On the relationship between logarithmic TAQ and logarithmic THH

Abstract

We provide a new description of logarithmic topological André-Quillen homology in terms of the indecomposables of an augmented ring spectrum. The new description allows us to interpret logarithmic TAQ as an abstract cotangent complex, and leads to a base-change formula for logarithmic topological Hochschild homology. The latter is analogous to results of Weibel-Geller for Hochschild homology of discrete rings, and of McCarthy-Minasian and Mathew for topological Hochschild homology. For example, our results imply that logarithmic THH satisfies base-change for tamely ramified extensions of discrete valuation rings.

Detecting and describing ramification for structured ring spectra

“…Calculations with log structures that are generated by more than one element are challenging because the methods above do not work. For a thorough investigation of log-étaleness and for related calculations, see Lundemo [31].…”

Motives and homotopy theory in logarithmic geometry

“…The following is a variant of the definition of log topological Hochschild homology pursued in the second-named author's thesis [Lun21,Lun22], and is also closely related to [Rog09, Section 13].…”

“…Remark 3.3. The definition is motivated by its relationship with the (spectral) log cotangent complex in [Lun21], in analogy with Example 2.13. Note that if (R, P ) → (A, M ) is a map of ordinary pre-log rings, this recovers the log Hochschild homology HH((A, M )/(R, P )) introduced in Definition 2.17 (it is enough to spell out the replete base change).…”