On Serre dimension of monoid algebras and Segre extensions

Abstract

Let $R$ be a commutative noetherian ring of dimension $d$ and $M$ be a commutative$,$ cancellative$,$ torsion-free monoid of rank $r$. Then $S$-$dim(R[M])\le max\{1,dim(R[M])-1\}=max\{1,d+r-1\}$. Further$,$ we define a class of monoids $\{{\mathfrak{M}}_n{\}}_{n\ge 1}$ such that if $M\in {\mathfrak{M}}_n$ is seminormal$,$ then $S$-$dim(R[M])\le dim(R[M])-n=d+r-n,$ where $1\le n\le r$. As an application, we prove that for the Segre extension $S_{mn}(R)$ over $R,$ $S$-$dim(S_{mn}(R))\le dim(S_{mn}(R))-\left[\frac{m+n-1}{min\{m,n\}}\right]=d+m+n-1-\left[\frac{m+n-1}{min\{m,n\}}\right]$.

Accepted for publication in JPAA