Corollary 2

Let k be a field, B a finitely generated graded polynomial k-algebra, and A a graded subalgebra of B. The following conditions are equivalent: (i) B is a free graded $A$-module; (ii) B is a flat A-module; (iii) one has $\operatorname{Tor}_1^A(k, B) = 0$; (iv) the algebra A is a finitely generated graded polynomial k-algebra, and every algebraically free generating sequence of A consisting of homogeneous elements is B-regular.

The implications (i) $\Rightarrow$ (ii) and (ii) $\Rightarrow$ (iii) are clear, and the implication (iii) $\Rightarrow$ (i) follows from A, X, p. 144, prop. 8, a).

(iv) ⇒ (iii): let us suppose condition (iv) satisfied, and let x be an algebraically free generating sequence of A consisting of homogeneous elements. The sequence x, being A-regular, is completely secant for A (A, X, p. 157, prop. 5), so that the A-module Tor_1^A(k, B) is isomorphic to H_1(x, B) (A, X, p. 159, remark 3); but the latter is zero, since the sequence x is B-regular.

Corollaries 1 and 2 imply lemma 5 of LIE, V, § 5, No. 5.