The asymptotic surjective number and F-surjective signature

@misc{smirnov2026asymptotic,
  title = {{The asymptotic surjective number and F-surjective signature}},
  author = {Ilya Smirnov and Kevin Tucker},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.22104},
  eprint = {2610.22104},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.22104},
}

TY  - UNPB
TI  - The asymptotic surjective number and F-surjective signature
AU  - Ilya Smirnov
AU  - Kevin Tucker
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.22104
UR  - https://arxiv.org/abs/2610.22104
AB  - We study two invariants measuring the asymptotic abundance of surjections between modules. We prove that Hashimoto's asymptotic surjective number is attained over local rings, hence it is a rational number. We also prove that its associated function is constructible in families and on spectra; on the latter, it is moreover lower semicontinuous on the support of the target. We establish the existence of the F-surjective signature, a common generalization of F-signature and dual F-signature, and prove that it is superadditive in its source and varies lower semicontinuously across the spectrum under mild assumptions. As applications, we obtain new rationality results for dual F-signature and prove a transformation inequality under finite maps, with consequences for quotient singularities and reduction mod p.
ER  -