On a conjecture about dominant dimensions of algebras

TL;DR

This paper provides counterexamples to a conjecture on dominant dimensions of algebras, showing that the dominant dimension of certain endomorphism algebras can differ from the original, but identifies classes where the conjecture holds.

Contribution

It constructs specific counterexamples to Chen and Xi's conjecture and identifies classes of algebras where the conjecture remains valid.

Findings

Counterexamples to the conjecture are constructed.

The conjecture is false in general.

Higher Auslander algebras satisfy the conjecture.

Abstract

For every $n \geq 1$, we present examples of algebras $A$ having dominant dimension $n$, such that the algebra $B=End_A(I_0 \oplus \Omega^{-n}(A))$ has dominant dimension different from $n$, where $I_0$ is the injective hull of $A$. This gives a counterexample to conjecture 2 of Chen and Xi. While the conjecture is false in general, we show that a large class of algebras containing higher Auslander algebras satisfies the property in the conjecture.