LAUSR

Let T be a triangulated category with triangulation ∆, ξ ⊆ ∆ a proper class of triangles and C an additive full subcategory of T . We provide a method for constructing a proper C(ξ)-resolution (respectively, coproper C(ξ)-coresolution) of one term in a triangle in ξ from those of the other two terms. By using this construction, we show the stability of the Gorenstein category GC(ξ) in triangulated categories. Some applications are given. Introduction. Triangulated categories were introduced by Grothendieck and Verdier in the early 1960s as the proper framework for homological algebra in an abelian category. Since then, triangulated categories have found important applications in algebraic geometry, stable homotopy theory and representation theory. Examples for this may be found in duality theory, Hartshorne [9] and Iversen [12], or in the fundamental work on perverse sheaves, Beilinson, Bernstein and Deligne [3]. Relative homological algebra has been formulated by Hochschild in categories of modules and later by Heller and Butler and Horrocks in more general categories with a relative abelian structure. Let T be a triangulated category with triangulation ∆. Beligiannis [4] developed a homological algebra in T which parallels the homological algebra in an exact category in the sense of Quillen. To develop the homology, a class of triangles ξ ⊆ ∆, called proper class of triangles, is specified. This class is closed under translations and satisfies the analogous formal 2010 AMS Mathematics subject classification. Primary 18G10, 18G25, 18E30.