TITLE: Descent properties of topological Hochschild homology
QUESTION [10 upvotes]: 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 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.

REPLY [5 votes]: 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.