The Stacks project

Ah, yes, of course. Silly me! I will add this as an example soon.

In Lemma 0CD9, it seems that there is a redundant condition that $H^0(C,\mathcal{O}_C) = K$. If $C$ is a projective curve, it is proper. In Tag 0A22, a curve $C/K$ is defined to be a variety of dimension 1 relative to $K$, and is therefore, in particular irreducible. An irreducible proper scheme has only trivial sections, hence the condition $H^0(C,\mathcal{O}_C) = K$ is redundant. Am I missing anything?

A typo: The dimension of $V$ should be $r$ instead of $s$ in lemma 0B0S.

it might be worth noting that the properties of excellence and quasi-excellence can be checked zariski-locally

In the definition of the descent data 023V, the same notation is used for the morphism $f:X\rightarrow S$ and the morphism of descent data.

In the proof of $(3)\Rightarrow(1)$ in "suppose given a factorization $\overline{f}=g_0h_0$ with $g_0,h_0\in A/I[T]$ monic" I would add the (implicit) assumption that $(g_0,h_0)$ is the unit ideal. Then it is clear that we have a decomposition of $B/IB$, which should be $$B/IB=A/I[T]/(g_0)×A/I[T]/(h_0)$$ (the text is missing the quotient by I in the second factor)

There are two instances of "wich" instead of "which" in the first two lemmas on this page.

When showing that the colimit is a henselian pair the following "Since $A^h/I^h=A/IB$" should instead read "Since $A^h/I^h=A/I$".

Right before "the second equality by Lemma ..." in the proof, $O_{X}$ should be $O_{S'}$.

There seems to be typos on the superscript for the $\mathop{\mathrm{Ext}}$-spectral sequence: "$E _ 1^{p, q} = \mathop{\mathrm{Ext}} \nolimits ^ p _ {\mathcal{O}_\mathcal {C}}( \mathcal{K} ^ {-q}, g ^ {-1}\mathcal{I}) \Rightarrow \mathop{\mathrm{Ext}} \nolimits ^ {p + q} _ {\mathcal{O} _\mathcal {C}}( \mathcal{K}^\bullet , g^{-1}\mathcal{I}) = 0.$" should be "$E _ 1^{p, q} = \mathop{\mathrm{Ext}} \nolimits ^ q _ {\mathcal{O}_\mathcal {C}}( \mathcal{K} ^ {-p}, g ^ {-1}\mathcal{I}) \Rightarrow \mathop{\mathrm{Ext}} \nolimits ^ {p + q} _ {\mathcal{O} _\mathcal {C}}( \mathcal{K}^\bullet , g^{-1}\mathcal{I}) = 0.$" Similarly the non-zero terms in the $E _ 1$-page should be $E _ 1^ {q,0}=\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O} _\mathcal {C}}( \mathcal{K}^{-q}, g^{-1}\mathcal{I})$.

Should it be Q_{\alpha}=Q\cap P_{\alpha} in condition (6)?

In the explanation of $\pi$ right after introducing the notation, $i$ should be $c$.

Typo: "The subpresheaf Isom(x,y) whose values over V is the set of isomorphisms..." should be "The subpresheaf Isom(x,y) whose value over V is the set of isomorphisms...".

(Sorry: a finite scheme is not necessarily affine. However, schemes with 1 or 2 points always are.)

A minor point, but shouldn't there be a resource for limits and colimits in the bibliography, or an appendix of sorts? It's a bit jarring that the chapter discusses elementary set theory, but assumes colimits are already known.

I don't know if this may be a triviality, but I think it's interesting to note that the analogue of the result for schemes, "a morphism of schemes is separated iff its fibers are separated schemes," is not true. Even though separated morphisms of schemes have separated fibers, the converse doesn't hold: one considers $X=$ the affine line with doubled origin over a field $k$ and $f:X\to\mathbb{A}_k^1$, the morphism squashing the double origin to the single origin of $\mathbb{A}_k^1$. Then, one verifies that $f^{-1}(t)$ is a singleton for every point $t\in\mathbb{A}^1_k$ distinct from the origin (in the case $t=\eta$ is the generic point, $f^{-1}(\eta)$ is the generic point of $X$) and $f^{-1}(0)$ are the two origins of $X$. (Finite schemes are always affine, whence separated.)

Typo: through the end there is a "neighbourood" missing an 'h'.

Is the nontrivial specialization here means proper specialization? I searched the whole project, and Lemma 0CXG is the first time this concept appears.

I was wondering if in the statement and at the end of the proof, the shift by $-r$ was meant instead of that by $r$.

$1$ is defined as an element of $H^0(\mathcal{Hom}(\mathcal{A}, \mathcal{O}_Y))$, but I guess it is just an element of $\mathcal{Hom}^0(\mathcal{A}, \mathcal{O}_Y)$.