Let me first ask an intuitive version of the question:

Let $$Sp$$ be the homotopy category of spectra. Let $$E$$ be a ring spectrum. Let $$D:Sp \to Sp$$ be the Spanier-Whitehead dual functor (maybe we need to restrict ourselves to compact object and that's fine). I want to define an algebraic Spanier-Whitehead duality functor $$D_E: E_*E\text{-}coMod \to E_*E\text{-}coMod$$ such that it commutes with the $$E$$-hurewicz map $$h_E$$, i.e. $$h_E \circ D = D_E \circ h_E$$.

The question under what conditions on $$E$$ there exists $$D_E$$? For example, when $$E = H\mathbb{F}_p$$ we know that $$D_E = Hom_{\mathbb{F}_p}(M, \mathbb{F}_p)$$.

Now let me make this question a little more precise. Maybe we need to replace $$E_*E\text{-}coMod$$ with finite objects in its derived category.

My gut feeling says, that if $$E$$ satisfies

1. Flatness: $$E_*E$$ is flat over $$E_*$$,
then it might just be enough to construct $$D_E$$. Is it though? If so can someone sketch a proof? If not do I need additional conditions?
So now the question is how to define dual objects in the (derived) category of $$E_*E$$ comodule and under what conditions on $$E$$ is it compatible with the Hurewicz map.
• A minor remark - if $DE$ itself makes sense (e. g. for finite $E$), then, I believe, the function spectrum $\mathscr F(-,DE)$ would work – მამუკა ჯიბლაძე Mar 2 at 17:37
• I want $D_E$ to be defined in the algebraic world (on `compact' objects). I should be able to define $D_E$ even for those $E_*E$ comodule $M$ which may not be $E$ homology of some spectrum. – Prasit Mar 2 at 19:14