The Extension dimension of syzygy module categories

TL;DR

This paper studies the extension dimensions of syzygy module categories over Artin algebras, showing invariance under derived, stable, and separable equivalences for large enough indices.

Contribution

It proves that extension dimensions of syzygy categories are invariant under various algebra equivalences, extending understanding of their structural properties.

Findings

Extension dimensions are identical for large $i$ under derived equivalence.

Extension dimension is invariant under stable and separable equivalences.

The invariance holds for all nonnegative integers $i$.

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 $i$.