Remark

Let A be a regular noetherian ring of infinite dimension (VIII, § 5, Exerc. 6 c)). We shall see below (No. 7, Th. 2 and § 4, No. 1, Prop. 1) that one has di_{A_m}(A_m) = dh(A_m) < +\infty for every maximal ideal m of A ; Prop. 3 therefore implies di_A(A) = dh(A) = +\infty . Consequently, the mappings p \mapsto di_{A_p}(N_p) and p \mapsto dh(A_p) are not in general upper semi-continuous.