Linearity of topological periodic cyclic homology

Question

Let $A$ be an $E_\infty$ ring spectrum, $B$ a ring spectrum. Then if I understand correctly, $TP(A)$ is a ring spectrum by the lax monoidal property of $TP$. Suppose there is a map of ringed spectra from $A \to B$. I would like to understand when $TP(B)$ is a module over $TP(A)$. Namely, I think the condition should be that the induced map

$$ A\otimes_\mathbb{S} B \to B$$

is a map of ringed spectra. Is there standard terminology for such maps?

I am novice in topological cyclic homology but have been studying the notes:

https://www.uni-muenster.de/IVV5WS/WebHop/user/nikolaus/Papers/Lectures.pdf

https://swc-math.github.io/aws/2019/2019MorrowNotes.pdf

Answer 1

If you want a full module structure (rather than just an "action map" $TP(A)\otimes TP(B)\to TP(B)$, which is enough for some arguments), the reasonable notion would be for $B$ to be an $A$-module in $\mathbb E_1$-rings.

This is equivalent to asking for $B$ to be an $\mathbb E_1$-ring in (the symmetric monoidal $\infty$-category of) $A$-modules, i.e. an $A$-algebra (this is a kind of centrality condition for the map $A\to B$). There is a canonical way of turning, e.g. an $\mathbb E_2$-map $A\to B$ into an $\mathbb E_1$-$A$-alebra (more generally, of turning an $\mathbb E_{n+1}$-map from $A$ into an $\mathbb E_n$-$A$-algebra), but "of course" these are in general different.

(the difference between an algebra map, and an algebra over the source occurs already in classical algebra, just to clarify - the higher algebraic subtleties only show up in the intermediate $\mathbb E_n$-stages).