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 popular with some workers in the subject. One approach uses cohomology of Čech complexes, and the other uses direct limits of homology modules of Koszul complexes. We shall have occasion later in the book to use the Čech complex approach to local cohomology, and we present the basic ideas of this approach in this chapter. This work leads naturally to the connection between local cohomology and direct limits of homology modules of Koszul complexes, and so some aspects of this are included in this chapter.

However, we do not use Koszul complexes as a tool anywhere in this book: in fact, the only references in later chapters to results from Section 5.2 are in Exercise 12.4.4, which shows that the main Theorem 5.2.9 of Section 5.2 can also be used in graded situations to calculate gradings on graded local cohomology modules. We have included Section 5.2 because it would be remiss of us to write a book on local cohomology without mentioning the links with Koszul complexes.

Links between local cohomology and Koszul complexes and Čech cohomology are described in A. Grothendieck's foundational lecture notes [18, §2]; related ideas are present in J.-P. Serre's fundamental paper [56, §61].