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Comparison of Logarithmic Hochschild Homologies

  • Joseph Stahl
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Abstract

We construct a comparison morphism HHRog→HHOls between Rognes' logarithmic Hochschild homology and Olsson's logarithmic Hochschild homology which generalizes the comparison between Gabber's log cotangent complex 𝕃Gab and Olsson's log cotangent complex 𝕃Ols. In particular, Olsson's comparison is recovered as the first associated graded piece of our comparison morphism HHRog→HHOls. 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.

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Cite

BibTeXRIS
BibTeX
@misc{stahl2026comparison,
  title = {{Comparison of Logarithmic Hochschild Homologies}},
  author = {Joseph Stahl},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.31879},
  eprint = {2610.31879},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.31879},
}
RIS
TY  - UNPB
TI  - Comparison of Logarithmic Hochschild Homologies
AU  - Joseph Stahl
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.31879
UR  - https://arxiv.org/abs/2610.31879
AB  - We construct a comparison morphism HH^Rog→HH^Ols between Rognes' logarithmic Hochschild homology and Olsson's logarithmic Hochschild homology which generalizes the comparison between Gabber's log cotangent complex 𝕃^Gab and Olsson's log cotangent complex 𝕃^Ols. In particular, Olsson's comparison is recovered as the first associated graded piece of our comparison morphism HH^Rog→HH^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.
ER  -