The Stacks project

I think it is interesting to give reasons to care for the definition of a $2$-morphism in the category of triangulated categories (given before Definition 13.3.4), instead of just staying with plain natural transformations. I will call a $2$-morphism as the one before Definition 13.3.4 a trinatural transformation (as other authors already do). Given a pre-triangulated category $\mathcal{D}$, I will denote $\operatorname{t}(\mathcal{D})$ to the category of triangles of $\mathcal{D}$, and $\operatorname{dt}(\mathcal{D})\subset\operatorname{t}(\mathcal{D})$ to the full subcategory of distinguished triangles. A triangulated functor $F:\mathcal{D}\to\mathcal{D}'$ of pre-triangulated categories induces functors $\operatorname{t}(F):\operatorname{t}(\mathcal{D})\to\operatorname{t}(\mathcal{D}')$ and $\operatorname{dt}(F):\operatorname{dt}(\mathcal{D})\to\operatorname{dt}(\mathcal{D}')$. If $G:\mathcal{D}\to\mathcal{D}'$ is another triangulated functor, then a trinatural transformation $\eta:F\Rightarrow G$ induces natural transformations $\operatorname{t}(\eta):\operatorname{t}(F)\Rightarrow\operatorname{t}(G)$ and $\operatorname{dt}(\eta):\operatorname{dt}(F)\Rightarrow\operatorname{dt}(G)$. That is, if $A\to B\to C\to A[1]$ is a (distinguished) triangle in $\mathcal{D}$ then we have a morphism $\require{AMScd} \begin{CD} F(A)@>>> F(B) @>>> F(C)@>>> F(A)[1]\\ @VVV@VVV@VVV@VVV\\ G(A)@>>> G(B) @>>> G(C)@>>> G(A)[1] \end{CD}$ of (distinguished) triangles in $\mathcal{D}'$. We remark this is possible only if $\eta$ is trinatural, just naturality is not enough. (Note the assignments $\operatorname{t}(-)$ and $\operatorname{dt}(-)$ become $2$-functors from pre-triangulated categories to categories.) In particular, if $\eta$ is a trinatural isomorphism, then the triangle on top of the last diagram is distinguished if and only if the bottom one is.

I would also include (somewhere) the fact that if $M$ is finite, $j$ has torsion kernel and cokernel because $j\otimes\mathrm{Frac}(R)$ is an isomorphism. (This is inspired by comment #9794).

According to section 15.88, this should include a definition of torsion modules.

After Definition 0AV4 of the reflexive hull, it is claimed that the reflexive hull functor is left adjoint to the inclusion functor, which I've failed to write down a proof. I know that any homomorphism $f\colon M\to N$ into a reflexive module $N$ factors through $M^{\ast\ast}$, but how do I show that such induced map $\tilde{f}\colon M^{\ast\ast}\to N$ is unique?

Typo: extra ')' at the beginning of the proof.

Where it reads "which is the direct sum of the coprojections $\mathcal O_S\to \mathcal O_S^{\oplus n}$ corresponding to elements of $i$", shouldn't it be "corresponding to the elements $i\in I$"?

(Unimportant remark) in the first paragraph does "the construction of the previous section" refer to the relative spec construction of sections 27.3-4 (two sections ago)?

For the interested reader, a proof in terms of ind-completions may be read here.

It seems that 04GU is used silently to get the morphism $\mathrm{Spec}(R^{\mathrm{sh}})\to \Spec(A)$.

In the proof of 00LN, is the induction on $\ell$ descending or ascending?

Where is it needed that $g$ is etale?

Line 2 of the proof: ... we can choose a prime $p'$ (not if but) in...?

The $R'_{p'}$ in (1) should be $(R'_{p'})^{\wedge}$.

For (7) and (8), didn't you mean as a $R_{f_i}$-algebra? It seems proven in this case, and used that way in the proof and in 10.36.12 for instance.

It doesn't seem like the proof sketch of Lemma 48.7.5 works in the generality that the lemma is stated. For instance, one could take $S' = (\mathbb{Z}, 0, \emptyset), T' = (\mathbb{Z}[x]/(x^2), (x), 0),$ and $S$ being the divided power polynomial algebra in one variable over $S'.$ Then, using the universal property of divided power polynomial algebras, $T$ should be the divided power algebra in one variable over $T'.$ But $T$ is not a thickening, since the ideal $\mathbb{Z}[x]/(x)^2\langle y \rangle_+$ is not locally nilpotent (which is counter to the statement given in the proof sketch).

Maybe I'm missing some hypotheses that were implicitly given in the chapter (but which weren't explicitly stated), or misunderstanding the definition of a thickening?

Typo in statement (2): a a finite...

An exponent "e" is missing from the last displayed formula towards the end of the proof of Lemma 0EXX. (The "$y_r$" on the left hand side should be a "$y_r^e$".)

Typo: $K^{a - 2} \xrightarrow {d^{a - 2}} K^{a - 1} \xrightarrow {d^{a - 1}} K^ a$

For flatness of the ring map one needs $M$ to be non-zero.

Is Lemma 30.4 missing a properness hypothesis? This is needed to invoke 087G.