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.







Niny Johanna Arcila-Maya

William W. Elliott Assistant Research Professor

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