Commutative Ring Theory

21 Soundable Subsets of a Spectrum and Depth

Gabriel PICAVET

Departement de Mathematiques

Universite BLAISE PASCAL ( Clermont II)

U.F.R. AUBIERE

63177 AUBIERE -CEDEX FRANCE

This note represents a summary of the author’s talk given at the Fes algebra meeting. The results contained in this paper will be published in a forthcoming paper together with their proofs and various other developments.

0. Preliminaries on Regular Sequences and Flat Topology

1. 1 The following result is standard :PROPOSITION 1 : Let A be a Noetherian ring, M a finitely generated A-module and I an ideal of A. Then I contains a M-regular sequence of length n + 1 if and only if ${\mathrm{Ext}}_A^i(A/I,M)=0$, for all i < n.The proof can be done by induction on n. When n = 0, the inductive ...