Orthogonal graded completion of modules

The construction and study of the orthogonal completion functor is an important step in the orthogonal completeness theory developed by K. I. Beidar and A. V. Mikhalev. The research of the graded orthogonal completion begun by the author is continued in this work. We consider associative rings graded by a group and modules over such rings graded by a polygon over the same group. Note that the graduation of a module by a group is a partial case of a more general and natural construction.
For any topology $\mathcal F$ of a graded ring $R$ consisting of graded right dense ideals and containing all two-sided graded dense ideals, the functor $O^\mathrm{gr}_\mathcal F$ of the graded orthogonal completion is constructed and studied in this paper. This functor maps the category of right graded $R$-modules into the category of right graded $O^\mathrm{gr}_\mathcal F(R)$-modules. The important feature of the graded case is that the graded modules $Q^\mathrm{gr}_\mathcal F(M)$ and $O^\mathrm{gr}_\mathcal F(M)$ (where $M$ is a right graded $R$-module) may not be orthogonal complete. A criterion for the orthogonal completeness is proved. As a corollary we get that these modules are orthogonal complete in the case of a finite polygon. The properties of the functor $O^\mathrm{gr}_\mathcal F$ and a criterion of its exactness are also established.