The Deligne–Milnor formula

Let $S$ be a strictly henselian trait with closed point $s$, generic point $\eta$, and geometric generic point $\bar\eta$. Let $p:U\to S$ be flat, with $U$ regular and purely of relative dimension $n$, smooth except at an isolated closed point $x\in U_s$. Define

$$\mu_{U/S}:=\operatorname{length}_{\operatorname{\mathcal O}_{U,x}}\left(\underline{\operatorname{Ext}}^1(\Omega^1_{U/S},\operatorname{\mathcal O}_U)_x\right).$$

For a prime $\ell$ different from the residue characteristics, let $\Phi_x$ be the $\mathbb{Q}_{\ell}$-vector space of vanishing cycles in degree $n$, with inertia action, and define its total dimension by $\operatorname{dimtot}(\Phi_x)=\dim(\Phi_x)+\operatorname{Sw}(\Phi_x)$. Deligne–Milnor formula. In the above situation,

$$\mu_{U/S}=(-1)^n\operatorname{dimtot}(\Phi_x).$$

The formula is known when $S$ has pure characteristic, when the relative dimension is zero, when the singularity is ordinary quadratic, and when $n=1$; it remains open in mixed characteristic beyond these cases.