Recent comments—The Stacks project

Comments 821 to 840 out of 9050 in reverse chronological order.

On Apr 09, 2024 Stacks project left comment #8892 on Lemma 5.8.16 in Topology

Sorry, but I could not find this in EGA part I. Can you be more precise with the reference?

On Apr 09, 2024 Stacks project left comment #8891 on Example 15.8.5 in More on Algebra

Thanks very much! I put these in their own example environment. Changes are here.

On Apr 09, 2024 Stacks project left comment #8890 on Section 37.53 in More on Morphisms

Thanks and fixed here.

On Apr 09, 2024 Stacks project left comment #8889 on Lemma 18.32.2 in Modules on Sites

Thanks and fixed here.

On Apr 09, 2024 Stacks project left comment #8888 on Lemma 20.6.1 in Cohomology of Sheaves

Thanks and fixed here.

On Apr 09, 2024 Stacks project left comment #8887 on Section 97.15 in Criteria for Representability

Thanks and fixed here.

On Apr 08, 2024 Laurent Moret-Bailly left comment #8886 on Lemma 4.14.10 in Categories

More precisely, if one of the double colimits exists then it is the total colimit.

On Apr 03, 2024 jhzg left comment #8885 on Section 4.14 in Categories

How to prove Lemma 002M? Is this lemma certain to be correct？

On Mar 28, 2024 Awllower left comment #8883 on Lemma 11.6.2 in Brauer groups

I do not understand why the proof tries to show that $A$ is a finite central simple algebra over the center of $A$ , which should be $k$ by the assumptions.

On Mar 27, 2024 Siyuan Zheng left comment #8882 on Definition 8.3.1 in Stacks

Sorry, I realized the map in the diagram comes from the triple fiber product which is different from the map coming from the double fiber product.

On Mar 27, 2024 Siyuan Zheng left comment #8881 on Definition 8.3.1 in Stacks

$\varphi_{ij}$ is defined to be a map from $\mathrm{pr}_0^\star X_i$ to $\mathrm{pr}_1^\star X_j$ . However, in the diagram, the map from $\mathrm{pr}_0^\star X_i$ to $\mathrm{pr}_{1}^\star X_j$ is labeled as $\mathrm{pr}_{ij}^\star\varphi_{ij}$ .

On Mar 27, 2024 Ryan Rueger left comment #8880 on Section 6.11 in Sheaves on Spaces

| Also we will say $s=s′$ in $\mathcal{F}_x$ for two local sections of $\mathcal{F}$ defined in an open neighbourhood of $x$ to denote that they have the same image in $\mathcal{F}_x$ .

I think it would be clearer to write "... that they are in the same equivalence class of $\mathcal{F}_x$ ".

I understand that the terminology of an "image" of a section is used often e.g. in tag01CY.

So it would be good idea to then additionally explain that we say "image" of a section to mean the "class" of the section in the stalk in this context.

I think being explicit here is useful, because sections are often set-theoretic maps (e.g. sections of the structure sheaf of a variety), and the term "image of section" is somewhat overloaded. Of course it is still discernable from context when we write "image $s \in \mathcal{L}_x$ " (like in the linked Lemma (Tag 01CY)). It is clear that we don't literally mean the set-theoretic image, but something else. Still, I believe it is worth describing this explicitly.

On Mar 26, 2024 Shuai Wei left comment #8879 on Lemma 26.24.3 in Schemes

Correction: If we take $U ' = b(V)$ , which is \textbf{open} in $U$ , it works. Actually, we see that $Z \to V \to b(V)$ is a composition of closed immersions and hence a closed immersion.

On Mar 26, 2024 Shuai Wei left comment #8878 on Lemma 26.24.3 in Schemes

If we take $U ' = b(V)$ , which may not open in $U$ , it works. Actually, we see that $Z \to V \to b(V)$ is a composition of closed immersions and hence a closed immersion.

On Mar 26, 2024 Kubrick left comment #8877 on Lemma 13.27.8 in Derived Categories

I think $C = \mathop{\mathrm{Ker}}(Z_{p - 1} \to Z_ {p-2})$ .

On Mar 26, 2024 ZW left comment #8876 on Remark 64.18.5 in The Trace Formula

I think one can prove that such $M$ is $\ell$ -adically complete, and then use induction to show that $M$ is finitely generated by $M/\ell M = M_1$ . Maybe this is what is meant by Nakayama.

On Mar 25, 2024 Zhenhua Wu left comment #8874 on Lemma 10.50.17 in Commutative Algebra

Sorry for the last comment. Actually from the definition of ideals of $\Gamma$ we can see that $\emptyset,\Gamma_{> 0}$ and $\Gamma_{\geq0}$ can all be ideals. They correspond to $\{0\},\mathfrak{m}$ and $A$ respectively. But $A$ is not a prime ideal, so we shouldn't allow $\Gamma_{\geq0}$ to be a prime ideal of $\Gamma$ .

On Mar 24, 2024 Eoin left comment #8873 on Lemma 37.13.11 in More on Morphisms

In the proof, in the cone of the first equation, the terms $NL_{X/Y}$ and $NL_{X/Z}$ are swapped.

On Mar 24, 2024 Eoin left comment #8872 on Lemma 42.59.10 in Chow Homology and Chern Classes

In the proof, the name g is being used for both a morphism $g:Y'\rightarrow Y$ and a morphism $g:P\rightarrow Y$ . Similarly with $g'$ .

On Mar 22, 2024 Zhenhua Wu left comment #8871 on Lemma 10.50.14 in Commutative Algebra

$a+b$ could happen to be $0$ which is out of the domain of definition. We could replace $\Gamma$ by $\Gamma\cup \{\infty\}$ and set $v(0)=\infty$ to avoid that. Actually the three properties also work in $v:K\to \Gamma\cup \{\infty\}$ .