Recent comments—The Stacks project

Comments 1721 to 1740 out of 9050 in reverse chronological order.

On Nov 11, 2022 Anonymous left comment #7868 on Section 1.1 in Introduction

Sorry that I have to give a comment here since there is no other place. When I download the book, bold font is not shown in the pdf file (for example: Lemma, section, tag,...). Can you fix it ? Thanks.

On Nov 10, 2022 Anonymous left comment #7867 on Lemma 69.12.3 in Cohomology of Algebraic Spaces

It seems to me that the proof implicitly uses some comparison results between modules on small Zariski and small étale sites. I wonder if adding some links to these results might be good (e.g. Lemma 35.8.10 and Lemma 35.10.2 or Lemma 35.10.3).

On Nov 10, 2022 Yuchong Zhang left comment #7866 on Section 10.12 in Commutative Algebra

The first line of the proof of Lemma 10.12.15 should be $f'(a/s, m)=am/s$ .

On Nov 08, 2022 James left comment #7865 on Lemma 39.9.10 in Groupoid Schemes

Here it could be noted that $g = \text{dim}(A)$ .

On Nov 06, 2022 Luis left comment #7863 on Section 4.2 in Categories

What is the point of the "proof" of Lemma 05SG? It is excessively pedantic. Why not just omit the proof altogether?

On Nov 02, 2022 Adrian left comment #7862 on Section 48.2 in Duality for Schemes

There is missing x to denote stalk in Lemma 0ECM, (3).

On Nov 01, 2022 Sándor left comment #7861 on Section 12.15 in Homological Algebra

There should be a variant of the "upper" canonical truncation that starts with im d^n instead of coker d^{n-1}. That one inherits the cohomology of the original complex starting at n+1 (accordingly Hatshorne in R&D denotes this with an index ">n" instead of "\geq n"). The advantage of that definition is that then one has a short exact sequence of the canonical truncations: 0 --> \sigma_{\leq n} A^\cdot --> A^\cdot --> \sigma_{>n} A^\cdot -->0. A similar ses works with the above definitions of the stupid truncation (though one still needs a shift, but that's not a big deal), but it does not work with the current definition of the upper canonical truncation.

On Oct 31, 2022 Rankeya Datta left comment #7860 on Lemma 15.41.4 in More on Algebra

It's strange to say let $A \to B \to C$ be regular maps, because that makes it seem like the composition is also regular, which is what one wants to show. Perhaps it is better to say let $A \to B$ and $B \to C$ be regular maps.

On Oct 27, 2022 Xiaolong Liu left comment #7857 on Section 44.6 in Picard Schemes of Curves

After the Lemma 0B9U, we may consider $\mathrm{Pic}_{X/k,\sigma}:(Sch/k)^{opp}\to(Ab)$ instead of $\mathrm{Pic}_{X/k,\sigma}:(Sch/S)^{opp}\to(Ab)$ .

On Oct 26, 2022 Rachel Webb left comment #7856 on Lemma 101.45.7 in Morphisms of Algebraic Stacks

How is it used that $f$ is separated?

On Oct 25, 2022 Shota Inoue left comment #7855 on Lemma 15.124.1 in More on Algebra

For the comment #7854, $a'=fa$ is a typo: it should be $fa'=a$ .

On Oct 25, 2022 Shota Inoue left comment #7854 on Lemma 15.124.1 in More on Algebra

The alphabet $a$ already appears in the sentence "To see this suppose ...". So the $a$ in the last displayed equation should be replaced by another symbol (say $a'$ ), and $fa=b$ should be $a'=fa$ (as we want to show the implication $A\cap fB\subset fA$ ).

On Oct 25, 2022 Shota Inoue left comment #7853 on Lemma 15.124.1 in More on Algebra

Typo in the proof of the converse implication: $A=\operatorname{Ker}(F\to P)$ should be replaced by $A=\operatorname{Ker}(B\to P)$ .

On Oct 24, 2022 Sean left comment #7851 on Section 12.19 in Homological Algebra

There is some inconsistency in the choice of symbol for indexing, even in the same definition or proof. Here is a list:

n (or m) * Definition 0121 (1) * Definition 0121 (5) * Definition 0121 (6) * Definition 0121 (7) * First paragraph of proof of Lemma 05SJ

i * Definition 0121 (3) * Definition 0121 (8) * Definition 0121 (9) * Definition 0123 and just below it * Third paragraph of proof of Lemma 05SJ

p * Proof of Lemma 0122 * Second paragraph of proof of Lemma 05SJ * Proof of Lemma 0129 * Proof of Lemma 05SM * Just below the proof of Lemma 05SN * Lemma 05SP and its proof * Proof of Lemma 0127 * Lemma 05QH

n and p * Proof of Lemma 05QH

On Oct 24, 2022 nkym left comment #7850 on Lemma 15.64.2 in More on Algebra

Two alphas in the proof, one from P to K and the other from P to E.

On Oct 22, 2022 DatPham left comment #7849 on Definition 12.31.2 in Homological Algebra

I am a bit confused by the sentence following the above definition, namely that the Mittag-Leffler condition ensures that the $\lim$ -functor is exact if one works with the abelian category of abelian sheaves on a site. Isn't it true that $R\lim \mathbf{Z}/l^n$ is not just $\mathbf{Z}_l$ in the étale topos?

On Oct 21, 2022 Gus Schmidt left comment #7848 on Section 26.14 in Schemes

It may be nice to at least mention that the topology on $X$ in Lemma 01JB is just the standard quotient topology as referenced https://stacks.math.columbia.edu/tag/08ZK.

On Oct 21, 2022 Reimundo Heluani left comment #7846 on Section 59.46 in Étale Cohomology

There's a typo in the proof of 04FV, the scheme $M$ should be defined as the product over $X$ : $M := U \times_X U'$

On Oct 20, 2022 Stéphane left comment #7845 on Section 9.10 in Fields

Being algebraically closed is defined (identically) in both 9.10.1 and 9.10.3.

On Oct 18, 2022 Shang Li left comment #7844 on Section 71.13 in Divisors on Algebraic Spaces

In (5), L should be an invertible O_X module.