Proposition 3

Let A be a noetherian local ring, M a nonzero finitely generated A-module, and d the dimension of M. The following conditions are equivalent:

(i) the A-module M is Macaulay; (ii) one has $\operatorname{prof}(M) = d$; (iii) one has $\operatorname{Ext}_A^i(\kappa_A, M) = 0$ for every integer $i < d$; (iv) one has $\operatorname{Ext}_A^i(N, M) = 0$ for every A-module N of finite length and every integer $i < d$; (v) one has $\operatorname{Ext}_A^i(N, M) = 0$ for every finitely generated A-module N and every integer $i < d - \dim(M \otimes_A N)$; (vi) there exists an M-regular sequence of elements of $m_A$ of length d.

Condition (i) is equivalent to the equality $\operatorname{prof}(M) = d$, that is, to condition (ii), or again to the inequality $\operatorname{prof}(M) \geq d$, that is, to (iii) and to (vi) (§ 1, No. 4, Th. 2). The implications (v) $\Rightarrow$ (iv) and (iv) $\Rightarrow$ (iii) are obvious.

Lastly, let us suppose $M$ Macaulay and let $N$ be a finitely generated A-module. Put $F = \operatorname{Supp}(N)$; by II, § 4, No. 4, Prop. 18, one has $\operatorname{Supp}(M) \cap F = \operatorname{Supp}(M \otimes_A N)$, so that

(No. 1, cor. to Prop. 1 and No. 2, cor. to Prop. 2). The implication (i) $\Rightarrow$ (v) then follows from Remark 1 of § 1, No. 5.

In the rest of this number we shall say that a finitely generated module M over a noetherian local ring A is pure if, for every prime ideal p associated with M, one has dim(A/p) = dim(M). This also means that M has no embedded associated prime ideals and that the irreducible components of the support of M all have the same dimension. Every Macaulay module over a noetherian local ring is pure (No. 2, Prop. 2 and its corollary).