The Stacks project

In Tag 0DNJ, the proof has two \beta_{M,0}'s. One is \beta_{M,0}(\beta) (which should probably be \beta_{M,0}(\eta)) and the other is \beta_{M,0} without any input (which should probably read \beta_{M,0}(0)).

@Vincent: The diagram in this definition is of sheaves on $S_i \times_S S_j \times_S S_k$, so the maps on the pairwise fiber products all have to be pulled back to the triple product. In particular, we have a map $\phi_{ik}$ of sheaves on the product $S_i \times_S S_k$ of the first and last factors, and to compare this to our other maps of sheaves we take the induced map $\text{pr}_{02}^*\phi_{ik}$ between the pullbacks of these sheaves along the natural projection $\text{pr}_{02}: S_i \times_S S_j \times_S S_k \to S_i \times_S S_k$.

(N.B. There isn't a $\text{pr}_{02}^*\phi_{ij}$, though - if we're pulling back $\phi_{ij}$ we have to use the projection $\text{pr}_{01}$ onto the first two factors instead.)

I noticed an additional small optimization to this lemma: the flatness hypothesis can be weakened to instead require only the set-theoretic statement, namely, that the irreducible components of $X$ map to the generic point (rather than all associated points). This came up for me in a situation where I knew that $X_s$ and $X_\eta$ were reduced, was not sure whether $X$ was flat, and specifically wanted a technical lemma to imply that $X$ was reduced (en route to proving flatness, in fact).

Here's how the proof goes under the weakened hypothesis: after the same initial setup, we have $x' \in X$ lying over $s$ and we wish to show $A = \mathcal{O}_{X, x'}$ is reduced. Now by the set-theoretic hypothesis, $\pi$ is not in any minimal prime ideal of $A$. The equation $a = \pi^n a'$ thus implies (since $a$ is nilpotent) that $a'$ is in every minimal prime ideal, hence $a'$ is again nilpotent. This contradicts reducedness of $A/\pi A$ as in the current proof.

Should one require that $y\to H(x_0)$ is the fixed morphism $y\to H(x)$?

In the proof of Lemma 14.18.5, it is claimed that $ker(d_0)\cap...\cap ker(d_j)=N(U_{n+1})\oplus \bigoplus_{i=j+1}^n s_i(ker(d_n)\cap...\cap ker(d_{i-1}))$. But I think it should be $ker(d_0)\cap...\cap ker(d_j)=N(U_{n+1})\oplus \bigoplus_{i=j+1}^n s_i(ker(d_0)\cap...\cap ker(d_{i-1}))$ because it is proven above that $x=s_0(z_0)+...+s_n(z_n)+x_n$ and element in $ker(d_0)\cap...\cap ker(d_j)$ should be elements whose first $j$ terms in the summation expression are zero.

In the proof of Lemma 14.18.5, it is claimed that $ker(d_0)\cap...\cap ker(d_j)=N(U_{n+1})\oplus \bigoplus_{i=j+1}^n s_i(ker(d_n)\cap...\cap ker(d_{i-1}))$. But I think it should be $ker(d_0)\cap...\cap ker(d_j)=N(U_{n+1})\oplus \bigoplus_{i=j+1}^n s_i(ker(d_0)\cap...\cap ker(d_{i-1}))$ because it is proven above that $x=s_0(z_0)+...+s_n(z_n)+x_n$ and element in $ker(d_0)\cap...\cap ker(d_j)$ should be elements whose first $j$ terms in the summation expression are zero.

In III, Remarque 3.2.5 of "Cohomologie cristalline des schémas de caractéristique $p > 0$", Berthelot writes that $$u_{X/S}:(X/S)_{\mathrm{cris}}\to \mathrm{Sh}(X_{\mathrm{Zar}})$$ may not be a morphism of ringed topoi. So perhaps one has to prove that in Situation 07MF, $u_{X/S}$ is a morphism of ringed topoi?

Typo : it seems that in the last two sentences the $f$'s are missing : "Thus $f(x)|_{T_ i}$ is an obeject $\mathcal{Z}$" and "we conclude that $f(x)$ is an object of $\mathcal{Z}$ as well".

In the sentence

"For a free $\Lambda$-module, we have $\text{End}_\Lambda(\Lambda^{\oplus m}) = \text{Mat}_n(\Lambda)$.",

I suggest we replace $\text{Mat}_ n(\Lambda )$ with $\text{Mat}_ m(\Lambda )$.

Sorry for the above empty comment, now I know the $A\bigotimes_RB$-module structure on $\mathrm{Tor}_i^R(A,B)$. We just need to choose $R$-resolutions $F_\bullet\to A$ and $P_\bullet\to B$, then any $a\in A$ induces a homomorphism $F_\bullet\to F_\bullet$, also $b\in B$ induces a homomorphism $P_\bullet\to P_\bullet$. They commute on the bicomplex $F_\bullet\otimes_RP_\bullet$.

Second sentence of proof: we will show that $M$ is a glueable module for $(R\to R',f)$

Second paragraph of the proof: remove extra "the" in "The long the exact sequence of Tors"

Two misprints:

1. In the statement of the lemma it should be "the subfield of elements separable algebraic over $\kappa(p)$".
2. In the first paragraph of the proof it should be "the compositum of $\kappa(q)$ and $\kappa_1^{sep}$".

I don't think that "descending chain condition" is defined anywhere in the project. a quick search suggests that this is the first place it is mentioned in the stacks project. similar for ascending chain condition. Is this something that should be defined in the project?

Small typos : the three arrows "$y \to f(x)$" should be "$y \to f(x')$".

I personally believe in $(45)$ it should be "a" Noetherian $R$-module instead of "an". In section $10.51$ there is a sentence says "Let $R$ be a Noetherian ring" . It might be a special case of words using . Therefore, I mention it in case it is a typo.

*nerve not 'never'

This lemma is important and sort of hard to find. I suggest the following slogan (or variant) 'Cohomology of a stack can be computed on the Cech never of a presentation'.

Three points:

1. [line 4] To conclude $p$ is the unique minimal prime of $A$, we also need the faithful flatness (not just the flatness) of $A\rightarrow A^h$.
2. [line 10] It should be $A/p[f_1,f_2] \subset A'$ instead of $A[f_1,f_2]\subset A'$.
3. [line 25] To conclude that $B/pB$ is a domain, just note that the inclusion $A/p \subset A'$ together with the flatness of $A\rightarrow B$, give the inclusion $B/pB \subset A' \otimes_A B=B'$, and it has already been proved that $B'$ is a domain.