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# Homology, Homotopy and Applications

## Volume 17 (2015)

### Number 2

### On homotopy categories of Gorenstein modules: Compact generation and dimensions

Pages: 13 – 24

DOI: http://dx.doi.org/10.4310/HHA.2015.v17.n2.a2

#### Author

#### Abstract

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

Published 3 December 2015