Abstract

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.