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.

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.

Question: What is the finest topology on $\mathrm{CAlg}$ (commutative ring spectra) for which $THH$ 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.

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.

Question: What is the finest topology on $\mathrm{CAlg}$ for which $THH$ 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.

Question: What is the finest topology on $\mathrm{CAlg}$ (commutative ring spectra) for which $THH$ 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.