Some criteria for Gorensteinness via Gorenstein projective cotorsion pairs

Let $R$ be a noetherian algebra over a Cohen–Macaulay ring admitting a canonical module, and assume that $R$ is maximal Cohen–Macaulay over the base ring. We provide a characterization of when $R$ is left weakly Gorenstein. We further show that the category of finitely generated Gorenstein projective $R$-modules coincides with the left $\text{Ext}$-orthogonal class of the thick subcategory generated by finitely generated $R$-modules of finite projective or finite injective dimension. As a consequence, finitely generated Gorenstein projective $R$-modules generate a hereditary cotorsion pair. Moreover, we show that a Cohen–Macaulay local ring is Gorenstein if and only if the right $\text{Ext}$-orthogonal class of finitely generated Gorenstein projective modules coincides with the category of finitely generated modules of finite projective dimension.