The Cohen-Macaulay property of invariant rings over ring of integers of a global field-II
Tony J. Puthenpurakal
TL;DR
This work studies the Cohen-Macaulayness of invariant rings $S=R^G$ with $R=A[X,Y,Z]$ and $A$ the ring of integers of a global field, under Veronese filtrations. It combines local-to-global methods, base-change to Hilbert class fields, and Ellingsrud-Skjelbred spectral sequences with detailed representation-theoretic analysis of MCM modules to prove that $S^{<ml>}$ is Cohen-Macaulay for all large $l$ when each Sylow $p$-subgroup of $G$ has exponent $p$ (or when $p$ is unramified in $K$). The results yield that $S^{<m>}$ is generalized Cohen-Macaulay, and they further relate CM to Gorenstein properties, showing equivalences with the Gorenstein property of the corresponding invariant ring over the field of fractions. Alongside, the paper provides injectivity results for class groups and explores representation-theoretic aspects of invariant rings in mixed vs equi-characteristic settings, thereby extending CM-invariant theory from $\\mathbb{Z}$ to rings of integers in global fields.
Abstract
Let $A$ be the ring of integers of a number field $K$. Let $G \subseteq GL_3(A)$ be a finite group. Let $G$ act linearly on $R = A[X,Y, Z]$ (fixing $A$) and let $S = R^G$ be the ring of invariants. Assume the Veronese subring $S^{<m>}$ of $S$ is standard graded. We prove that if for all primes $p$ dividing $|G|$, the Sylow $p$-subgroup of $G$ has exponent $p$ then for all $l \gg 0$ the Veronese subring $S^{<ml>}$ of $S$ is Cohen-Macaulay. We prove a similar result if for all primes $p$ dividing $|G|$, the prime $p$ is unramified in $K$.