Chapter 4 - LOCAL COHOMOLOGY AND KOSZUL COMPLEXES

Summary

Both subjects are well known and standard tools in commutative algebra. Nevertheless, it is not easy to find all that we shall need covered in one single text, see however the recent book [HIO, Ch. VII]. Since both topics will play an important role in later chapters, we have decided to present a bare-bones account. Thus proofs will be merely outlined or sketched, and in some cases we shall just provide the reader with a reference.

Local cohomology was introduced by Grothendieck, using the language of sheaves, in [Gro 67]. In his hands and those of others it became an important technique in commutative algebra [HK], [Sc 82a], algebraic geometry [Ha], [Li], the theory of invariants [HR 74], [HR 76], analytic geometry [ST] and the theory of singularities [Gre]. In section 4.1 we content ourselves with a quick introduction by way of commutative algebra which is due to Sharp [Sh 70].

Koszul complexes or exterior algebra complexes have been with us even longer. They and related complexes play a key part not only in commutative algebra but for instance in complex algebraic geometry and in differential geometry (de Rham complex). Here we give a frankly utilitarian account, not emphasizing the exterior algebra structure like in [Bo 80], see section 4.2.

In section 4.3 finally we consider limits of Koszul complexes and explain how these serve to read off local cohomology.

LOCAL COHOMOLOGY

To define local cohomology and show that it can also be written as a direct limit of Ext-functors, one only need assume one's ring to be commutative - though noncommutative versions have been given [Go, Ch. 60].