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

Journal Article

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.

### Cited Authors

• Arcila-Maya, N; Bethea, C; Opie, M; Wickelgren, K; Zakharevich, I