It's well-known that if $P$ is a finitely generated saturated submonoid of $\mathbb{Z}^n$, then the monoid algebra $k[P]$ is Cohen-Macaulay (at the maximal ideal $m = (P-0)k[P]$). However, all proofs of this fact I've seen so far use homological characterizations of the Cohen-Macaulay property. Since the definition of Cohen-Macaulayness (there exists a regular sequence in $m$ of maximal length) does not involve homological algebra, I would like to ask if such a straightforward proof is known.

Do you know of a straightforward argument showing the existence of a regular sequence?

If the cone defining $P$ is simplicial, I think one can just take the ray generators. Otherwise, it is easy to see that one cannot pick a regular sequence of monomials. Maybe in the general case, $n = dim(k[P])$ general linear combinations of the ray generators suffice?

isa homological property. It is exactly reflected by the vanishing and nonvanishing of certain Koszul homology groups. $\endgroup$