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arXiv · 2003.09457

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

Niny Arcila-Maya·Candace Bethea·Morgan Opie·Kirsten Wickelgren·Inna Zakharevich
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Abstract

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.

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Niny Arcila-Maya, Candace Bethea, Morgan Opie, Kirsten Wickelgren, Inna Zakharevich. 2020-03-20. Compactly supported $\mathbb{A}^{1}$-Euler characteristic and the Hochschild complex. https://arxiv.org/abs/2003.09457

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