Perfectoid Pure and Signature Pairs

@misc{cai2026perfectoid,
  title = {{Perfectoid Pure and Signature Pairs}},
  author = {Hanlin Cai and Anne Fayolle and Linquan Ma and Karl Schwede and Ilya Smirnov and Kevin Tucker and Joe Waldron},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.22059},
  eprint = {2610.22059},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.22059},
}

TY  - UNPB
TI  - Perfectoid Pure and Signature Pairs
AU  - Hanlin Cai
AU  - Anne Fayolle
AU  - Linquan Ma
AU  - Karl Schwede
AU  - Ilya Smirnov
AU  - Kevin Tucker
AU  - Joe Waldron
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.22059
UR  - https://arxiv.org/abs/2610.22059
AB  - We extend perfectoid purity and related invariants to divisor pairs in mixed characteristic. The resulting purity condition agrees with F-purity of pairs in characteristic p as defined by Hara--Watanabe. For normal (respectively, demi-normal) log ℚ-Gorenstein pairs, we show that perfectoid pure implies log canonical (respectively, semi-log canonical). We similarly establish adjunction for perfectoid pure pairs and inversion of adjunction for principal pairs in complete intersections along a Cartier divisor. We also define perfectoid pure thresholds and prove localization and base change results, including a formula for local extensions with regular closed fiber. We establish upper bounds in terms of analytic spread, dimension, and log canonical thresholds. Finally, we also introduce perfectoid signatures for divisor pairs in mixed characteristic whose positivity characterizes the BCM-regularity of log ℚ-Gorenstein pairs. We prove a transformation rule under finite extensions and show continuity and convexity for scalings of a Cartier divisor.
ER  -