Corollary 5

*Suppose the ring $A_0$ semisimple. If $\operatorname{Tor}_{n+1}^A(A_0, A_0) = 0$, one has $\mathrm{dp}_A(M) \leq n$ for every graded $A$-module bounded below.

Denote by $A^\circ$ the graded ring opposite to $A$: we have $(A^\circ)_0 = (A_0)^\circ$, so $(A^\circ)_0$ is semisimple (VIII, § 5, No. 1, remark 3). Since $\operatorname{Tor}_{n+1}^{A^\circ}(A_0^\circ, A_0^\circ) = 0$, one has $\mathrm{dp}_A(A_{0s}^\circ) \leq n$ by cor. 4: this implies $\operatorname{Tor}_{n+1}^{A^\circ}(M_0, A_0^\circ) = 0$ for every $A$-module $M$, hence $\operatorname{Tor}_{n+1}^A(A_0, M) = 0$ and we apply cor. 4.