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

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