$t$-structure on modules over highly structured ring spectra

Question

Let $Sp$ be a suitably nice model of a suitably nice category of spectra. By this I mean that $Sp$ is a symmetric monoidal closed stable model category, where we can make sense of homotopy "groups" and so on. I'm not sure how to formalise this, but I'm thinking about the cases where $Ho(Sp)$ is the classical stable homotopy category, or the $G$-equivariant stable homotopy category for some (finite) group $G,$ or the Morel-Voevodsky $\mathbb{A}^1$-stable category of $\mathbb{P}^1$-spectra over a perfect field.

Now let $A$ be a commutative ring object in $Sp.$ Then by general theory the category $A-Mod$ of module objects in $Sp$ over $A$ has essentially the same formal properties as $Sp.$ However one notion which I have not seen established for $A-Mod$ is that of a $t$-structure. So

Question Does the category $A-Mod$ admit a (nice) $t$-structure? Is this written down anywhere?

In a bit more detail:

The category $Ho(Sp)$ affords a $t$-structure. Let's concentrate on the $G$-equivariant case, $G$ finite, since this seems to be middle ground in difficulty between ordinary stable homotopy and motivic stable homotopy. We can define homotopy presheaves $\underline\pi_n: Ho(G-Sp) \to Fun(\mathcal{O}, Ab),$ where $\mathcal{O}$ is just the subcategory of $Ho(G-Sp)$ consisting of the suspensions of $G/H_+$ for subgroups $H$ of $G.$ One proves that $(G/H_+)$ generate $Ho(G-Sp)$ and so the functors $\underline\pi_*$ form a conservative system. We tentatively put

$Ho(G-Sp)^{\le 0} = \{X | \underline\pi_n(X) = 0 \text{ if } n>0\}$

and similarly for $Ho(G-Sp)^{\ge 0}.$ Unless I'm mistaken one can prove this specifies a $t$-structure. Namely we can use the fact that the sphere spectrum is connective to kill off homotopy groups using cones (and homotopy colimits I guess) thus giving one of the truncation functors, and the rest should follow. One may moreover prove that $Ho(G-Sp)^{\ge 0}$ is generated under homotopy colimits and extensions by $\mathcal{O}.$ In particular the smash product of connective spectra is connective.

So here is a more refined question: Let $A$ be a connective commutative highly structured ring spectrum in the $G$-equivariant stable homotopy category, for $G$ a finite group. Do $Ho(A-Mod)^{\ge0} = \{X : \underline\pi_n(X)=0 \text{ for } n < 0\}$ and $Ho(A-Mod)^{\le 0} = \dots$ specify a $t$-structure on $Ho(A-Mod)$? Is the derived smash product over $A$ right (t-)exact?

A more philosophical question to end: It seems to me that if this is true, it should be rather formal and so valid in much greater generality. What is the natural setting of such an observation, and where can I learn about it?

Answer 1

I don't have time to think carefully. However, I answered this question for the sphere $G$-spectrum here: A heart for stable equivariant homotopy theory. I see no problem in generalizing that answer to any connective commutative ring $G$-spectrum $A$. Using cell $A$-modules, one can kill the higher homotopy groups of an $A$-module, etc. The Mackey functor $\underline{\pi}_0 A$ is a Green functor, so one can form module Mackey functors over it, and the heart should be the Eilenberg-MacLane $G$-spectra of such modules, with appropriate structure as $A$-module $G$-spectra.