The Stacks project

In Lemma blowup commute with flat base change it should be "X_i' are the blowups of X_i in Z_i" and not viceversa

Where it says any group scheme over a field is separated it refers to Lemma 79.9.4 whereas it should be Lemma 39.7.3.

$f$ is a morphism of scheme and not of algebraic spaces!

In the proof, $s$ is a section of $L^{\otimes n}$ and $t$ is a section of $N^{\otimes m}$, so $r$ is not in general a section of $M$ (or even of a tensor power of $M$ if $n \neq m$). Should one first replace $s$ by $s^{\otimes m}$ and $t$ by $t^{\otimes n}$? Then both are sections of the $nm$-th tensor powers, and $r$ is a section of $M^{\otimes nm}$, with $$(X \times Y)_r = X_s \times Y_t.$$

Potential typo: the last "if and only if" of the third sentence is supposed to be somehow separated from the one before, it seems. Maybe it should be "Thus $\mathrm{Ker}(f)$ is $0$ if and only if $\mathrm{Mor}_A(z,x) \to \mathrm{Mor}_A(z,y)$ is injective for all $z$, i.e., if and only if $f$ is a monomorphism." or something similar.

Very nitpicky, but I am not sure that the second sentence after the diagram in the proof of Lemma 12.3.4 is grammatically correct. Instead of "suppose given morphisms", I think it's supposed to be "suppose we are given morphisms" or "suppose there are morphisms". I apologize if this is stupid.

Paragraph 2: "continuous functors" should be "continuous functions"

In the last sentence of the proof, what if neither $j\leq k$ nor $k \leq j$?

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?