Chapter IX

Complete noetherian local rings

In this chapter, all rings are assumed commutative; the algebras are associative, commutative and with a unit element. We denote by $1_A$ the unit element of a ring $A$.

If $A$ is a ring and $\mathfrak{p}$ a prime ideal of $A$, we denote by $\kappa(\mathfrak{p})$ the residue field of the local ring $A_{\mathfrak{p}}$. If the ring $A$ is local, we denote by $m_A$ its maximal ideal and by $\kappa_A$ or $\kappa(m_A)$ its residue field.

A ring homomorphism $\rho : A \to B$ is said to be flat (resp. faithfully flat) if it makes $B$ into a flat (resp. faithfully flat) $A$-module. Recall (I, § 3, no 5, prop. 9) that if $A$ and $B$ are local, $\rho$ is faithfully flat if and only if it is flat and local.