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

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 ${\chi }_{{\mathbb{A}}^1}^c:K_0({\mathbf{Var}}_k)\to \text{GW}(k)$ from the Grothendieck group of varieties to the Grothendieck--Witt group of bilinear forms. We also provide example computations.

arXiv comment24 pages. Accepted for publication in Topology and its Applications

References in corpus (7)

• The six operations in equivariant motivic homotopy theory
• Hermitian K-theory, derived equivalences and Karoubi's Fundamental Theorem
• A quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula
• A^1-Euler classes: six functors formalisms, dualities, integrality and linear subspaces of complete intersections
• Tensor-triangulated categories and dualities
• The Grothendieck ring of varieties and algebraic K-theory of spaces
• The relation between Grothendieck duality and Hochschild homology

Cited by in corpus (1)

• On the ${\mathbb{A}}^1$-Euler characteristic of the variety of maximal tori in a reductive group