Chapter 5 - (PRE-) REGULAR SEQUENCES AND DEPTH

Summary

The theory of regular sequences became prominent in the middle fifties in the hands of such authors as Auslander-Buchsbaum, Rees, Serre and Kaplansky. It provides a link between ideal-theoretic properties of a ring and its modules on the one hand and homological concepts on the other. The reason is that, while reflecting certain ideal-theoretic conditions, regular sequences behave well with respect to homological invariants. These sequences are usually treated for finitely generated modules over noetherian rings. In section 5.1 we take a more general and formal point of view and consider the notion of a pre-regular sequence, a weak form of regular sequence. In section 5.2 we prove that after completion the two notions coincide, which permits an immediate proof of a theorem in Bourbaki. In section 5.3 then, we introduce the standard notion of depth and tie it up with an important variant, the Ext-depth. The well-known results in the noetherian case will be easy consequences of our more general considerations.

(PRE-) REGULAR SEQUENCES

Suppose A is a ring and M an A-module. We shall often work with a fixed sequence x₁,…,x_n of elements in A and standardly write a for the ideal (x₁,…,x_n).