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Let $X$ be an Hausdorff space, which is locally compact and locally connected. Let $R$ be a commutative and unitary ring and let $D(X)$ be the derived category of unbounded complexes of sheaves of $R$-modules.

Question: is $D(X)$ a stable homotopy category, and is it unital algebraic in the sense of Hovey, Palmieri and Strickland (def 1.1.4 in their AMS memoir "Axiomatic Stable Homotopy Theory")?

Edit: A. Neeman proved that if $M$ is a non-compact manifold of $dim\geq 1$, and $R=\mathbb{Z}$, the only compact object of the category $D(M)$ is the zero object (see his paper "On the derived category of sheaves on a manifold"). Thus in that case $D(M)$ cannot be an algebraic stable homotopy category.

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There are results about quasicoherent sheaves on schemes in the paper "The derived category of quasi-coherent sheaves and axiomatic stable homotopy" by Alonso, Jeremías, Pérez and Vale. This refers to another paper "Localization in categories of complexes and unbounded resolutions" by Alonso, Jeremías and Souto, which works with more general abelian categories satisfying Grothendieck's AB5 condition (filtered colimits are exact). You should be able to recover what you need from these.

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  • $\begingroup$ Thank you Neil! I will look at the references. Looking back at your memoir a strategy would be to apply theorem 2.3.2, that is to find a set of small, dualizable objects that detect acyclic sheaves. $\endgroup$ – David C Jan 27 '17 at 17:04

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