Characteristic cycles and non-acyclicity classes for constructible étale sheaves

Let $f:X\to Y$ be a morphism between smooth schemes over a perfect field $k$. Let $\mathcal{F}$ be a constructible sheaf on $X$ and $Z$ a closed subscheme of $X$ such that $f$ is $SS(\mathcal{F})$-transversal outside $Z$. We construct a class supported on the non-transversal locus $Z$ by using the characteristic cycle $CC(\mathcal{F})$ defined by T. Saito. This class is a geometric counterpart of the non-acyclicity class introduced by the third author and Zhao in [Invent. Math. 240 (2025), 123-191]. Under certain conditions, the formation of this class is compatible with base change and proper pushforward. It also satisfies the Milnor formula and a conductor formula. We conjecture that the image of this geometric class under the cycle class map is the non-acyclicity class. This conjecture can be viewed as a (relative version of) Milnor-type formula for non-isolated singularities. We prove this conjecture when $k$ is algebraically closed, $X$ is quasi-affine and $Y={\mathbb{A}}_k^1$.