Globalize the étale-local cyclotomic Morita comparison

Construct a coherent global cyclotomic equivalence between $THH(A)$ and $THH(Z)$ for an Azumaya algebra $A$ over a commutative ring $Z$ that is not globally split, by coherently gluing the étale-local Morita trace maps and identifying the resulting global map with the underlying-spectrum equivalence $\Phi_A$.

Background

The paper constructs a global equivalence of underlying THHTHH spectra, denoted ΦA\Phi_A, normalized by the reduced trace. Separately, after an étale splitting, it obtains cyclotomic Morita trace equivalences τP\tau_P.

When the Brauer class is nonzero, the authors do not establish that the local maps τP\tau_P can be coherently glued across the Čech nerve, nor that any such gluing agrees with ΦA\Phi_A beyond π0\pi_0. A global cyclotomic comparison is therefore unresolved in general.

References

In general, transforming the $\acute{e}$tale-local cyclotomic statement (endowed with $\tau_P$ degreewise on a cover) to a single global equivalence of cyclotomic spectra two obstacles occur. The first is gluing the $\tau_P$ coherently across the cover (the original \v Cech problem of Remark \ref{remcoherence}). The second is, even having that, identifying the result with the independently-constructed global $\Phi_A$. We do not resolve these obstacles in this paper.

— Azumaya descent and étale-local cyclotomic comparison: a noncommutative filtration  (2610.02870 - Lê et al., 2 Oct 2026) in Remark \ref{remcoherence}, Section 3.1 and the proof of Theorem \ref{thmazumaya}

We do not construct or transport a filtration in the derived case in this paper (see Table \ref{tabsummary}).

— Azumaya descent and étale-local cyclotomic comparison: a noncommutative filtration  (2610.02870 - Lê et al., 2 Oct 2026) in Section 5, immediately following Theorem \ref{thmderived}; see Table \ref{tabsummary}