Proposition 8

Let $M$ be a graded $A$-module bounded below.

a) The following conditions are equivalent : (i) $M$ is isomorphic to a graded $A$-module of the form $A \otimes_{A_0} N$, where $N$ is a graded projective $A_0$-module (resp. graded free) ; (ii) $M$ is a projective $A$-module (resp. graded free) ; (iii) $M/J_0 M$ is a projective $A_0$-module (resp. graded free) and $\mathrm{Tor}_1^A(A_0, M) = 0$.

b) Assume furthermore that $M$ has a generator system consisting of homogeneous elements of bounded degrees. Then the following conditions are equivalent : (i) The graded $A$-module $M$ has a finite composition series whose quotients are isomorphic to graded $A$-modules of the form $A \otimes_{A_0} N$, where $N$ is a graded $A_0$-module that is flat ; (ii) $M$ is a flat $A$-module ; (iii) $M/J_0 M$ is a flat $A_0$-module and $\mathrm{Tor}_1^A(A_0, M) = 0$.

In each case, there is obviously (i) ⇒ (ii) ⇒ (iii). Therefore, it remains to prove (iii) ⇒ (i).

a) The graded $A$-module $A \otimes_{A_0} M/J_0 M$ is a projective $A$-module since $M/J_0 M$ is a projective $A_0$-module. The canonical homomorphism of $A$-modules

is surjective ; hence there exists a graded $A$-homomorphism of degree zero such that $p \circ s(a \otimes x) = ax$ for $a \in A$ and $x \in M/J_0 M$.

By Lemma 5, $s$ is an isomorphism of $A$-modules from $A\otimes_{A_0}M/J_0M$ onto $M$, whence (i).

b) By the hypothesis on $M$, there exist integers $a,b$ with $a\leq b$ such that $M$ is generated by $\displaystyle\bigoplus_{a\leq i\leq b}M_i$. We argue by induction on the positive integer $b-a$.

If $b-a=0$, then $M$ is generated by $M_a$ and the canonical $A_0$-homomorphism

is bijective; we then deduce from the $A$-homomorphism $A\otimes_{A_0}M_a\longrightarrow M$ defined by the $A$-module structure of $M$ a graded $A$-homomorphism satisfying the condition of Lemma 5. Then, by Lemma 5, $s$ is bijective, whence (i).

In the general case, let $M^{(a)}$ be the (graded) sub-$A$-module of $M$ generated by $M_a$ and $M'$ the quotient $M/M^{(a)}$. We have an exact sequence

whence, since $\operatorname{Tor}_1^A(A_0,M)=0$ by hypothesis, exact sequences

But the canonical homomorphism $M_a\longrightarrow M^{(a)}/J_0M^{(a)}$ is bijective. It follows that the homomorphism $1\otimes f:M^{(a)}/J_0M^{(a)}\longrightarrow M/J_0M$ is injective and that its image is a direct factor sub-$A_0$-module of $M/J_0M$. It then follows from the exact sequence (7) that $\operatorname{Tor}_1^A(A_0,M')=0$ and that the $A_0$-module $M'/J_0M'$ is flat since isomorphic to a direct factor of $M/J_0M$. By the induction hypothesis (which applies to $M'$ since it is generated by the $M'_i$ for $a<i\leq b$), $M'$ satisfies condition (i), hence is flat. We then deduce from the exact sequence (6) that $\operatorname{Tor}_1^A(A_0,M^{(a)})=0$, but $M^{(a)}/J_0M^{(a)}$ is identified with $M_a$ which is a flat $A_0$-module (as a direct factor submodule of $M/J_0M$) ; by what has already been proved, the graded $A$-module $M^{(a)}$ is isomorphic to $A\otimes_{A_0}M_a$, hence also satisfies (i), which completes the proof.