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We describe a procedure for constructing morphisms in additive categories, combining Auslander's concept of a morphism determined by an object with the existence of flat covers. Also, we show how flat covers are turned into projective covers and we interpret these constructions in terms of adjoint functors. [ABSTRACT FROM AUTHOR]

The Hom closed colocalizing subcategories of the stable module category of a finite group are classified. Along the way, the colocalizing subcategories of the homotopy category of injectives over an exterior algebra, and the derived category of a formal commutative differential graded algebra, are classified. To this end, and with an eye towards future applications, a notion of local homology and cosupport for triangulated categories is developed, building on earlier work of the authors on local cohomology and support. [ABSTRACT FROM AUTHOR]

For a right artinian ring A we show that for every n ≥ 0 there exists a pure-injective A-module P[subn] such that the A-modules of projective dimension at most n are precisely the direct factors of A-modules having a finite filtration in products of copies of P[subn]. This is a consequence of a general description of certain contravariantly finite resolving subcategories of Mod A. It leads in addition to a one-to-one correspondence between equivalence classes of (not necessarily finitely generated) cotilting modules and resolving subcategories of Mod A which are closed under products and admit finite resolutions and special right approximations. As an application it is shown that every finitely presented partial cotilting module over an artin algebra admits a complement. [ABSTRACT FROM AUTHOR]