# If $A\in \mbox{Rings}\subset E_\infty\mbox{-rings}$, what is the equivalence betweenobjectsof $\mathcal{D}(\mbox{Mod}_A)\simeq\mbox{ModSpectra}_A$?\mathcal{D}(\mbox{Mod}_A)$and$A$-modulespectra? In Lurie's "A Survey of Elliptic Cohomology", he writes on page 14 that if$A$is an ordinary commutative ring considered as an$E_\infty$-ring, then$A$-module spectra are the same thing as objects of the derived category of$A$-modules. This is mysterious to me. On the one hand, to an$A$-module spectrum$M$we might associate the$A$-modules$\pi_n(M)$, but I don't know of any interesting maps between these; perhaps this will just end up being the homology of any representative chain complex of$A$-modules. But then, I certainly don't see a natural way of getting from an obvious candidate for object of$\mathcal{D}(\mbox{Mod}_A)$to an inverse equivalence$\mathcal{D}(\mbox{Mod}_A) \rightarrow \mbox{ModSpectra}_A$.A$-module spectrum.
Incidentally, what does this induce on the level of categories? The obvious first guess is that $A$-module spectra actually form a topological category and that passing to $\mathcal{D}(\mbox{Mod}_A)$ applies $\pi_0$.
# If $A\in \mbox{Rings}\subset E_\infty\mbox{-rings}$, what is the equivalence $\mathcal{D}(\mbox{Mod}_A) \simeq \mbox{ModSpectra}_A$?
In Lurie's "A Survey of Elliptic Cohomology", he writes on page 14 that if $A$ is an ordinary commutative ring considered as an $E_\infty$-ring, then $A$-module spectra are the same thing as objects of the derived category of $A$-modules. This is mysterious to me. On the one hand, to an $A$-module spectrum $M$ we might associate the $A$-modules $\pi_n(M)$, but I don't know of any interesting maps between these; perhaps this will just end up being the homology of any representative chain complex of $A$-modules. But then, I certainly don't see an obvious candidate for an inverse equivalence $\mathcal{D}(\mbox{Mod}_A) \rightarrow \mbox{ModSpectra}_A$.