Constructible Witt theory of schemes

TL;DR

This paper develops a constructible Witt theory for schemes using recent six-functor formalism advances, linking algebraic and topological Witt theories for smooth complex varieties with finite coefficients.

Contribution

It introduces a new constructible Witt theory framework for schemes based on étale sheaves and recent formalism, connecting algebraic and topological Witt theories.

Findings

Constructible Witt theory defined for schemes with finite characteristic coefficients.

Identification of algebraic and topological Witt theories for smooth complex varieties.

Provides a cohomological invariant framework for schemes.

Abstract

We study the constructible Witt theory of \'etale sheaves of $\Lambda$-modules on a scheme $X$ for coefficient rings $\Lambda$ having finite characteristic not equal to 2 and prime to the residue characteristics of the scheme $X$. Our construction is based on the recent advances by Cisinski and D\'eglise on six-functor formalism for derived categories of \'etale motives and offers a background for the study of constructible Witt theory as a cohomological invariant for schemes. In the case of smooth complex algebraic varieties and finite coefficient rings, we show that the algebraic constructible Witt theory studied in this paper can be identified with the topological constructible Witt theory.