The stable category of Gorenstein-projective modules over a monomial algebra

Let $\mathrm{\Lambda }$ be an arbitrary monomial algebra. We investigate the stable category ${\underset{}{\mathrm{Gproj}}}^{\mathbb{Z}}\mathrm{\Lambda }$ of graded Gorenstein-projective $\mathrm{\Lambda }$-modules and the orbit category ${\underset{}{\mathrm{Gproj}}}^{\mathbb{Z}}\mathrm{\Lambda }\mathrm{/}(1)$ induced by ${\underset{}{\mathrm{Gproj}}}^{\mathbb{Z}}\mathrm{\Lambda }$ and the degree shift functor $(1)$. We prove that ${\underset{}{\mathrm{Gproj}}}^{\mathbb{Z}}\mathrm{\Lambda }$ is triangle equivalent to the bounded derived category of a path algebra of Dynkin type $\mathbb{A}$ and that ${\underset{}{\mathrm{Gproj}}}^{\mathbb{Z}}\mathrm{\Lambda }\mathrm{/}(1)$ is triangle equivalent to the stable module category of a self-injective Nakayama algebra. Both the path algebra and the self-injective Nakayama algebra will be given explicitly. The latter result provides an explicit description of the stable category of (ungraded) Gorenstein-projective $\mathrm{\Lambda }$-modules.