$n\mathbb{Z}$-Gorenstein cluster tilting subcategories

Abstract

Let $\Lambda$ be an artin algebra. In this paper, the notion of $n\mathbb{Z}$-Gorenstein cluster tilting subcategories will be introduced. It is shown that every $n\mathbb{Z}$-cluster tilting subcategory of ${\rm{mod}}{\mbox{-}}\Lambda$ is $n\mathbb{Z}$-Gorenstein if and only if $\Lambda$ is an Iwanaga-Gorenstein algebra. Moreover, it will be shown that an $n\mathbb{Z}$-Gorenstein cluster tilting subcategory of ${\rm{mod}}{\mbox{-}}\Lambda$ is an $n\mathbb{Z}$-cluster tilting subcategory of the exact category ${\rm{Gprj}}{\mbox{-}}\Lambda$, the subcategory of all Gorenstein projective objects of ${\rm{mod}}{\mbox{-}}\Lambda$. Some basic properties of $n\mathbb{Z}$-Gorenstein cluster tilting subcategories will be studied. In particular, we show that they are $n$-resolving, a higher version of resolving subcategories.