The extraordinary derived category (Q1093001)

The derived category, introduced by Grothendieck and Verdier [\textit{J.-L. Verdier}, Lect. Notes Math. 569, 262-311 (1977; Zbl 0407.18008)] is framework in which homological contructions and calculations can be made. But there are also extraordinary (or generalized) homology theories arising from spectra as developed by \textit{G. W. Whitehead} [Trans. Am. Math. Soc. 102, 227-283 (1962; Zbl 0124.383)] following examples of K- theory and bordism. The author defines a category \({\mathcal D}(E)\), associated to any ring spectrum E having an \(A_{\infty}\)-homotopy associativity structure, as being the homotopy category of right E-module spectra and E-linear maps. When E is the sphere spectrum, it is equivalent to the stable homotopy category. Considering the case of an Eilenberg-MacLane ring spectrum, the present paper shows how \({\mathcal D}(E)\) is the generalized derived category associated with E in the sense of Grothendieck-Verdier. One can expect (like what happens for algebraic K-theory) that this category can be for generalized homology theories (arising from spectra or sheaves of spectra) what is the derived category for ordinary homology.