The derived category of quasi-coherent sheaves and axiomatic stable homotopy

prove in this paper that for a quasi-compact and semi-separated (nonnecessarily noetherian) scheme X, the derived category of quasi-coherent sheaves over X, D(A(qc)(X)), is a stable homotopy category in the sense of Hovey, Palmieri and Strickland, answering a question posed by Strickland. Moreover we show that it is unital and algebraic. We also prove that for a noetherian semi-separated formal scheme x, its derived category of sheaves of modules with quasi-coherent torsion homologies D-qct(x) is a stable homotopy category. It is algebraic but if the formal scheme is not a usual scheme, it is not unital, therefore its abstract nature differs essentially from that of the derived category D-qc(X) (which is equivalent to D(A(qc)(X))) in the case of a usual scheme. (c) 2008 Elsevier Inc. All rights reserved.