Serre Dimension of Monoid Algebras

Abstract

Let $R$ be a commutative Noetherian ring of dimension $d$, $M$ a commutative cancellative torsion-free monoid of rank $r$ and $P$ a finitely generated projective $R[M]$-module of rank $t$. $(1)$ Assume $M$ is $\Phi$-simplicial seminormal. $(i)$ If $M\in \CC(Φ)$, then {\it Serre dim} $R[M]\le d$. $(ii)$ If $r\le 3$, then {\it Serre dim} $R[int(M)]\le d$. $(2)$ If $M\subset \BZ_+^2$ is a normal monoid of rank $2$, then {\it Serre dim} $R[M]\le d$. $(3)$ Assume $M$ is $c$-divisible, $d=1$ and $t\ge 3$. Then $P\cong \wedge^t P\op R[M]^{t-1}$. $(4)$ Assume $R$ is a uni-branched affine algebra over an algebraically closed field and $d=1$. Then $P\cong \wedge^t P\op R[M]^{t-1}$.

To appear in Proc. Indian Acad. Sci. (Math. Sci)