Azumaya descent and étale-local cyclotomic comparison: a noncommutative filtration
Abstract
Bhatt-Morrow-Scholze motivic filtration on topological Hochschild homology (THH) is extended to Azumaya algebras over quasisyntomic rings. We prove the Azumaya descent theorem: THH of such an algebra is equivalent, as an underlying spectrum, to THH of its center, normalized on π0 by the reduced trace instead of the unit map. This comparison ΦA is not shown to be étale-locally cyclotomic. But what is cyclotomic étale-locally is a separate Morita trace, agreeing with ΦA only on π0. The resulting transported filtration is exactly the center BMS filtration, well defined up to the automorphism ambiguity of ΦA. It is insensitive to the Brauer class, reflected instead by K-theoretic and module-categorical pairings THH does not see. We establish the analogous statement for derived Azumaya algebras over connective 𝔼∞-rings, under a connectivity hypothesis.
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BibTeX
@misc{le2026azumaya,
title = {{Azumaya descent and étale-local cyclotomic comparison: a noncommutative filtration}},
author = {Hông Vân Lê and Alexander Zuevsky},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02870},
eprint = {2610.02870},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02870},
note = {CC0},
}
RIS
TY - UNPB
TI - Azumaya descent and étale-local cyclotomic comparison: a noncommutative filtration
AU - Hông Vân Lê
AU - Alexander Zuevsky
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.02870
UR - https://arxiv.org/abs/2610.02870
AB - Bhatt-Morrow-Scholze motivic filtration on topological Hochschild homology (THH) is extended to Azumaya algebras over quasisyntomic rings. We prove the Azumaya descent theorem: THH of such an algebra is equivalent, as an underlying spectrum, to THH of its center, normalized on π_0 by the reduced trace instead of the unit map. This comparison Φ_A is not shown to be étale-locally cyclotomic. But what is cyclotomic étale-locally is a separate Morita trace, agreeing with Φ_A only on π_0. The resulting transported filtration is exactly the center BMS filtration, well defined up to the automorphism ambiguity of Φ_A. It is insensitive to the Brauer class, reflected instead by K-theoretic and module-categorical pairings THH does not see. We establish the analogous statement for derived Azumaya algebras over connective 𝔼_∞-rings, under a connectivity hypothesis.
ER -