Cohen-Macaulay rings

In this chapter we introduce the class of Cohen–Macaulay rings and two subclasses, the regular rings and the complete intersections. The definition of Cohen–Macaulay ring is sufficiently general to allow a wealth of examples in algebraic geometry, invariant theory, and combinatorics. On the other hand it is sufficiently strict to admit a rich theory: in the words of Hochster, ‘life is really worth living’ in a Cohen–Macaulay ring ([183], p. 887). The notion of Cohen–Macaulay ring is a workhorse of commutative algebra. Regular local rings are abstract versions of polynomial or power series rings over a field. The fascination of their theory stems from a unique interplay of homological algebra and arithmetic. Complete intersections arise as residue class rings of regular rings modulo regular sequences, and, in a sense, are the best singular rings. Their exploration is dominated by methods related to the Koszul complex. Cohen–Macaulay rings and modules Let R be a Noetherian local ring, and M a finite module. If the ‘algebraic’ invariant depth M equals the ‘geometric’ invariant dim M , then M is called a Cohen–Macaulay module: Definition 2.1.1 Let R be a Noetherian local ring. A finite R -module M ≠ 0 is a Cohen–Macaulay module if depth M = dim M . If R itself is a Cohen–Macaulay module, then it is called a Cohen–Macaulay ring . A maximal Cohen–Macaulay module is a Cohen–Macaulay module M such that dim M = dim R . In general, if R is an arbitrary Noetherian ring, then M is a Cohen–Macaulay module if M m is a Cohen–Macaulay module for all maximal ideals m ∈ Supp M .