  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.
