  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 $i$.
