Corollary 4

Assume that $A_0$ is semisimple. Let $M$ be a graded $a$-module bounded below and let $n$ be an integer $\geq 0$. In order that $\mathrm{dp}_A(M) \leq n$, it is necessary and sufficient that

If $\mathrm{dp}_A(M) \leq n$, then $\operatorname{Tor}_{n+1}^A(A_0, M) = 0$ (X, p. 135, Lemma 1). Conversely, let $\dots \to L_1 \to L_0 \to M \to 0$ be an exact sequence of graded $a$-modules bounded below such that $L_0, \dots, L_{n-1}$ are graded free (X, p. 56, Prop. 11); according to Cor. 3 of X, p. 131, the equality $\operatorname{Tor}_{n+1}^A(A_0, M) = 0$ implies $\operatorname{Tor}_1^A(A_0, K) = 0$; since $K/J_0 K$ is a projective $A_0$-module because $A_0$ is semisimple (X, p. 140, Prop. 6), $K$ is projective by Prop. 8 (X, p. 144), and $\mathrm{dp}_A(M) \leq n$.