$\newcommand{\Mod}{\mathbf{Mod}} \newcommand{\map}{\mathrm{map}_{E_\infty-A}}$ 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 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.

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    $\begingroup$ The map $\Gamma(\Bbb A^2, \mathcal{O}) \to \Gamma(\Bbb A^2 \setminus 0, \mathcal{O})$, when derived, induces such a map of $E_\infty$ rings. This is discussed a little here: mathoverflow.net/questions/268614/… $\endgroup$ Aug 25 '20 at 21:39
  • $\begingroup$ @TylerLawson : thanks ! Do the references you give in the beginning of that answer provide a proof of that precise statement or do they dicsuss "this phenomenon" in general ? And I'm guessing spectral algebraic geometry provides other examples of that type ? (Since you seem to say that usual quasi-affines becoming affines is a general phenomenon, and I'm guessing that similar results should hold there) $\endgroup$ Aug 25 '20 at 21:56
  • $\begingroup$ The references do discuss how quasi-affines become affine, and in particular Lurie shows that quasicoherent sheaves on a quasi-affine are equivalent to modules over the (derived) global section ring. This gets back at your original motivation because q-c sheaves on an open subscheme of an affine are more easily seen to be a full subcategory of q-c sheaves on the affine itself. $\endgroup$ Aug 25 '20 at 22:56
  • $\begingroup$ @TylerLawson : do you know if there are examples like this where everything stays connective ? (I don't know any spectral algebraic geometry, so I don't really have any intuition about this question) $\endgroup$ Aug 26 '20 at 16:29
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    $\begingroup$ The following question discusses how one can have an affine open subscheme Spec(R) of Spec(A) such that R is not a localization. I believe that this should give you an example which is not only connective, but purely algebraic. mathoverflow.net/questions/133470/… $\endgroup$ Aug 26 '20 at 17:40

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 for the stated relation to "smashing maps".

  • $\begingroup$ Thank you ! Do you know references for these $L_n$ or even just $L_1$? $\endgroup$ Oct 31 '20 at 19:51
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    $\begingroup$ Ravenel's paper (Localization with respect to certain periodic homology theories, 1984) and his Annals of Mathematics Study (Nilpotence and periodicity in stable homotopy theory, 1992) might be useful if you want to see a detailed discussion. Section 8 of the paper is specifically about L_1 and Chapter 7 of the book is about the Bousfield localizations. Earlier work on the image of J by Adams, on v_1-periodic homotopy by Mahowald and Miller, and on localizations by Bousfield is also relevant. $\endgroup$ Oct 31 '20 at 19:56
  • $\begingroup$ Thank you for the references and for the example ! $\endgroup$ Oct 31 '20 at 20:28

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