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