Homological dimension formulas for trivial extension algebras

TL;DR

This paper derives formulas for projective, injective, and global dimensions of modules over trivial extension algebras, providing modern expressions and conditions for Iwanaga-Gorenstein properties using derived functors.

Contribution

It establishes new formulas for homological dimensions of modules over trivial extension algebras using derived functors, generalizing classical results.

Findings

Formulas for projective and injective dimensions using derived functors

Characterization of global dimension for trivial extension algebras

Necessary and sufficient conditions for Iwanaga-Gorenstein property

Abstract

Let $A= \Lambda \oplus C$ be a trivial extension algebra. The aim of this paper is to establish formulas for the projective dimension and the injective dimension for a certain class of $A$-modules which is expressed by using the derived functors $- \otimes^{\mathbb{L}}_{\Lambda}C$ and $\mathbb{R}\text{Hom}_{\Lambda}(C, -)$. Consequently, we obtain formulas for the global dimension of $A$, which gives a modern expression of the classical formula for the global dimension by Palmer-Roos and L\"ofwall that is written in complicated classical derived functors. The main application of the formulas is to give a necessary and sufficient condition for $A$ to be an Iwanaga-Gorenstein algebra. We also give a description of the kernel $\text{Ker} \varpi$ of the canonical functor $\varpi: \mathsf{D}^{\mathrm{b}}(\text{mod} \Lambda) \to \text{Sing}^{\mathbb{Z}} A$ in the case $\text{pd} C < \infty$.