Chapter X

Depth, regularity, duality

In this chapter, all rings are assumed commutative, the algebras are associative, commutative and unitary, and the algebra homomorphisms are unitary. If $A$ is a local ring, its maximal ideal is denoted by $m_A$, and its residue field by $\kappa_A$. If $p$ is a prime ideal of a ring $A$, the residue field of the local ring $A_p$ is denoted by $\kappa(p)$; it is identified with the field of fractions of the integral ring $A/p$. We denote by $\overline{\mathbf{Z}}$ the subset $\mathbf{Z} \cup \{-\infty, +\infty\}$ of $\overline{\mathbf{R}}$. Thus, for every $n \in \mathbf{Z}$, one has the relations $-\infty < n < +\infty$ and $n + \infty = \infty + n = \infty + \infty = \infty$, $n - \infty = -\infty + n = -\infty - \infty = -\infty$.