On homotopy categories of Gorenstein modules: Compact generation and dimensions

Let $A$ be a virtually Gorenstein algebra of finite CM-type. We establish a duality between the subcategory of compact objects in the homotopy category of Gorenstein projective left $A$-modules and the bounded Gorenstein derived category of finitely generated right $A$-modules. Let $R$ be a two-sided noetherian ring such that the subcategory of Gorenstein flat modules $R\mbox{-}\mathcal{GF}$ is closed under direct products. We show that the inclusion $K(R\mbox{-}\mathcal{GF})\to K(R\mbox{-}{\rm Mod})$ of homotopy categories admits a right adjoint. We introduce the notion of Gorenstein representation dimension for an algebra of finite CM-type, and give a lower bound by the dimension of its bounded Gorenstein derived category.

Keywords

Gorenstein projective module, Gorenstein flat module, compactly generated homotopy category, Gorenstein representation dimension

2010 Mathematics Subject Classification

18G25

Full Text (PDF format)

Published 3 December 2015