Remark 1

Consider the chain of inequalities of Prop. 12.

a) In order that one have prof_A(J ; M) = [J/m_AJ : κ_A], it is necessary and sufficient that J can be generated by an M-regular sequence (No. 3, Cor. 2 of Th. 1 and A, X, p. 160, Cor. 1).

b) The equality dim(M) − dim(M/JM) = [J/m_AJ : κ_A] means that J can be generated by a secant sequence for M (VIII, § 3, No. 2).

c) If M is Macaulay, one has profA(J ; M) = dim(M) − dim(M/JM) (§ 2, No. 2, cor. of Prop. 2).*