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

TL;DR

This paper establishes that the derived category of quasi-coherent sheaves on certain schemes forms a stable homotopy category, highlighting its algebraic and unital properties in the case of schemes and the differences in formal schemes.

Contribution

It proves that the derived category of quasi-coherent sheaves on quasi-compact semi-separated schemes is a stable homotopy category, answering a longstanding question.

Findings

Derived category of quasi-coherent sheaves is a stable homotopy category.

It is unital and algebraic for schemes.

For formal schemes, it is algebraic but not unital.

Abstract

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.