Let $D(\mathbf{Z})$ be the derived category of abelian groups, and let $D(\mathbf{Z}\_p)$ be the derived category of modules over the p-adic integers. Bousfield localization gives a full subcategory of $L\_{\mathbf{Z}/p} D(\mathbf{Z})$ of ``$p$-complete'' complexes that is subtly different from $D(\mathbf{Z}\_p)$: the compact generator is $\mathbf{Z}/p$ instead of $\mathbf{Z}\_p$. 


I could apply $L\_{\mathbf{Z}/p}$ to $D(\mathbf{Z}\_p)$ instead. Are the Bousfield localizations $L\_{\mathbf{Z}/p}(D(\mathbf{Z}))$ and $L\_{\mathbf{Z}/p}(D(\mathbf{Z}\_p))$ equivalent?



p-complete Z_p-modules
Yes, they are equivalent. Perhaps the simplest way to see this is that, in general, Bousfield localization factors through the category of modules over the local sphere (where sphere means unit of the tensor product). So any Z/p-local object in D(Z) is automatically a module over Z\_p (in D(Z)), and from there it is not difficult to complete the proof. 


I should say that I have a student, Gabriel Valenzuela, who is thinking about related issues, and Charles Rezk has a preprint. L\_Z/p D(Z) is the derived category of the (non-Grothendieck) abelian category of Ext p-complete abelian groups, also called L-complete abelian groups. 



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                   Mark Hovey

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