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PreprintCC BY-SA 4.0arXiv

The WittVectors package for Macaulay2

  • Anne Fayolle
  • Abhay Goel
  • Devlin Mallory
  • Eamon Quinlan-Gallego
  • Teppei Takamatsu
Open at sourcePDFDOICite

Abstract

We implement a Macaulay2 package for computations involving rings of truncated Witt vectors Wn(R) of a finitely generated Fp-algebra R. This package includes ring operations (addition, multiplication, Frobenius, Verschiebung, etc.) on elements of Wn(R), conversion between tuple representatives and ghost map representatives of elements of truncated Witt vectors over polynomial rings, and explicit computation of Wn(R) as a finite-type algebra over Z/pn. This functionality is based on an algorithm we developed for performing arithmetic operations in rings of finite length p-typical Witt vectors over finitely generated Fp-algebras. In addition, we implement an algorithm to calculate lifts of Frobenius from Fp-algebras to flat lifts over Z/p2Z, and an algorithm to find the quasi-F-splitting height of a local or graded complete intersection ring.

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Cite

BibTeXRIS
BibTeX
@misc{fayolle2026wittvectors,
  title = {{The WittVectors package for Macaulay2}},
  author = {Anne Fayolle and Abhay Goel and Devlin Mallory and Eamon Quinlan-Gallego and Teppei Takamatsu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.37931},
  eprint = {2609.37931},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.37931},
  note = {CC BY-SA 4.0},
}
RIS
TY  - UNPB
TI  - The WittVectors package for Macaulay2
AU  - Anne Fayolle
AU  - Abhay Goel
AU  - Devlin Mallory
AU  - Eamon Quinlan-Gallego
AU  - Teppei Takamatsu
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.37931
UR  - https://arxiv.org/abs/2609.37931
AB  - We implement a Macaulay2 package for computations involving rings of truncated Witt vectors W_n(R) of a finitely generated F_p-algebra R. This package includes ring operations (addition, multiplication, Frobenius, Verschiebung, etc.) on elements of W_n(R), conversion between tuple representatives and ghost map representatives of elements of truncated Witt vectors over polynomial rings, and explicit computation of W_n(R) as a finite-type algebra over Z/p^n. This functionality is based on an algorithm we developed for performing arithmetic operations in rings of finite length p-typical Witt vectors over finitely generated F_p-algebras. In addition, we implement an algorithm to calculate lifts of Frobenius from F_p-algebras to flat lifts over Z/p^2Z, and an algorithm to find the quasi-F-splitting height of a local or graded complete intersection ring.
ER  -