Lemma 5

Let $M$ be a graded $A$-module bounded below, and let

be a graded $A$-homomorphism such that $1 \otimes_A s : A_0 \otimes_A (A \otimes_{A_0} M/J_0 M) \to A_0 \otimes_A M$ is the canonical isomorphism. Then $s$ is surjective. If $\mathrm{Tor}_1^A(A_0, M) = 0$, $s$ is bijective.

We have an exact sequence

and the graded $A$-modules $\mathrm{Ker}\ s$ and $\mathrm{Coker}\ s$ are bounded below. From the exact sequence $A_0 \otimes_A (A \otimes_{A_0} M/J_0 M) \xrightarrow{1 \otimes s} A_0 \otimes_A M \to A_0 \otimes_A \mathrm{Coker}\ s \to 0$, we deduce that $A_0 \otimes_A \mathrm{Coker}\ s = 0$, hence $s$ is surjective (Lemma 4). We then have an exact sequence If $\mathrm{Tor}_1^A(A_0, M) = 0$, then $A_0 \otimes_A \mathrm{Ker}\ s = 0$ and $s$ is injective (Lemma 4).