# Interesting "epimorphisms" of $E_\infty$-ring spectra

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

• 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/… Aug 25, 2020 at 21:39
• @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) Aug 25, 2020 at 21:56
• 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. Aug 25, 2020 at 22:56
• @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) Aug 26, 2020 at 16:29
• 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/… Aug 26, 2020 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".
• Thank you ! Do you know references for these $L_n$ or even just $L_1$? Oct 31, 2020 at 19:51