On Serre dimension of monoid algebras and Segre extensions (2201.06364v1)

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