Typo after Definition 02YY: "Note that if T is a schemes".

The hypothesis "in the situation above" has unfortunately quite a few possible interpretations. To author it is probably quite logical which ones are meant. Does it mean for example that one already assumes the possibility to resolve singularities of Y (absolutely) as a prerequisite to have embedded resolution ? I think with a more precisely stated hypothesis, this Lemma could become more usefull as a source for reliable citation.

The hypothesis "in the situation above" has unfortunately quite a few possible interpretations. To author it is probably quite logical which ones are meant. Does it mean for example that one already assumes the possibility to resolve singularities of Y (absolutely) as a prerequisite to have embedded resolution ? I think with a more precisely stated hypothesis, this Lemma could become more usefull as a source for reliable citation.

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The issue mentioned by the above commenter is a good example of the too simple to be simple phenomenon; a cute way to correct the statement is to modify the hypothesis to that for any finite (possibly empty!) family $( x _ i ) _ { i \in N }$ of objects of $\mathcal{I}$, there exists an object $y \in \text{Ob} ( \mathcal{I} )$ and family of morphisms $( f _ i : x _ i \to y ) _ { i \in N }$. (But, alas, it is perhaps more idiomatic to just stipulate additionally that $\mathcal{I}$ is nonempty. What can one do?)

It's also worth noting that (IINM) this claim holds not just in $\textit{Ab}$ but more generally in the category of models of any algebraic theory (in the sense of universal algebra)!

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 $dx_{1},\ldots, dx_{n}$.