Constructible Witt theory of schemes
Onkar Kamlakar Kale, Girja S Tripathi
TL;DR
The paper defines and analyzes constructible Witt groups W^i_c(X_{et}, Λ) for schemes X with Λ of finite characteristic not equal to 2, building on Balmer’s framework for triangulated categories with duality and Cisinski–Déglise’s six-functor formalism for locally constructible motives. It shows that for smooth complex varieties and finite Λ with 2 invertible, the algebraic constructible Witt theory agrees with the topological theory on X^{an}, providing a robust bridge between algebraic and analytic settings. It then identifies the Witt theory of Spec ℝ with a Z/2Z-equivariant Witt theory of Λ-modules and establishes algebraic analogues of signatures, including pushforwards that yield real and complex Witt-valued invariants. The work also develops localization and functoriality properties, discusses homotopy-invariance limitations, and outlines future directions linking constructible Witt theory to motivic homotopy and algebraic cobordism, with explicit comparisons to real/complex cases and potential for deeper cobordism interpretations.
Abstract
We study the constructible Witt theory of étale sheaves of $Λ$-modules on a scheme $X$ for coefficient rings $Λ$ 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 Déglise on six-functor formalism for derived categories of étale 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.