Homological dimension formulas for trivial extension algebras

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 $\text{pd} C < \infty$.