Triangle equivalences involving Gorenstein FP-injective modules

ABSTRACT

Recently, Hu et al. [16] introduced the notion of Gorenstein FP-injective modules and investigated a notion of Gorenstein FP-injective dimension for complexes. Let R be a left coherent ring and D^b(R-Mod) the bounded derived category of left R-modules. It is proved that the quotient triangulated category of the subcategory of D^b(R-Mod) consisting of complexes with both finite Gorenstein FP-injective dimension and FP-projective dimension by the bounded homotopy category of FP-projective-injective left R-modules is triangle-equivalent to the stable category $\underline{\mathcal{𝒢}\mathcal{ℱ}\mathcal{ℐ}\cap \mathcal{ℱ}\mathcal{𝒫}}$ of the Frobenius category of all Gorenstein FP-injective and FP-projective left R-modules. Note that a similar argument is also valid for the case of Gorenstein injective left R-modules. We extend a triangle equivalence established by Beligiannis involving Gorenstein injective left R-modules from rings with finite left Gorenstein global dimension to arbitrary rings.

KEYWORDS:

• Dimension
• Gorenstein FP-injective module
• Gorenstein injective module
• stable category
• triangle equivalence

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13D05
• 13D07
• 18G15
• 18G25