Quotients of powers of maximal ideal in a local ring.

Question

I need a reference to a (hopefully) known result:

I think I can quite easily prove that if $L$ is a commutative Noetherian local ring with maximal ideal $M$ then for any positive integer $n$, $L$-module $M^{n-1}/M^n$ is isomorphic to a direct sum $F \oplus F \oplus \dots \oplus F$ of finite number of instances of field $F = L/M$. Moreover, the number of copies of $F$ equals the number of generators of $M^{n-1}$.

It would be very surprising if such result would not have been previously known. I would appreciate any pointers. Thank you very much!

Answer 1

This is a more general fact about finitely generated $L$-modules. If $X$ is a finitely generated $L$-modules, then $X/MX$ is a direct sum of a finite number of copies of $F$, and the number of such copies is the same as the cardinality of any minimal generating set for $X$. Your result is then just the special case $X=M^n$.

I don't know a specific reference for this exact statement, but it is a well-known corollary of Nakayama's lemma (and I would consider it reasonable to just state that it is true "by Nakayama's lemma" without needing any more specific reference). The fact that $X/MX$ is a direct sum of copies of $F$ is trivial: $X/MX$ is a module over the ring $L/M=F$, but $F$ is a field so any $F$-module (i.e., vector space) is a direct sum of copies of $F$. Then, by Nakayama's lemma a subset of $X$ generates it iff its image in $X/MX$ generates $X/MX$. So, a minimal generating set for $X$ has the same cardinality as a minimal generating set for $X/MX$, which is just the dimension of $X/MX$ as a vector space over $F$.