The Extension dimension of syzygy module categories

Abstract: In this paper, our primary focus is on investigating the extension dimensions of syzygy module categories associated with Artin algebras, particularly under various equivalences. We demonstrate that, for sufficiently large $i$, the $i$-th syzygy module categories of derived equivalent algebras exhibit identical extension dimensions. Furthermore, we establish that the extension dimension of the $i$-th syzygy module category is an invariant under both stable equivalence and separable equivalence for each nonnegative integer $i$.