We prove that, in certain situations, intersection numbers on formal schemes that come in profinite families vary locally constantly in the parameter. To this end, we define the product $S\times M$ of a profinite set $S$ with a locally noetherian formal scheme $M$ and study intersections thereon. Our application is to the arithmetic fundamental lemma of W. Zhang where the result helps to overcome a restriction in its recent proof. Namely, it allows to spread out the validity of the AFL identity from an open to the whole set of regular semisimple elements.

中文翻译：

---

交叉点数的局部恒定性

我们证明，在某些情况下，有限族的形式方案上的交点数在参数中不断局部变化。为此，我们定义了产品$\mathrm{小号}\times 米$有限集的$\mathrm{小号}$具有本地诺特式的正式方案$米$并研究其上的交叉点。我们的应用是 W. Zhang 的算术基本引理，其结果有助于克服其最近证明中的限制。即，它允许将 AFL 身份的有效性从开放扩展到整个常规半简单元素集。