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 - d i m ( R [ M ] ) ≤ m a x { 1 , d i m ( R [ M ] ) − 1 } . Further, we define a class of monoids { M n } n ≥ 1 such that if M ∈ M n is seminormal, then S - d i m ( R [ M ] ) ≤ d i m ( R [ M ] ) − n = d + r − n , where 1 ≤ n ≤ r . As an application, we prove that for the Segre extension S m n ( R ) over R , S - d i m ( S m n ( R ) ) ≤ d i m ( S m n ( R ) ) − [ m + n − 1 m i n { m , n } ] = d + m + n − 1 − [ m + n − 1 m i n { m , n } ] .