Local rings of embedding codepth at most 3 have only trivial semidualizing complexes

TL;DR

This paper proves that local rings with embedding codepth at most 3 have at most two semidualizing complexes, simplifying the classification of these complexes in such rings.

Contribution

It establishes a bound on the number of semidualizing complexes in local rings with small embedding codepth, identifying only the trivial and dualizing complexes.

Findings

Local rings of embedding codepth ≤ 3 have at most two semidualizing complexes.

The only semidualizing complexes are the ring itself and a dualizing complex if it exists.

This result constrains the structure of semidualizing complexes in low codepth rings.

Abstract

We prove that a local ring $R$ of embedding codepth at most 3 has at most two semidualizing complexes up to shift-isomorphism, namely, $R$ itself and a dualizing $R$-complex if one exists.