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PreprintarXiv licencearXiv

Test ideals can be computed by boundaries

  • Rahul Ajit
  • with Appendix joint with Daniel Apsley
Open at sourcePDFDOICite

Abstract

Let R be a normal F-finite ring of characteristic p>0. We show that one effective ℚ-divisor Δ, with KR+Δ ℚ-Cartier of index prime to p, computes the big test ideal. This answers a question of Schwede and Smith. More generally, we prove their conjecture that test module of a Cartier algebra acting on an ideal is the test module of a single map ψ in the algebra and even the image of ψ. We prove a global version for schemes quasi-projective over an F-finite local ring as well. In particular, every such normal, locally strongly F-regular scheme admits an effective boundary Δ for which (X,Δ) is locally strongly F-regular and KX+Δ is ℚ-Cartier of index prime to p.

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Cite

BibTeXRIS
BibTeX
@misc{ajit2026test,
  title = {{Test ideals can be computed by boundaries}},
  author = {Rahul Ajit and with Appendix joint with Daniel Apsley},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.38163},
  eprint = {2610.38163},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.38163},
}
RIS
TY  - UNPB
TI  - Test ideals can be computed by boundaries
AU  - Rahul Ajit
AU  - with Appendix joint with Daniel Apsley
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.38163
UR  - https://arxiv.org/abs/2610.38163
AB  - Let R be a normal F-finite ring of characteristic p>0. We show that one effective ℚ-divisor Δ, with K_R+Δ ℚ-Cartier of index prime to p, computes the big test ideal. This answers a question of Schwede and Smith. More generally, we prove their conjecture that test module of a Cartier algebra acting on an ideal is the test module of a single map ψ in the algebra and even the image of ψ. We prove a global version for schemes quasi-projective over an F-finite local ring as well. In particular, every such normal, locally strongly F-regular scheme admits an effective boundary Δ for which (X,Δ) is locally strongly F-regular and K_X+Δ is ℚ-Cartier of index prime to p.
ER  -