The WittVectors package for Macaulay2

Abstract

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 $\mathbb{Z}/p^n$. 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 $\mathbb{Z}/p^2\mathbb{Z}$, and an algorithm to find the quasi-$F$-splitting height of a local or graded complete intersection ring.

15 pages, comments welcome!