Chapter 7 - COMPARING HOMOLOGICAL INVARIANTS

Summary

In this chapter we compare various homological invariants. All three sections have in common that certain results are proved by using double complexes of modules. In the first two, we are dealing with a particular double complex in distinct ways, Theorems 7.1.2 and 7.2.14. In the first case we get identities, the second leads to inequalities. Of course, double complexes have been employed more extensively in commutative algebra by such authors as Foxby [Fo 77a], [Fo 79], and P. Roberts [Ro 76], [Ro 80b], while one might also introduce the full machinery of derived categories as mentioned in section 1.2. We shall however restrict ourselves to a single type of down to earth complex as discussed in that section, freely using the Matlis dual to obtain pairs of results.

The content of 7.1 and 7.2 is in large part taken from Bartijn's thesis [Ba], which in turn owes much to the works of Foxby. Section 7.3 explains certain ideas of P. Roberts [Ro 76], who used a double complex to relate the annihilators of local cohomology modules to the exactness of finite free complexes. These results will play a key role in Chapter 11.

Observe that only homological notions figure in this chapter, like Ext-depth, various homological dimensions and grade. Though the notion is doubtless familiar to most readers, in this book we take the somewhat perverse view that Krull dimension is a more subtle invariant and only introduce it in the next chapter.