By J. P. Harnad, S. Shnider
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In other words, Y is f -local if, as far as mapping into Y is concerned, f looks like an equivalence. 2 Definition. A map X → X of R-modules is said to be an f -local equivalence if it induces an equivalence HomR (X , Y ) → HomR (X, Y ) for every f -local R-module Y . An f -localization of X is a map : X → Lf (X), such that Lf (X) is f -local and is an f -local equivalence. 3 Remark. It is not hard to see that any two f -localizations of X are equivalent, so that we can speak loosely of the f -localization of X.
If Lf is smashing then the category of f -local R-modules is equivalent, from a homotopy point of view, to the category of Lf (R)modules. In particular, the homotopy category of f -local R-modules is equivalent to the homotopy category of Lf (R)-modules. 9 Examples. Let R = Z, pick a prime p, and let f be the map Z − → Z. Then Lf is smashing, and Lf (X) ∼ Z[1/p] ⊗Z X. 5], which is the total left derived functor of the p-completion functor. In particular, Lf (Z) ∼ Zp and Lf (Z/p∞ ) ∼ ΣZp . Since ΣZp is not equivalent to Zp ⊗Z Z/p∞ , Lf is not smashing in this case.
More generally, the homotopy category of any Quillen model category   can be built by formally inverting maps. In a slightly different direction, the process of localization with respect to a map (§2) has recently developed into a powerful tool for making homotopy-theoretic constructions [2, §4]; roughly speaking, localizing with respect to f involves converting an object X into a new one, Lf (X), with the property that, as far as mapping into Lf (X) goes, f looks like an equivalence. 2).
Geometrical and Topological Methods in Gauge Theories by J. P. Harnad, S. Shnider