Generalizing Witt vector construction

TL;DR

This paper introduces a generalized construction of Witt vector functors parameterized by polynomials over integers, extending classical Witt vector theory and exploring their structural properties.

Contribution

It develops a new class of Witt vector functors based on polynomial parameters, broadening the scope of classical Witt vector constructions.

Findings

Constructed functors from $ ext{Z}[q]$-algebras to rings

Analyzed functorial properties like induction and restriction

Explored structural aspects such as classification and unitalness

Abstract

The purpose of this this paper is to generalize the functors arising from the theory of Witt vectors duto to Cartier. Given a polynomial $g(q)\in \mathbb Z[q]$, we construct a functor ${\overline {W}}^{g(q)}$ from the category of $\mathbb Z[q]$-algebras to that of commutative rings. When $q$ is specialized into an integer $m$, it produces a functor from the category of commutative rings with unity to that of commutative rings. In a similar way, we also construct several functors related to ${\overline { W}}^{g(q)}$. Functorial and structural properties such as induction, restriction, classification and unitalness will be investigated intensively.