Chapter VIII

Dimension

In this chapter, all rings are assumed to be commutative, the algebras are associative, commutative and with unit.

Let $A$ be a ring. If $a$ is an ideal of $A$ and $n$ a negative integer, we put $a^n = A$. For every prime ideal $p$ of $A$, we denote by $\kappa(p)$ the residue field of the local ring $A_p$; it is canonically isomorphic to the field of fractions of the ring $A/p$ (II, § 3, No. 1, prop. 2). If $A$ is local, we denote by $m_A$ its maximal ideal and $\kappa_A = A/m_A = \kappa(m_A)$ its residue field.

Let $\rho : A \to B$ be a homomorphism of rings. We denote by $^a\rho$ the continuous mapping of Spec(B) into Spec(A) associated with $\rho$ (II, § 4, No. 3). We say (V, § 2, No. 1, def. 1) that a prime ideal $q$ of $B$ is above the prime ideal $p$ of $A$ if $p$ is the image of $q$ by $^a\rho$, that is to say if $\rho^{-1}(q) = p$. Let $M$ be a $B$-module; when we consider $M$ as an $A$-module, it will always be a question of the $A$-module structure $\rho_*(M)$ defined by the external law $(a, m) \mapsto \rho(a).m$ (A, II, p. 30).

Let $M$ be an $A$-module. If $U$ (resp. $V$) is an additive subgroup of $A$ (resp. $M$), we denote by $UV$ or $U.V$ the additive subgroup of $M$ generated by the products $uv$, for $u \in U$ and $v \in V$. If $S$ is a subset of $A$, we denote by $SM$ the submodule $\sum_{s \in S} sM$ of $M$; the ideal $\mathfrak{S}$ of $A$ generated by $S$ is equal to $SA$ and one has $\mathfrak{S}.M = SM$.