Independence of perfectoid signature from the Noether–Cohen normalization and parameters

Establish whether the perfectoid signature of a mixed-characteristic complete Noetherian normal local domain, including the perfectoid signature of a divisor pair, is independent of the chosen Noether–Cohen normalization and the chosen regular system of parameters.

Background

The perfectoid signature is defined using a Noether–Cohen normalization A contained in a complete normal local domain R together with a chosen system of parameters. The resulting normalized length is used to measure BCM-regularity and is extended in the paper to divisor pairs.

The construction is shown to be independent of the auxiliary finite normal extension used to make the divisor integral, but the authors explicitly leave unresolved whether changing the Noether–Cohen normalization or the parameter system changes the resulting signature. This issue applies already in the non-pair setting and therefore also affects the pair construction.

References

Note that, like the non-pair case (), we do not know whether perfectoid signature is independent of the choice of $A$ or \underline{x}.

— Perfectoid Pure and Signature Pairs  (2610.22059 - Cai et al., 15 Oct 2026) in Section 1, subsection “Perfectoid signature pairs” (Introduction); see also Section 18, opening discussion of perfectoid signature for pairs