Chapter 6 - EXACTNESS OF COMPLEXES AND LINEAR EQUATIONS OVER RINGS

Summary

The Acyclicity Lemma made its surprise appearance in the joint thesis of Peskine and Szpiro [PS 73, Ch. I, Lemme 1.8] which injected new vigour into the entire field. It played an important role in their treatment of the Homological Conjectures - ostensibly the subject of this book. They considered noetherian local rings and their maximal ideals, using local cohomology. It was quickly realized that Ext-functors work equally well, and the lemma was extended in various directions, [No 76, Ch. 5, Th. 21], [Fo 77b, Lemma 1.3], [Sc 82, Kor. 2.3.2]. Here, however, we take a different tack, considering several specializations of the abstract Acyclicity Lemmas 1.1.1 and 1.1.2. This approach, due to A.-M. Simon, allows for greater flexibility as we shall see, and establishes that Ext-depth can also be measured by the Koszul complex, 6.1.6.

In section 6.2 we treat the Buchsbaum-Eisenbud criterion for a finite free complex to be exact, a tool which has proved its value in many a concrete situation. As a rather minor application, we show in section 6.3 how it affects the solution of systems of linear equations over rings. Since the ring is seldom in doubt, we usually drop the subscript A from notations like E-dp_A (α,M) and dp_A (a,M).

By considering in the following noetherian local rings, their maximal ideals and finitely generated modules, the reader may recover the original Acyclicity Lemma's.