5 - Other approaches

Summary

Although we have now developed enough of the basic algebraic theory of local cohomology so that we could, if we wished, start right away with serious calculations with local cohomology modules, there are two other approaches to the construction of local cohomology modules which are useful, and popular with many workers in the subject. One approach uses cohomology of Čech complexes, and the other uses direct limits of homology modules of Koszul complexes. Links between local cohomology and Koszul complexes and Čech cohomology are described in A. Grothendieck's foundational lecture notes [25, §2]; related ideas are present in J.-P. Serre's fundamental paper [77, §61]. Among other texts which discuss links between local cohomology and the Čech complex or Koszul complexes are those by W. Bruns and J. Herzog [7, §3.5], D. Eisenbud [10, Appendix 4], M. Herrmann, S. Ikeda and U. Orbanz [32, §35], P. Roberts [70, Chapter 3, §2], J. R. Strooker [83, §4.3] and J. Stückrad and W. Vogel [84, Chapter 0, §1.3].

We shall make very little use in this book of the descriptions of local cohomology modules as direct limits of homology modules of Koszul complexes. However, we will use the approach to local cohomology via cohomology of Čech complexes, and so we present the basic ideas of this approach in this chapter. As this work leads naturally to the connection between local cohomology and direct limits of homology modules of Koszul complexes, we also present some aspects of that connection.