$\newcommand{\Mod}{\mathbf{Mod}} \newcommand{\map}{\mathrm{map}\_{E\_\infty-A}}$ Suppose $i:A\to B$ is a map of $E\_\infty$-ring spectra. It induces a functor of $\infty$-categories $\Mod\_B\to\Mod\_A$ by restriction of scalars.


A reasonable question is to ask when this is fully faithful; studying the counit of the restriction-extension of scalars adjunction, it's pretty easy to check that this is the case if and only if $B\otimes\_A B\to B$ (the "multiplication" map) is an equivalence.


By studying its sections, if I'm not mistaken, one checks that this is the case if and only if the two inclusions $i\_0,i\_1: B\to B\otimes\_A B$ are equivalent as maps of $E\_\infty$-$A$-algebras.


For this it suffices that $A\to B$ be an "epimorphism" of $E\_\infty$-$A$-algebras (and I think it's actually equivalent), that is, that $\map(B,-)\to \map(A,-)$ be an inclusion of components; since $\map(A,-) \simeq \*$, this amounts to saying that $\map(B,C)$ is empty or contractible for all $C$.


For instance, this happens if $B$ is a localization of $A$ at a certain set of classes $S\subset \pi\_\*(A)$ (for instance $\mathbb{S\to Q, Z\to Q}, ku\to KU,$ etc.)


My question is:



> 
> Are there interesting cases where this happens but it's *not* a localization in the above sense ?
> 
> 
> 


In the $1$-categorical case, [this question](https://mathoverflow.net/questions/109/what-do-epimorphisms-of-commutative-rings-look-like) was asked about epimorphisms of commutative rings (for which $\Mod\_B\to \Mod\_A$ is fully faithful if and only if $A\to B$ is an epimorphism), and there are examples that are neither quotients nor localizations.


Here, quotients usually do not satisfy this property, as "$x=0$" becomes additional structure (e.g. $\mathbb F\_p\otimes\_\mathbb Z\mathbb F\_p \simeq \mathbb F\_p[\epsilon], |\epsilon|=1$ as $E\_1$-algebras), so it seems reasonable to ask what "epimorphisms" can look like in this setting.



Interesting "epimorphisms" of $E_\infty$-ring spectra
If $A$ is an $E\_\infty$ ring spectrum and $i : A \to B$ is any map of $A\_\infty = E\_1$ ring spectra such that the multiplication $\mu : B \wedge\_A B^{op} \to B$ is an equivalence, then $B \simeq LA$ where $L$ is some smashing Bousfield localization on the category of $A$-modules. In particular, $B$ will be $E\_\infty$ and $i$ is an $E\_\infty$ map. Taking $A = S$ and $L = L\_n$ to be the Bousfield localization with respect to the Johnson-Wilson theory $E(n)$, for $0 < n < \infty$, gives examples that are not given by algebraic localization at any set $S$ of classes in $\pi\_\*(A)$. The case $n=1$ corresponds to localization at ($p$-local) topological $K$-theory, with $B = L\_1 S$ closely related to the image-of-$J$ spectrum. See Definition 1.18 of Ravenel's 1984 Amer. J. Math. paper for the notion of a smashing localization, and Proposition 9.3.3 in [my AMS Memoir](https://www.mn.uio.no/math/personer/vit/rognes/papers/galois.pdf) for the stated relation to "smashing maps".


