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XE. This gives the terminal example of an E*-equivalence out of X and is automatically natural in X. Moreover, the full subcategory SE C S of E*-local spectra is equivalent to the category of fractions S [6-1], and SE thus captures the part of stable homotopy theory seen by E*.

3 Theorem. A spectrum Y is K*-local if and only if A : 7r*(Y; Z/p) 7r*+q(Y; Z/p) for each prime p. 4 Theorem. A map f : X - Y of spectra. is a K*-equivalence if and only if f* : it*X®Q = r*Y®Q and f* : 7r*(X; Z/p)[A-1] - it*(Y; Z/p)[A-1] for each prime p. Thus the K*-localization makes all mod-p homotopy groups A-periodic (or v1-periodic in BP-parlance) with x*(XK; Zlp) - i*(X; Zlp)[A-1] The preceding results give a fairly good homotopy theoretic understanding of K*-local spectra, but they don't go very far toward classifying these spectra algebraically.

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