Short Proof of a Conjecture Concerning Split-By-Nilpotent Extensions

TL;DR

This paper provides a concise proof of a conjecture relating to how properties of algebra modules are preserved under split-by-nilpotent extensions, specifically showing that tilted algebra properties are inherited.

Contribution

It offers a short proof confirming that if B is a tilted algebra, then the split extension C also retains the tilted algebra property, advancing understanding of algebra extensions.

Findings

If B is tilted, then C is tilted

Properties of mod B are inherited by mod C in split-by-nilpotent extensions

Provides a concise proof of a conjecture by Assem and Zacharia

Abstract

Let C be a finite dimensional algebra with B a split extension by a nilpotent bimodule E. We provide a short proof to a conjecture by Assem and Zacharia concerning properties of mod B inherited by mod C. We show if B is a tilted algebra, then C is a tilted algebra.