There's a typo in the following part of the proof of Lemma 097X, where $T=V(JD)$ should be $T=V(JD' )$ with a prime ' (there, the ring $D$ has not been defined yet):

"the map $T=V(JD)\to \mathop{\mathrm{Spec}}(B)$ is surjective too."

There's a typo in the proof of Lemma 097U, where the last prime ' should be deleted:

"By Algebra, Lemma 04D1 we can write $\overline B=B/IB$ for some étale ring map $A\to B′$"

There are other B' later in the proof, which are fine.

Proposed simpler proof: $\bigcap \mathfrak{m}^n =0$ by lemma 00IP, hence the map $R \rightarrow \bigoplus \mathfrak{m}^n / \mathfrak{m}^{n+1}$ is an inclusion. By lemma 00NO $\bigoplus \mathfrak{m}^n / \mathfrak{m}^{n+1}$ is an integral domain, and hence so is $R$.

Typo in Theorem 03Y3 assumption (b): It should be 'representable by algebraic spaces' instead of 'representable by algebraic paces'.

In 01M9, I believe that $\widetilde M$ is nonzero but the ring of global sections is zero.

I am afraid there is an ambiguity in (46) and (47). Listing them as two different items suggests that they denote different notions. However, in Definition 0518 they are explained to have the same meaning.

I think $G$ should be additive, since Definition 12.12.1 requires the cohomological $\delta$-functors to be additive.

It might by helpfull to clearify that $\bar{k}$ denotes a separable algebraic closure of $k$ in 33.6.4. At leat this is the notation that is used in 33.8.8. Of course this is equivalent to the fact that the base change to an algebraic closure is reduced. Maybe one could use the common notation $k_{s}$ instead.

In the fifth line of the proof, $U$ surjects to $V\times_Y X$ instead of "$U=V\times_Y X$"

I am wondering about the title of this section. Is the word étale missing in the title (étale local on source-and-target)?

In the first example, is there a typo that $f_1 ,..., f_m \in R[x_1,..., x_m]$ should be $f_1 ,..., f_m \in R[x_1,..., x_n]$?

Typo: In the last three lines, $a^{\ast}\mathcal{F}$ is written as $\varphi ^*\mathcal{F}$ for twice.

8808 It is defined in the proof of Lemma 01SD, it is the fibred product of $V$ and $f^{-1}(U)$ along the maps $V \to U$ (given by the restriction of $S' \to S$) and $f^{-1}(U) \to U$ (the restriction of $f\colon X \to S$).

About the argument in Lemma 01SD: you have the following commutative cube $$\xymatrix{ f'^{-1}(V) \ar[rr] \ar[dd] \ar[rd] & & f^{-1}(U) \ar@{-->}[dd] \ar[rd] & \\ & X_{S'} \ar[rr] \ar[dd]^<<<<<{f'} & & X \ar[dd]^f \\ V \ar@{-->}[rr] \ar[rd] & & U \ar[rd] & \\ & S' \ar[rr] & & S }$$ The front, left and right faces are pullbacks by construction. By the pasting law for pullbacks, therefore, so is the back one, which tells you that $f'^{-1}(V) \cong V \times_U f^{-1}(U)$. And the latter is affine by Lemma https://stacks.math.columbia.edu/tag/01JQ . (Might it be worth adding the above explanation in Lemma 01SD?)

It appears that the proof of uniqueness of the contraction $c$ in Lemma 0E8A uses Lemma 0E89 instead of Lemma 0E7C.

Something's wrong with the phrasing of Definition 43.15.1 (tag 0AZV). I think this needs to be rewritten along the following lines: 'In the situation above, [...] if d > dim(Supp(M)), and equal to d! times [...] if d = dim(Supp(M)). For d = dim(Supp(M)) we therefore have [...]'

Is this a typo?? (the sentence after the statement of Lemma 0DLS)

This holds even if $G$ is not smooth over $S$! (←$Y$??)

In the statement of (3) should $Z$ be $Supp(\mathcal{F})$?

Hello, I do not really understand why Lemma 15.78.2 implies that $1:K\to K$ factors through $\sigma_{\geq n}K\to K$. (Prove of Compact implies Perfect)

Is there a reference for the claim (in Remark 0FMN) that the de Rham cohomology sheaves $\mathcal{H}^q_{dR}(X/Y)$ are locally free? Thanks!

In the following sentence, the last $M$ should be an $N$:

"If $M$ is $I$-power torsion, then the image of any map $M\to N$ factors through $M [I^\infty ]$"