Catenarity in Module-finite Algebras

The main theorem says that any module-finite (but not necessarily commutative) algebra Λ over a commutative Noetherian universally catenary ring R is catenary. Hence the ring Λ is catenary if R is Cohen-Macaulay. When R is local and Λ is a Cohen-Macaulay R-module, we have that Λ is a catenary ring, dim Λ = dimΛ/Q + htΛQ for any Q ∈ Spec Λ, and the equality n = htΛQ− htΛP holds true for any pair P ⊆ Q of prime ideals in Λ and for any saturated chain P = P0 ⊂ P1 ⊂ · · · ⊂ Pn = Q of prime ideals between P and Q.