We equate various Euler classes of algebraic vector bundles, including those of [12] and one suggested by M. J. Hopkins, A. Raksit, and J.-P. Serre. We establish integrality results for this Euler class and give formulas for local indices at isolated zeros, both in terms of the six-functors formalism of coherent sheaves and as an explicit recipe in the commutative algebra of Scheja and Storch. As an application, we compute the Euler classes enriched in bilinear forms associated to arithmetic counts of d-planes on complete intersections in $\mathbb P^n$ in terms of topological Euler numbers over $\mathbb {R}$ and $\mathbb {C}$ .

我们将代数向量丛的各种 Euler 类等同起来，包括 [12] 中的那些以及 MJ Hopkins、A. Raksit 和 J.-P 建议的一类。塞尔。我们为这个 Euler 类建立了完整性结果，并给出了孤立零点处的局部指数的公式，既根据相干层的六函子形式主义，也作为 Scheja 和 Storch 的交换代数中的显式配方。作为一个应用程序，我们根据 $\mathbb {R}$ 和 $\mathbb上的拓扑欧拉数来计算与 $\mathbb P^n$中完整交叉点上的d平面的算术计数相关联的双线性形式的欧拉类{C}$ 。