Precovers, localizations, and stable homotopy (Q953478)

A category \(\mathcal E\) is Frobenius if it is an exact category, the classes of injective and projective objects coincide, and for all \(X\) there is a deflation (an analogue of epic) \(P(X)\to X\) and an inflation (an analogue of monic) \(X\to I(X)\), and both \(P(X)\) and \(I(X)\) are in the category of pro/injective objects. The author calls \(P(X)\) a projective cover of \(X\) and \(I(X)\) an injective hull of \(X\). For a triangulated category \(\mathcal T\) of \(\mathcal E\), the author constructs a homotopy approximation of a resolution in \(\mathcal T\) and objects \(L_i\in\mathcal T\). Given a class of objects \(\mathcal R\) in \(\mathcal E\) which is a precovering class in \(\mathcal E\), is closed under shifts in \(\mathcal T\) and such that, for every \(R\in\mathcal R\), the injective hull \(I(R)\in\mathcal R\); let \(X\) be an object in \(\mathcal E\) with finite \(\mathcal R\)-dimension, the main result then states that there is an isomorphism of the morphism sets: \(\mathcal T(R, L_0)\cong \mathcal T(R,X)\), for every \(R\in\mathcal R\). Furthermore, if it is assumed that every object in \(\mathcal T\) has finite \(\mathcal R\)-dimension and if \(\mathcal S\) denotes the smallest full localizing subcategory of \(\mathcal T\) that contains \(\mathcal R\), then the inclusion functor \(i:\mathcal S\to \mathcal T\) has a right adjoint \(i'\). The original motivation for this construction came from modular representation theory and the author gives some examples that illustrate uses of these localization theorems.