Relative homological algebra and purity in triangulated categories
As the author points out in the introduction: ``Triangulated categories were introduced by Grothendieck and Verdier in the early sixties as the proper framework for doing homological algebra in an abelian category. Since then triangulated categories have found important applications in algebraic geometry, stable homotopy theory, and representation theory. Our main purpose in this paper is to study a triangulated category, using relative homological algebra which is developed inside the triangulated category.'' To develop relative homological algebra in a triangulated category, one has to specify a class of triangles (among the distinguished ones) satisfying the expected properties. In contrast to relative homological algebra of abelian categories, the absolute theory in a triangulated category is trivial. The author shows that a proper class of triangles is uniquely determined by what he calls phantom maps. Although there are nontrivial phantomless examples, the phantom maps provide an important invariant. This paper contains a wealth of results, which cannot be reported here. The following table of contents may give a first impression: Proper classes of triangles and phantom maps. The Steenrod and Freyd category of a triangulated category. Projective objects, resolutions, and derived functors. The phantom tower, the cellular tower, homotopy colimits, and compact objects. Localization and the relative derived category. The stable triangulated category. Projectivity, injectivity, and flatness. Phantomless triangulated categories. Brown representation theorems. Purity. Applications to derived and stable categories. The author is fully aware of both the representation theoretic and the homotopy theoretic contexts in which triangulated categories are being used and he presents his results in such a way that researchers from both areas will find them very useful. Among the applications given here are many improvements of results in the literature, both from algebra and from topology.
scientific article; zbMATH DE number 1448646
scientific article