Homological dimension formulas for trivial extension algebras

• Published in: arXiv.org
• Main Authors: Minamoto, Hiroyuki, Yamaura, Kota
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 04.10.2017
• Subjects: Algebra; Homology
• ISSN: 2331-8422

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 \(\text{pd} C < \infty\).