Abstract
In this chapter we shall use the tools forged in Chapter 17 to explore the basic facts about Cohen-Macaulay rings, which are rings R in which depth(I, R) = codim I for every ideal I (it is enough to assume this when I is a maximal ideal). These rings are important because they provide a natural context, broad enough to include the rings associated to many interesting classes of singular varieties and schemes, to which many results about regular rings can be generalized. The geometric meaning of the Cohen-Macaulay property is somewhat obscure but has a good expression in terms of maps to regular varieties, as we shall see in Theorem 18.16 and Corollary 18.17.
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