A Regularity Criterion for Semigroup Rings

Abstract

An analogue of the Kunz-Frobenius criterion for the re- gularity of a local ring in a positive characteristic is established for general commutative semigroup rings. Let S be a commutative semigroup (we always assume that S contains a neutral element), and K a Þeld. For every m 2Z+ the assignment x 7! x m , x 2 S, induces a K-endomorphism om of the semigroup ring R = K(S). Therefore we can consider R as an R-algebra via om, and especially as an R-module. Let R (m) denote R with its R-module structure induced by om. If S is Þnitely generated, then R (m) is obviously a Þnitely generated R-module. In this note we want to give a regularity criterion for S in terms of the homological properties of R (m) that is analogous to Kunz's (1) characteriza- tion of regular local rings of a characteristic p > 0 in terms of the Frobenius functor. Our criterion, which generalizes the result of Gubeladze (2, 10.2), requires only a mild condition on S and we provide a 'pure commutative al- gebraic' proof. (In (2) the result was stated for seminormal simplicial ane semigroup rings and derived from the main result of (2) that K1-regularity implies the regularity for such rings.) Theorem 1. Let S be a Þnitely generated semigroup, K a Þeld, R = K(S), and m 2 Z+, m > 0. Suppose that S has no invertible element 6 1 and is generated by irreducible elements. Then the following conditions are equivalent: (a) R (m) has a Þnite projective dimension; (b) R (m) is a free module; (c) S is free, in other words, S oZ n for some n 2Z+.