On syzygy categories over Iwanaga-Gorenstein algebras: Reduction, minimality and finiteness

Abstract

We study 2-Calabi-Yau tilted algebras which are non-commutative Iwanaga-Gorenstein algebras of Gorenstein dimension 1. In particular, we are interested in their syzygy categories or equivalently the stable categories of Cohen-Macauley modules $\underline{\text{CMP}}$. First we show that if an algebra $A$ is Iwanaga-Gorenstein of Gorenstein dimension 1 then its stable category is generated under extensions by its radical $\text{rad}\,A$. Next, for a 2-Calabi-Yau tilted algebra $A$ we provide an explicit relationship between the $\underline{\text{CMP}}$ category of $A$ and its quotient $A/Ae_iA$ by an ideal generated by an idempotent $e_i$. Consequently, we obtain various equivalent characterizations of when the $\underline{\text{CMP}}$ category remains the same after passing to the quotient. We also obtain applications to two classes of algebras that are CM finite, the dimer tree algebras and their skew group algebras.