I have some difficulty with a detail in the treatments of the proof of this theorem, most of which follow that of Theorem 11.1 in A-M (many more or less word for word). This is not really significant till Proposition 11.3 when it seems to me some details are either glossed over or missing. I would be grateful either for agreement or correction of my misconception.
For convenience I will use the notation and set-up in A-M and not repeat them.
The exact sequence $(1)$ seems fine where we have $L_{n+{k_s}}=\dfrac{M_{n+k_s}}{x_sM_n}$. The problem then comes with the definition $L=\oplus_nL_n$ ($L_i$ is not defined for $i<k_s$).
It would seem that this should be $L=\oplus_nL_{n+{k_s}}$. Its then I think fairly easy to show that $L=\dfrac{\bigoplus\limits_{n={k_s}}^\infty M_n}{x_sM_n}$. i.e. not quite a quotient module of $M$ as A-M states and not $L=\dfrac{M}{x_sM}$ as I have seen in some proofs.
The rest of the proof still goes through fine and indeed the preamble to 11.3 which shows that $d(L)=d(M)-1$. The step that I think is then missing is to note that $d(L)=d\left(\dfrac{M}{xM}\right)$.
Presumably we can just define $L'=\dfrac{\bigoplus\limits_{n=0}^{k-1} M_n}{xM_n}$ and note that $\dfrac{M}{xM}=L'+L$ and $d(L')=0$.
This might seem a trivial point on what could be just a typo in A-M but I have seen the argument repeated so many times elsewhere that I am concerned that I have not followed something correctly.
