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

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

2007-06-04

arxiv.org/abs/0706.0493v3

Mathematics

Algebraic Geometry

v2: 31 pages, some improvements in exposition; v3 updated bibliography, to appear Adv. Math

Scientific paper

We prove in this paper that for a quasi-compact and semi-separated (non necessarily 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 of a usual scheme.

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