The Regular property of Invariant Rings over Regular Domains
Shubham Jaiswal, Tony J. Puthenpurakal
TL;DR
The article proves a broad generalization of the Chevalley–Shephard–Todd theorem: if $A$ is a regular domain with field of fractions $K$, and a finite group $G\subseteq GL_n(A)$ acts linearly on $A[X_1,...,X_n]$ with $|G|$ invertible in $A$, and $G$ is generated by pseudo-reflections in $GL_n(K)$, then the invariant ring $(A[X_1,...,X_n])^G$ is regular. The proof advances through a local-to-global strategy, first establishing regularity for DVRs, then for regular local rings via induction on dimension and cohomology vanishing $H^1(G,R)=0$, and finally for regular domains by localizing at maximal ideals. A key facet is lifting regularity from the residue field using exact sequences and CST on the residue, together with a structural lemma about pseudo-reflections in the DVR/regular-local setting. This work broadens CST to arithmetic contexts and subsumes Mundelius’s Dedekind-domain case, yielding regular invariant rings in a wide class of algebro-geometric situations.
Abstract
The main result of this paper is a generalization of the theorem of Chevalley-Shephard-Todd to the rings of invariants of pseudo-reflection groups over regular domains. More precisely, let $A$ be a regular domain and let $K$ be its field of fractions. Let $G\subseteq GL_n(A)$ be a finite group. Let $G$ act linearly on $A[X_1,X_2,\dots, X_n]$ (fixing $A$). Assume that $|G|$ is invertible in $A$. We prove that if $G\subseteq GL_n(K)$ is generated by psuedo-reflections then $(A[X_1,X_2,\dots, X_n])^G$ is regular.