Serre's conjecture on Artin representations for regular local rings

Let $A$ be a regular local ring with maximal ideal $\mathfrak{m}$, and let $G$ be a finite group acting on $A$. For $\sigma\in G$, let $I_\sigma$ be the ideal generated by $\{a-\sigma(a)\mid a\in A\}$. Assume that $A^G$ is noetherian and $A$ is finite over $A^G$, that $A/I_\sigma$ has finite length for every $\sigma\in G\setminus\{1\}$, and that the induced map

$$A^G/(A^G\cap\mathfrak{m})\longrightarrow A/\mathfrak{m}$$

is an isomorphism. Define $a_G\colon G\rightarrow\mathbb{Z}$ by

$$a_G(\sigma):=\begin{cases}-\operatorname{length}_A(A/I_\sigma)&\text{if }\sigma\in G\setminus\{1\},\\-\displaystyle\sum_{\xi\in G\setminus\{1\}}a_G(\xi)&\text{if }\sigma=1.\end{cases}$$

Serre's conjecture. Let $\ell$ be a prime number invertible in $A$. The function $a_G$ is the character of a $\overline{\mathbb{Q}}_\ell$-representation of $G$, called the Artin representation, and this representation is $\mathbb{Q}_\ell$-rational.

This generalizes the realization and rationality of the Artin character for discrete valuation rings to higher-dimensional regular local rings. The paper proves the conjecture in the equal-characteristic case; the supplied context does not establish its status in the remaining cases.