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Lazardian Witt vectors

Alexander Lai De Oliveira

TL;DR

The paper develops a choice-free framework for universal residual perfections via Lazard's universal $ rac{1}{p}$-ring, introducing Lazardian Witt vectors and Lazardian jet algebras to realize a formal adjoint-equivalence structure between residual-perfection categories and perfect residue algebras. Central results include the construction of the Lazardian Witt vector functor $W^{(t)}_{m,q}$, its adjoint relationships with the residue functor $U$, and the jet-algebra adjunction $(oldsymbol{ abla},W)$, culminating in a Lazardian URP theorem that describes $A^u$ as $W(oldsymbol{ abla}(A)^{pf})$ in a canonical, choice-controlled manner. In equal characteristic, the universal residual perfection is computed explicitly as $k^{u}=( ext{HS}^m(k) ext{pf})[[oldsymbol{ ext{π}}]]/(oldsymbol{ ext{π}}^{m+1})$, connecting jet-derivation frameworks with residual perfections and extending classical Witt-vector theory to the Lazardian setting. These constructions illuminate how to canonicalize residual-perfection moduli, relate ramified Witt vectors to Lazardian structures, and integrate Hasse–Schmidt derivations into a broader, functorial Witt-vector landscape.

Abstract

We use Lazard's universal $π$-ring to construct a variation of the $π$-typical ramified Witt vector functors, which we call the Lazardian Witt vector functor. We then use the Lazardian Witt vector functor to construct the universal residual perfection of a Lazardian algebra.

Lazardian Witt vectors

TL;DR

The paper develops a choice-free framework for universal residual perfections via Lazard's universal -ring, introducing Lazardian Witt vectors and Lazardian jet algebras to realize a formal adjoint-equivalence structure between residual-perfection categories and perfect residue algebras. Central results include the construction of the Lazardian Witt vector functor , its adjoint relationships with the residue functor , and the jet-algebra adjunction , culminating in a Lazardian URP theorem that describes as in a canonical, choice-controlled manner. In equal characteristic, the universal residual perfection is computed explicitly as , connecting jet-derivation frameworks with residual perfections and extending classical Witt-vector theory to the Lazardian setting. These constructions illuminate how to canonicalize residual-perfection moduli, relate ramified Witt vectors to Lazardian structures, and integrate Hasse–Schmidt derivations into a broader, functorial Witt-vector landscape.

Abstract

We use Lazard's universal -ring to construct a variation of the -typical ramified Witt vector functors, which we call the Lazardian Witt vector functor. We then use the Lazardian Witt vector functor to construct the universal residual perfection of a Lazardian algebra.
Paper Structure (11 sections, 26 theorems, 90 equations)

This paper contains 11 sections, 26 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Ramified Witt vectors in the literature
  4. Conventions
  5. Acknowledgments
  6. Lazardian Witt vectors
  7. Additional structures
  8. Explicit adjoint equivalence
  9. Lazardian jet algebras
  10. Universal residual perfection for Lazardian $O$-algebras
  11. An explicit calculation in equal characteristic

Key Result

Theorem A

Let $O$ be an object of $\mathsf{CRP}_\mathcal{L}$, and let $A$ be a Lazardian $O$-algebra. Given a solid diagram \begin{tikzcd} \mathsf{Alg}_{O}\arrow[from = d, shift right = 1ex, "W"'] &\mathsf{CRP}_{O}\arrow[l,hook']\arrow[ d, shift left = 1ex, "U"]&\mathsf{CRP}_A \arrow[d, shift left = 1ex,

Theorems & Definitions (60)

  • Definition 1.1
  • Example 1.2
  • Theorem A
  • Theorem B
  • Corollary
  • proof
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Example 2.3
  • ...and 50 more