# Can one prove that toric varieties are Cohen-Macaulay by finding a regular sequence?

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?

• Well, for what it's worth, the existence of a regular sequence in an ideal that is of a given length is a homological property. It is exactly reflected by the vanishing and nonvanishing of certain Koszul homology groups. Mar 23, 2014 at 22:55
• I agree with @NeilEpstein: in principle any homological proof that a ring is CM could be translated in an existence proof for a regular sequence. Apr 6, 2014 at 19:55

## 1 Answer

In my paper arXiv:math/9802052 I give an argument in the graded case. The general case can be approached similarly (as sketched in the paper). The sequence in question is made from log-derivatives of a general element of given degree. The arguments are fairly self-contained and not cohomological.