Costar Subcategories and Cotilting Subcategories with Respect to Cotorsion Triples

Abstract:

      Let $\mathcal{A}$ be an abelian category, and $(\mathcal{X}, \mathcal{Z},\mathcal{Y})$ be a complete hereditary cotorsion triple. We introduce the definition of $n$-$\mathcal{Y}$-cotilting subcategories of $\mathcal{A}$, and give a characterization of $n$-$\mathcal{Y}$-cotilting subcategories, which is similar to Bazzoni characterization of $n$-cotilting modules. As an application, we prove that if $\mathcal{GP}$ is $n$-$\mathcal{GI}$-cotilting over a virtually Gorenstein ring $R$, then $R$ is an $n$-Gorenstein ring, where $\mathcal{GP}$ denotes the subcategory of Gorenstein projective $R$-modules and $\mathcal{GI}$ denotes the subcategory of Gorenstein injective $R$-modules. Furthermore, we investigate $n$-costar subcategories over arbitrary ring $R$, and the relationship between $n$-$\mathcal{I}$-cotilting subcategories with respect to cotorsion triple $(\mathcal{P}, R$-Mod, $\mathcal{I})$ and $n$-costar subcategories, where $\mathcal{P}$ denotes the subcategory of projective left $R$-modules and $\mathcal{I}$ denotes the subcategory of injective left $R$-modules.