# $\mathbb{A}^{1}$-Local Degree via Stacks

@article{Kobin2019mathbbA1LocalDV, title={\$\mathbb\{A\}^\{1\}\$-Local Degree via Stacks}, author={Andrew Kobin and Libby Taylor}, journal={arXiv: Algebraic Geometry}, year={2019} }

We extend results of Kass--Wickelgren to define an Euler class for a non-orientable (or non-relatively orientable) vector bundle on a smooth scheme, valued in the Grothendieck--Witt group of the ground field. We use a root stack construction to produce this Euler class and discuss its relation to other versions of an Euler class in $\mathbb{A}^{1}$-homotopy theory. This allows one to apply Kass--Wickelgren's technique for arithmetic enrichments of enumerative geometry to a larger class of… Expand

#### 2 Citations

The trace of the local $\mathbf{A}^1$-degree

- Mathematics
- 2019

We prove that the local $\mathbb{A}^1$-degree of a polynomial function at an isolated zero with finite separable residue field is given by the trace of the local $\mathbb{A}^1$-degree over the… Expand

An arithmetic enrichment of B\'ezout's Theorem

- Mathematics
- 2020

The classical version of Bezout's Theorem gives an integer-valued count of the intersection points of hypersurfaces in projective space over an algebraically closed field. Using work of Kass and… Expand

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