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 $-{\otimes }_{\Lambda }^{\mathbb{L}}C$ 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 Löfwall 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 $\text{pd}C<\infty$.

30 pages

References in corpus (2)

• Another Proof of a Theorem of Van den Bergh about Graded-Injective Modules
• Tilting and cluster tilting for preprojective algebras and Coxeter groups