Corollary 3

Let $A$ be an integrally closed ring, $K$ its field of fractions, $L$ a finite $K$-algebra such that $[L : K] 1_A$ is invertible in $A$, and $B$ a sub-$A$-algebra of $L$, finite over $A$.

a) The sub-$A$-*module* $Al_B$ of $B$ is a direct factor.

b) For every ideal $J$ of $A$, one has the inequality $\mathrm{prof}_A(J; A) \geq \mathrm{prof}_B(JB; B)$.

c) If $B$ is a Macaulay ring, the same is true of $A$.

The $K$-linear mapping $\mathrm{Tr}_{L/K} : L \to K$ maps $B$ into $A$ (V, $§ 1$, No. 6, corollary 2 to Prop. 17), and therefore defines by restriction an $A$-linear mapping $t : B \to A$. For every $x \in A$, we have $t(xl_B) = [L : K] x$, whence a).

By Proposition 4 of $§ 1$, No. 3, we have $\mathrm{prof}_A(J; B) = \mathrm{prof}_B(JB; B)$; but by a) and Remark 4 of $§ 1$, No. 1, we have $\mathrm{prof}_A(J; A) \geq \mathrm{prof}_A(J; B)$, whence b).

If the ring $B$ is noetherian, the same is true of $A$: in fact, by a) we have $aB \cap A = a$ for every ideal $a$ of $A$; thus every increasing sequence $(a_n)_{n \in \mathbf{N}}$ of ideals of $A$ is stationary since the sequence $(a_nB)_{n \in \mathbf{N}}$ is stationary. Under the hypotheses of c), the $A$-module $B$ is Macaulay (Corollary 2), and the same is true of the $A$-module $A$ (No. 1, Example 2).