2006.109: Injective objects in triangulated categories

2006.109: Grigory Garkusha and Mike Prest (2004) Injective objects in triangulated categories. J. Algebra Applications, 3 (4). pp. 367-389. ISSN 0219-4988

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DOI: 10.1142/S0219498804000927

Abstract

We extend ideas and results of Benson and Krause on pure-injectives in triangulated categories. Given a generating set of compact objects in a compactly generated triangulated category T we define notions of monomorphism, exactness and injectivity relative to this set. We show that the injectives correspond to injective objects in a localisation of the functor category Mod T c where Tc denotes the subcategory of compact objects of T. The paper begins by setting up the required localisation theory.

Benson and Krause [BK] showed that injective modules over the Tate cohomology ring of a finite group algebra kG, where k is a field of characteristic p and G is a p-group, correspond to certain pure-injective objects in the (compactly generated, triangulated) stable module category of kG. We generalise this to arbitrary compactly generated triangulated categories, replacing the trivial module k by any compact object and the Tate cohomology ring by the graded endomorphism ring of that object. We obtain the strongest results in the case that this graded endomorphism ring is coherent.

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