Catenarity in rings with abelian group action

Abstract

Let R be a commutative ringG a free abelian group of finite rank n and R#G the corresponding skew group ring. We fix an integer m; 0 ≤ m ≤ n and a free subgroup Gm of G of rank m. We prove that if R is noetherian, if specG (R#Gm ) is (R,G)-normally separated and if the Laurent polynomial ring is catenary, then the ring R#Gm is G-catenary. In the particular case where R is an affine algebra over an algebraically closed field, we prove that if R is G-locally finite, then the skew group ring R#Gm is universally catenary and universally G-catenary. Furthermore Tauvel's height formula is valid in all prime factors and G-prime factors of R#Gm.