# Associative Ring Spectra and Derived Completion

So, I was thinking before that this might have some nice, simple topos theoretic explanation, but Jacob disabused me of that notion. However, I'm still very interested in the following question:

Is there a criterion for determining when a morphism $f:A\to B$ of connective $A_\infty$-ring spectra is such that $A$ can be recovered as the (homotopy) limit of the (homotopical) Amitsur complex on $f$ in the $(\infty,1)$-category of $A_\infty$-ring spectra? For what it's worth, in the context of EKMM, such a criterion is given as an (unproven) proposition in a paper of Carlsson's (Proposition 3.3 here). There, it is stated that a morphism $f:A\to B$ of connective associative $\mathbb{S}$-algebras has the desired property if $\pi_0(f)$ is an isomorphism and $\pi_1(f)$ is onto. I'm interested in proving this in the $\infty$-categorical context (and really proving it at all).

If by effective monomorphism'' you mean the categorical dual of the condition of being an effective epimorphism, then it is not (or at least not obviously) equivalent to the statement that A can be recovered as the totalization of the cosimplicial A-module given by the tensor powers of B, because tensor product is not the same as coproduct when working with associative rings.
• Jacob have you (or anyone else) written down somewhere a proof that it's true under the conditions (on $\pi_0$ and $\pi_1$) that Carlsson gives? – Jonathan Beardsley Oct 17 '14 at 21:45