The Stacks project

See Definition 59.33.2 for notation used.

Thanks and fixed here.

Yes indeed. Fixed here.

Thanks and fixed here.

Typo: first line after Definition 34.3.15 "a sheaf $\mathcal{G}$ on the small site".

Thanks for answering my question and share your philosophy on how to deal with the $2$-morphisms! The canonical $2$-arrows sometimes really bug me a lot.

For the equalizer $\mathrm{eq}(a\rightrightarrows b)$, I don't see why $V \times _{\Delta _{V/T}, V \times _ T V, (a, b)} U \times _ T U$ is a sub-object of $U$. Do you possibly mean $V \times _{\Delta _{V/T}, V \times _ T V, a \times b} U$?

Thanks for the clarification. I was being silly!

OK, yes, good point -- that was an oversight. I fixed this by first shrinking the schemes $U_j$ such that $s$ and $s_j$ agree on all of $U_j \times_X Z$. I also improved (I hope) the notation a bit. See changes here.

Where the notation $V(\mathcal{I})$ for an ideal sheaf $\mathcal{I}$ over a ringed space $X$ means $\operatorname{supp}(\mathcal{O}_X/\mathcal{I})$.

In case this is any useful: the quasi-coherent ideal sheaf associated with the closed subscheme from 29.5.4 and 29.5.5 is $\operatorname{Ann}_{\mathcal{O}_X}\mathcal{F}$ from 17.23.1. Specifically:

Suppose $X=\operatorname{Spec}A$ is affine and let $M$ be an $A$-module. Then we have a containment of ideal sheaves of $\mathcal{O}_X$ $$\widetilde{\operatorname{Ann}_AM}\subset\operatorname{Ann}_{\mathcal{O}_X}\widetilde{M},$$ with equality if $M$ is $A$-finite.

Proof. Since $S^{-1}\operatorname{Ann}_AM\subset\operatorname{Ann}_{S^{-1}A}S^{-1}M$, in particular for $f\in A$ we have that every element of $(\operatorname{Ann}_AM)[f^{-1}]$ kills every section of $\widetilde{M}$ over $D(f)$. Thus, we have a containment $$\tag{1}\label{1} \widetilde{\operatorname{Ann}_AM}\subset\operatorname{Ann}_{\mathcal{O}_X}\widetilde{M}$$ of ideal sheaves of $\mathcal{O}_X$. Taking stalks at $x\in X$, we get containments $$\label{2}\tag{2} (\operatorname{Ann}_AM)_x\subset(\operatorname{Ann}_{\mathcal{O}_X}\widetilde{M})_x\subset\operatorname{Ann}_{\mathcal{O}_{X,x}}\widetilde{M}_x,$$ the first containment comes from \eqref{1} and the latter is 17.23.1.1. If $M$ is finite, then the first term equals the third one (i.e, $(\operatorname{Ann}_AM)_\mathfrak{p}=\operatorname{Ann}_{A_\mathfrak{p}}M_\mathfrak{p}$, where $\mathfrak{p}\subset A$ is the prime associated to $x$ [ref]), so \eqref{2} becomes an equality and thus \eqref{1} too. $\square$

Moreover, for $X$ any ringed space and $\mathcal{F}$ any quasi-coherent $\mathcal{O}_X$-module, it holds $\operatorname{supp}\mathcal{F}\subset V(\operatorname{Ann}_{\mathcal{O}_X}\mathcal{F})$, with equality if $\mathcal{F}$ is of finite type. The containment follows from 17.23.1.1, whereas equality follows from 17.23.2.

An email by Bogdan points out that the argument showing that $B'$ is rig-etale over $(A, (a_t))$ needs to be improved. I hope to fix this soon.

@#9295 Anytime :)

In the commit you linked there's a typo: 'Yoneday' instead of 'Yoneda.'

OK, the typo is that a subscript $0$ was missing on the $M$ in the sentence, right? I fixed that. But I think it immediately follows from that sentence that $\text{id} \to f_* f^*$ is is an isomorphism. I did add another reference to the first sentence of the last paragraph to clarify. Changes here.

In the end of proof of lemma, why does $s\mid_{Z'}$ Zariski locally come from a section of $\mathcal{F}\mid_{X′}$ ? And why are these local sections compatible such that they can glue altogether?

For the first question, let's be more careful(with the same notations). Let $W\subset X'$ be an open subscheme which factors through $U_{\alpha}$ for some $\alpha$. So $s_{\alpha}$ defines a section of $\mathcal{F}\mid_{W}$ named $t_{\alpha}$, we want to show its restriction is equal to $s\mid_{Z\times_{X}W}$. Note that $Z\times_{X}W$ has an etale cover $\coprod V_{j}\times_{X}W$, So we only need to check each piece $V_{j}\times_{X}W$. We look at the stalks and choose a geometric point $pt\in V_{j}\times_{X}W$.

On the one hand, the stalk of restriction of $t_{\alpha}$ is just the stalk of $s_{\alpha}$ at pt, which is considered as a geometric point of $U_{\alpha}$.

On the other hand, the stalk of $s\mid_{ V_{j}\times_{X}W}$ is just the stalk of $s_{j}$ at pt, which is considered as a geometric point of $U_{j}$.

Now the key point is we only know pt factors through $U_{\alpha}\times_{X}U_{j}$. But by definition $s_{\alpha}$ and $s_{j}$ coincide at some scheme $U$ which is etale over $X$ such that $V_{\alpha}\rightarrow X$ and $V_{j}\rightarrow X$ both factor through $U$. So $U$ can be strictly "smaller" than $U_{\alpha}\times_{X}U_{j}$.

Good catch! OK, I rewrote the proof to make it work in the non-Noetherian case. More interesting was the case of Lemma 31.28.4. Finally, Lemma 31.31.3 was unfixable and I needed to assume the Noetherian assumption. See these changes.

Thanks and fixed here.

Fixed.

Dear Cop 223, I understand what you are saying, but in the setup as in this chapter, I do not think this "helps".

Yes. Going to leave as is.