We prove several K\"unneth formulas in motivic homotopy categories and deduce a Verdier pairing in these categories following SGA5, which leads to the characteristic class of a constructible motive, an invariant closely related to the Euler-Poincar\'e characteristic. We prove an additivity property of the Verdier pairing using the language of derivators, following the approach of May and Groth-Ponto-Shulman; using such a result we show that in the presence of a Chow weight structure, the characteristic class for all constructible motives is uniquely characterized by proper covariance, additivity along distinguished triangles, refined Gysin morphisms and Euler classes. In the relative setting, we prove the relative K\"unneth formulas under some transversality conditions, and define the relative characteristic class.

中文翻译：

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迹的动机和可加性的 Künneth 公式

我们证明了动机同伦范畴中的几个 K\"unneth 公式，并在 SGA5 之后推导出这些范畴中的 Verdier 配对，这导致了可构造动机的特征类，一个与 Euler-Poincar\'e 特征密切相关的不变量。我们证明遵循 May 和 Groth-Ponto-Shulman 的方法，使用导数语言的 Verdier 配对的可加性属性；使用这样的结果，我们表明在存在 Chow 权重结构的情况下，所有可构造动机的特征类是唯一的特征为适当的协方差、沿区分三角形的可加性、精制的 Gysin 态射和欧拉类。在相对设置中，我们证明了一些横向条件下的相对 K\"unneth 公式，并定义了相对特征类。