Stacks project -- Comments

#10092 on tag 0B5I by Chris

Chris — Wed, 16 Apr 2025 02:22:32 GMT

Just to be clear, for Lemma 0B5I, this simple tensor is a priori an elment of $F(D_+(f))\otimes O_X(-nd)$ ?

#10091 on tag 0B5I by Chris

Chris — Wed, 16 Apr 2025 02:21:58 GMT

Just to be clear, for Lemma 0B5I, this simple tensor is a priori an elment of F(D_+(f))\otimes O_X(-nd)?

#10090 on tag 02ZA by Ryan Rueger

Ryan Rueger — Tue, 15 Apr 2025 05:02:17 GMT

The term "on the nose" is used here for the first time (according to the search function) and is used later on as well a few times. From context I gather it roughly means actual equality (instead of up to isomorphism)? I think it would be helpful for this notion to be defined somewhere concretely (or defined on-the-nose so to speak)

#10089 on tag 02ZA by Ryan Rueger

Ryan Rueger — Tue, 15 Apr 2025 05:00:40 GMT

#10088 on tag 01KH by Haodong Yao

Haodong Yao — Sun, 13 Apr 2025 05:40:45 GMT

In Lemma 01KO, (2) why do we bother saying that " $U\cap V$ is a finite union of affine opens in $X$ ", instead of saying that " $U\cap V$ is quasi-compact"? Is this to compare with Lemma 01KP, (1)(a) where we require $U\cap V$ to be affine?

Also personally I think it is more clear and parallel to rewrite Lemma 01KP in the form of Lemma 01KO, i.e. 3 things are equivelent : $f$ being separated is equivalent to for any pair of affine opens such that ... , and is equivalent to there is an affine open covering such that ... This is because in this form Lemma 01KO and Lemma 01KP would easily imply that being separated and quasi-separated is local on the target (which is also missing in this tag)

In Remark 0816, "Moreover, if $X\to Y$ and $Z\to Y$ are morphisms $\mathcal{C}$ ", do you want to say "are morphisms in $\mathcal{C}$ "?

#10087 on tag 0FPQ by Jhan-Cyuan Syu

Jhan-Cyuan Syu — Sat, 12 Apr 2025 01:01:13 GMT

In the statement, $(\mathcal{C},\mathcal{O})$ should be a ringed site.

#10086 on tag 02VW by student

student — Sat, 12 Apr 2025 04:05:57 GMT

In line with the proof of Lemma 02VX, which references Lemmas 01S1 and 01U9, a stronger statement holds: A surjective and flat morphism is a universal epimorphism in the category of schemes.

#10085 on tag 0EGL by Alex Scheffelin

Alex Scheffelin — Sat, 12 Apr 2025 12:57:07 GMT

$\kappa \subset N''/N'$ should be $\kappa \subset N'/N''$

#10084 on tag 0GA6 by Wenqi Li

Wenqi Li — Fri, 11 Apr 2025 02:18:27 GMT

In the paragraph before Lemma 51.22.1, "we denote $d = d(S)=...$ " all the $d$ should probably be $q$ .

#10083 on tag 0372 by Otmar Venjakob

Otmar Venjakob — Fri, 11 Apr 2025 05:07:26 GMT

The composition of $(f,g):(U,b,L)→(U′,b′,L′)$ with $(f′,g′):(U′,b′,L′)→(U′′,b′′,L′′)$ is probably given by $(f'∘f,g∘f∗(g′)),$ i.e., the order of $f$ and $f'$ should be changed