On graded projective modules

Question

If $R$ is a PID, then we know that any projective module over $R$ is free. Is there any graded version of this? That is, let us call a graded commutative ring $R=\oplus _{g \in G} R_g$ a graded PID if every homogeneous ideal (i.e. graded ideal) of $R$ is generated by a homogeneous element (I don't know if this is a standard definition. If there is a standard definition, please let me know). If $M=\oplus_{g\in G} M_g$ is a projective graded module over $R$ (note that graded-projective is equivalent to graded and projective), then obviously $M$ is free; my question would be: Is $M$ graded-free, i.e., does there exist a basis for $M$ over $R$ consisting of homogeneous elements? If this is not true in general, does it hold with some extra condition on $M$ or $R$, like $M$ being finitely generated, or something else?

Answer 1

This is true. More generally, we have the following:

Let $\psi\colon G\rightarrow H$ be an epimorphism of abelian groups. We denote by $\bullet_{[\psi]}$ the $\psi$-coarsening functor from the category of $G$-graded $R$-modules to the category of $H$-graded $R_{[\psi]}$-modules. Let $R$ be a $G$-graded ring, and let $M$ be a $G$-graded $R$-module. If $M$ is free, then so is $M_{[\psi]}$. If $R$ is principal, then the converse holds.

Note that the "free" and "principal" are understood in the graded sense.