihilator is $\mathfrak{p}R_\mathfrak{p}$ ".

On Mar 20, 2023 Won Seong left comment #8273 on Section 12.26 in Homological Algebra

There's a typo at the very first of this section 'cateogry'

On Mar 20, 2023 Zhenhua Wu left comment #8272 on Lemma 33.9.3 in Varieties

In (7), ''geometrically reduced and connected'' should be replaced by ''geometrically reduced and geometrically connected'' both in the statement and proof to reduce ambiguity.

On Mar 20, 2023 Wouter Rienks left comment #8271 on Lemma 50.15.7 in de Rham Cohomology

I think it should be $\mathbb{A}^{m + 1}_T = \Spec(P)$ , not $\Spec(B)$ .

On Mar 20, 2023 Xiaolong Liu left comment #8268 on Lemma 59.29.9 in Étale Cohomology

In (4) it should be $\mathcal{F}\mapsto\mathcal{F}_{\bar{s}}$ instead of $\mathcal{F}\mapsto\mathcal{F}_{\bar{x}}$ .

On Mar 17, 2023 Haohao Liu left comment #8265 on Lemma 20.11.1 in Cohomology of Sheaves

It seems that here "covering" means an open covering.

On Mar 16, 2023 Stacks Project left comment #8264 on Proposition 10.35.19 in Commutative Algebra

See Section 10.2.

On Mar 15, 2023 William Sun left comment #8262 on Proposition 10.35.19 in Commutative Algebra

$\mathfrak p = \mathfrak q\cap R$ is the retraction of the prime ideal $\mathfrak q\subset S$ under the (not necessarily injective) ring map $R\to S$ . It might be better to clarify the abuse of notation here.

On Mar 15, 2023 Haohao Liu left comment #8261 on Lemma 30.6.1 in Cohomology of Schemes

It seems to me that the proof works for every abelian sheaves $F=colim F_i$ .

On Mar 15, 2023 Matthieu Romagny left comment #8260 on Lemma 70.15.1 in Limits of Algebraic Spaces

typo in first sentence of proof: a morphism of algebraic spaces

On Mar 14, 2023 Xiao left comment #8259 on Section 110.34 in Examples

What is $M_i$ in the definition of $L_{n,i}=A\otimes_{\mathbb Q}...$ ? (The explanation below 05WK.)

On Mar 14, 2023 Runlei Xiao left comment #8258 on Lemma 42.2.4 in Chow Homology and Chern Classes

In Lemma 42.2.4. the $(M, N, \varphi , \psi )$ should change to $(M,N,0,0)$ if not the following exact sequence in your proof will be not well-defined.

On Mar 13, 2023 Haohao Liu left comment #8257 on Lemma 36.4.5 in Derived Categories of Schemes

Before applying 08D5, we should add "We can work locally on $S$ , so assume $S$ is quasi-compact."

On Mar 13, 2023 Nicolas Weiss left comment #8256 on Lemma 13.18.9 in Derived Categories

One of the typos pointed out above hasn't been fixed in the correction, namely that $B\rightarrow E$ is a q-iso (and not something about $B\rightarrow A$ ).

On Mar 13, 2023 Haohao Liu left comment #8255 on Definition 59.18.1 in Étale Cohomology

It seems that the existence of the fiber product $U_{i_0}\times_UU_{i_1}$ is implicitly assumed.