It should day $\mathfrak{m}_B/\mathfrak{m}_B^2$, currently the 2 is missing.

Trivial off by 1 error: in (1), should be "of dimension $2^{i-1}$". For example, if $i=1$ then $\mathfrak p_i = (x_1)$ and localising there gives a DVR. This does not affect anything anywhere, of course.

Disclosure: this was pointed out to me by an LLM when I was having it do a some surveying for https://adjectivesproject.org. Not sure how you feel about LLM-inspired contributions...

For the record, a characterization on ring epimorphisms is given in Corollary 4. I definitely could not find this in the existing literature on ring epis.

This result could fit to this section (although I don't know if it serves any purpose):

Monomorphisms are local on source and target in the following sence: Let $f:X\to Y$ be a morphism of schemes and let $X=\bigcup_{i\in I}U_i$ and $Y=\bigcup_{i\in I}V_i$ be open covers such that $f(U_i)\subseteq V_i$. Then $f$ is monic if and only if $f_{|U_i}:U_i\to V_i$ is a monomorphism for every $i$.

The proof is not difficult.

Hypothesis (2) may be relaxed to

(2') for any $x\in X$ the ring map $j_x^\sharp:\mathcal{O}_{Y,f(x)}\to\mathcal{O}_{X,x}$ is a ring epimorphism.

Moreover, it turns out monicity is equivalent to (1)+(2'), even in the category of ringed spaces, see here.

Regarding “it is clear that the collection $F_{Zar}$ of open subspaces of $F$ is a set (as $(Sch/S)_{fppf}$ is a site, hence a set).” The argument is foundations-dependent, but the fact the collection $F_{Zar}$ of open subspaces of $F$ is a set is a foundations-independent fact by Properties of Spaces, Lemma 66.4.8. Maybe this other route is worth mentioning?

Inside Definition 66.4.1 it is asserted the non-obvious fact that the collection of points of an algebraic space is a set. Shouldn't one justify this by invoking Lemmas 66.4.2 and 66.4.5?

In the proof, after “choose any field $\Omega''$ and embeddings $\Omega\to\Omega''$ and $\Omega'\to\Omega''$ agreeing on $L$” one could add “see Fields, Lemma 9.6.9.”

Right before Definition 66.4.1, one could replace “we omit the verification that this defines an equivalence relation” by “from Fields, Lemma 9.6.9, we see this defines an equivalence relation.”

Right before Lemma 26.13.3, after “this defines an equivalence relation, because given any set of field extensions $K_i/\kappa$ there exists some field extension $\Omega/\kappa$ such that all the field extensions $K_i$ are contained in the extension $\Omega$,” one could say “see Fields, Lemma 9.6.9, for the case when there are finitely many $K_i$.”

do we really need flatness? One could say that $E\otimes_kF$ is non-zero because it is a $k$-vector space of dimension $\dim_k(E)\cdot\dim_k(F)\gt 0$.

11687 infact you can do better. You can find a Noetherian affine scheme surjecting onto the algebraic space.

Maybe it is worth to remark that an algebraic space $X$ being quasi-compact and locally Noetherian is equivalent to existence of an étale surjection $U\to X$ from a Noetherian scheme $U$.

It is clear that the condition is sufficient. Conversely, suppose $X$ is quasi-compact and locally noetherian. Let $U\to X\leftarrow V$ be étale surjections from a quasi-compact scheme $U$ and a locally noetherian scheme $V$. Then $U\times_XV\to V$ is an étale surjection of schemes. Thus $U\times_XV$ is locally Noetherian. Since $U\times_XV\to U$ is also an étale surjection, $U$ is locally noetherian, i.e., $U$ is Noetherian.

In the proof of Lemma 02HT, the displayed equation $$S\otimes_{S^{\prime}}\Omega_{S^{\prime}/ R}=S\otimes_{R[x_{1},\ldots, x_{n}]}\Omega_{R[x_{1},\ldots, x_{n}]/ R}=\oplus_{j=1}^{m}Sdx_{j}$$ seems to contain a typo. The index should be $n$, not $m$, since $\Omega_{R[x_{1},\ldots, x_{n}]/ R}$ is free with basis $dx_{1},\ldots, dx_{n}$.

In Lemma 04KB (1), should it be " The support of $\sigma$ is closed in $U$ " , instead of $|X|$ ?

The proof of Lemma 052V doesn't need any conditon (like local ring homomorphism), right ?

Ugh, undefined control sequence should read $\check{H}^0(\mathscr{U},\mathscr{F})$.

In the last paragraph before the Theorem, "Given a section $s\in \mathcal{F}(U)$" should be "$s\in \v{H}^0(\mathcal{U},\mathcal{F}$" I think.

It looks like wo don't need the categories to be fibred for statement (1) to hold.