The Stacks project

In the proof of (4), it seems a reference to Lemma 032W is missing for the finiteness of $S_i$ over $\hat{R}/p_i$, from which the finiteness of $S$ over $\hat{R}$ is deduced.

In the proof of Lemma 03I2, I do not see why the first sentence starts the way it does. We do not use $R$ anywhere, nor the projections $t,s$. It is also a bit confusing to have $t$ be both the projection and then use it to define the map for the middle equality.

I believe the proof of Lemma 08XV is a bit confusing, due to a small clash of notations. The $R$-ideal from which we are mapping to $E$ should not be denoted by the same symbol as the set that indexes the direct sum.

$A$ is undefined

It is a little bit anxiety-inducing to see an argument by stalks applied to an infinite product. Maybe it would be clearer to observe that for any open $V \subseteq X$ subordinate to $\mathcal{U}$, the cochain complex of groups $0 \to \mathcal{F} (V) \to \mathfrak{C}^0 (\mathcal{U}, F) (V) \to \mathfrak{C}^1 (\mathcal{U}, F) (V) \to \cdots$ is homotopic to zero, so same is true on stalks a fortiori because $\mathcal{U}$ is an open cover of $X$.

Suggestion to alternate proof of lema 00GZ: a PID is normal, as PIDs are UFDs and UFDs are integrally closed by Gauss' lemma.

The writing should be improved a bit. First, as the comments above suggest, it should be made clear where normality is used. Second, "By linear algebra" should be replaced by a more transparent argument, which also highlights where separability is used (for instance, by separabiliity the bilinear form is non-degenerate and thus $x_1,\dots,x_n$ can be chosen to be orthonormal).

it should be defined as an ideal whose radical is the maximal ideal. (Sorry for two comments)

The notion of an ideal of definition is not defined in the context of a local ring on the stacks project.

The proof of the special case $I=0$ of Lemma 0GLX is not true as written. An equality in $S^{-1}M$ does not necessarily imply the same equality in $M_s$. A small modification solves the problem (see David Lui's comment in https://math.stackexchange.com/questions/3895881/geometric-nakayamas-lemma) : for each $i$, we can write $y_i = \sum (a_{i,j}'/s)x_j$ wich implies the existence of $t_i \in S$ such that $t_i \cdot (s \cdot y - \sum a'_{i,j} \cdot x_j) = 0$. Let $t$ be the product of the $t_i$ ; then this time $s' = st$ works.

It will be better if one wrote $\Psi\colon (B\langle x_t\rangle, IB\langle x_t \rangle+B\langle x_t\rangle_+,\delta)\to (D,\bar{J},\bar{\gamma})$: the statement just implicitly uses $\delta$.

Dear Branislav, In the conventions of the stacksproject, a field is a Dedekind domain.

Perhaps an unnecessary modification that can be ignored: in order to avoid repetition in the proof of points (4) and (5), we can say directly that (4) is a consequence of (5) (even possibly removing point (4)?).

Typo: "esthablished" in the last sentence of the proof should be established

Isn't it possible that $B$ be a field? For example, if $A$ is a DVR and $B$ its fraction field.

suggested slogan 'Excision exact triangle in \'etale cohomlogy'

In the 1st line of the proof, there seems to be a typo: change "$V\to U$ in $S(U)$" to "$V\to U$ in $S(V)$".

The same happens for the 2nd part of the proof.

Even though the canonical morphism $R\Gamma_I(M)\to M$ is not an equivalence, the functor $R\Gamma_Z(M)\to M$ should be right? The latter functor depends only on $V(I)=Z\subset Spec A$ as a topological space?

I agree with David Savitt's comment that (at least for students) it's good to explain why (2) and (3) exclude (4). For (2), Savitt's comment is enough, for (3) a bit more explanation is to observe that depth $\geq 2$ implies $\mathrm{Ext}^i(R/m,R)$ are $0$ for $i\leq 1$. This implies that $0\to R\to R'\to R'/R\to 0$ is injective and $R'$ has $m$-torsion, which contradicts the assumption.

Sorry, the first reference should be 15.59.5, not 20.26.6.