I am looking for a proof of the structure theorem of finitely generated graded modules over graded PID's:
Let $R$ be a graded PID and $M$ be a finitely generated graded module over $R$. Then $M$ decomposes uniquely as $$M \cong \bigoplus_{i=1}^n \Sigma^{\alpha_i} R\ \oplus\ \bigoplus_{j=1}^m \Sigma^{\gamma_i} R/d_j R$$ where $d_j \in R$ are homogeneous elements so that $d_j \mid d_{j+1}$ , $\alpha_i, \gamma_j \in \mathbb{Z}$ and $\Sigma^\alpha$ denotes an $\alpha$-shift upwards in grading.
But i can't find anything. Does anybody know where i could find a proof of this theorem? I know how to proof the usual structure theorem of finitely generated modules, is there an easy way to translate this result to the graded case ? I would guess that one obtains $M \cong R^n\ \oplus\ \bigoplus_{j=1}^m R/d_j R$ by the usal structure theorem , but the isomorphism is just a usal isomorphism of modules. To get a graded isomorphism one has to eventually shift the terms on the right side in grading.
It seems that the decomposition of graded $k[x]$-modules is important in the study of persistent homology; papers on the subject are the only place in the literature where I could find a treatment of graded PIDs.
An algorithm to find the graded Smith normal form of a matrix is discussed in Section 3.1.3 of the paper
Primoz Skraba and Mikael Vejdemo-Johansson, Persistence modules: Algebra and algorithms, arXiv:1302.2015.
Although the algorithm is stated only for graded $k[x]$-modules, the authors mention that it works over any graded PID. Once one has a graded Smith normal form, the structure theorem for finitely generated graded modules can be easily proved.