Serre Dimension of Monoid Algebras

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 $Φ$-simplicial seminormal. $(i)$ If $M\in \CC(Φ)$, then {\it Serre dim} $R[M]\leq d$. $(ii)$ If $r\leq 3$, then {\it Serre dim} $R[int(M)]\leq d$. $(2)$ If $M\subset \BZ_+^2$ is a normal monoid of rank $2$, then {\it Serre dim} $R[M]\leq d$. $(3)$ Assume $M$ is $c$-divisible, $d=1$ and $t\geq 3$. 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 $P\cong \wedge^t P\op R[M]^{t-1}$.

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