Characterizations of graded Pr\"ufer $\star$-multiplication domains

Let $R={⨁}_{\alpha \in \mathrm{\Gamma }}{R}_{\alpha }$ be a graded integral domain graded by an arbitrary grading torsionless monoid $\mathrm{\Gamma }$, and $\star$ be a semistar operation on $R$. In this paper we define and study the graded integral domain analogue of $\star$-Nagata and Kronecker function rings of $R$ with respect to $\star$. We say that $R$ is a graded Pr"{u}fer $\star$-multiplication domain if each nonzero finitely generated homogeneous ideal of $R$ is ${\star }_f$-invertible. Using $\star$-Nagata and Kronecker function rings, we give several different equivalent conditions for $R$ to be a graded Pr"{u}fer $\star$-multiplication domain. In particular we give new characterizations for a graded integral domain, to be a P$v$MD.