**Question:** What is the finest topology on $\mathrm{CAlg}$ (commutative ring spectra) for which *THH* (Topological Hochschild Homology) satisfies descent?


Adaptations of the arguments appearing in Section 3 of [BMS2](https://arxiv.org/abs/1802.03261) show that *THH* has flat descent for simplicial commutative rings. However, the methods therein cannot be used to check whether *THH* has flat (or any weaker form of) descent for commutative ring spectra; simplicial $R$-algebras have the special property of being the nonabelian derived category of the category of finitely generated polynomial $R$-algebras. 



Descent properties of topological Hochschild homology
In Theorem 1.2 of B. I. Dundas and J. Rognes: "Cubical and cosimplicial descent", Journal of the London Mathematical Society (2) 98 (2018) 439-460, DOI 10.1112/jlms.12141, we showed that for each $1$-connected map $\phi : A \to B$, of connective commutative $S$-algebras, the map from $THH(A)$ to the homotopy limit of the cosimplicial spectrum
$$
[q] \mapsto THH(B \wedge\_A \dots \wedge\_A B)
$$
(with $q+1$ copies of $B$) is an equivalence. Here $1$-connected means that $\pi\_0(\phi)$ is an isomorphism and $\pi\_1(\phi)$ is surjective. This should let you reduce the general descent question to the one for (simplcial) commutative rings.


