Comparison of Logarithmic Hochschild Homologies

Source abstract

We construct a comparison morphism $\operatorname{HH}^{\mathrm{Rog}}\to\operatorname{HH}^{\mathrm{Ols}}$ between Rognes' logarithmic Hochschild homology and Olsson's logarithmic Hochschild homology which generalizes the comparison between Gabber's log cotangent complex $\mathbb{L}^{\mathrm{Gab}}$ and Olsson's log cotangent complex $\mathbb{L}^{\mathrm{Ols}}.$ In particular, Olsson's comparison is recovered as the first associated graded piece of our comparison morphism $\operatorname{HH}^{\mathrm{Rog}}\to\operatorname{HH}^{\mathrm{Ols}}.$ In the development of this comparison, we also prove a descent result for the cotangent complex generalizing a result of Bhatt, and construct a notion of HKR completed Hochschild homology for a general morphism of derived stacks which is functorial in the morphism and generalizes existing constructions of Hochschild homology when the base is affine.