Notes on an Algebraic Stable Homotopy Category

Abstract

Ohkawa showed that the collection of Bousfield classes of the stable homotopy category of spectra is a set (Ohkawa in Hiroshima Math. J. 19:631–639, [8]). Let ${\mathcal C}$ be an algebraic stable homotopy category in the sense of Hovey, Palmieri and Strickland (Axiomatic Stable Homotopy Theory, American Mathematical Society, Providence, RI, [6]). We here show that Bousfield classes of ${\mathcal C}$ form a set by introducing a homology theory based on the generators of ${\mathcal C}$, in a similar manner as Dwyer and Palmieri did in Dwyer and Palmieri (Proc. Am. Math. Soc. 129(3):881–886, [3]). We also consider a relation between Bousfield classes of finite objects and supports of them on a collection of objects.