Compactly supported $\mathbb{A}^{1}$-Euler characteristic and the Hochschild complex

TL;DR

This paper establishes a relationship between the $ ext{A}^1$-Euler characteristic of smooth projective schemes and their Hochschild complexes, providing a new perspective and explicit computations within algebraic geometry.

Contribution

It demonstrates that the $ ext{A}^1$-Euler characteristic can be represented by the Hochschild complex with a canonical bilinear form and explores the compactly supported $ ext{A}^1$-Euler characteristic mapping from varieties to the Grothendieck--Witt group.

Findings

Representation of $ ext{A}^1$-Euler characteristic via Hochschild complex

Explicit computations of the $ ext{A}^1$-Euler characteristic

Exposition of the compactly supported $ ext{A}^1$-Euler characteristic

Abstract

We show the $\mathbb{A}^{1}$-Euler characteristic of a smooth, projective scheme over a characteristic $0$ field is represented by its Hochschild complex together with a canonical bilinear form, and give an exposition of the compactly supported $\mathbb{A}^{1}$-Euler characteristic $\chi^{c}_{\mathbb{A}^{1}}: K_0(\mathbf{Var}_{k}) \to \text{GW}(k)$ from the Grothendieck group of varieties to the Grothendieck--Witt group of bilinear forms. We also provide example computations.