While algebraic K-theory, TC, and THH all enjoy a common localization property, a key tool to study the resulting cofiber sequences - dévissage - is only available for algebraic K-theory. There have been two major attempts to circumvent this: The first, due to Hesselholt--Madsen and Blumberg--Mandell, involves a model of THH and TC of Waldhausen categories that produces new localization sequences as an instance of Waldhausen's fibration theorem. The second, due to Rognes--Sagave--Schlichtkrull, bypasses dévissage entirely. Instead, they use Rognes' logarithmic THH to generalize a classical residue sequence involving logarithmic differential forms to a cofiber sequence of cyclotomic spectra.
I will report on work in progress, in which I will address the problem of reconciling the localization property of THH with that of its logarithmic counterpart. When combined with forthcoming work of Ramzi--Sosnilo--Winges, this becomes closely related to the problem of realizing logarithmic THH as the THH of a stable infinity-category, providing a candidate category of "logarithmic modules" in specific cases.
