# The Cohen-Macaulay property of invariant rings over ring of integers of a
global field

Tony J. Puthenpurakal  Department of Mathematics, IIT Bombay, Powai, Mumbai
400 076<EMAIL_ADDRESS>
###### Abstract.

Let $A$ be the ring of integers of global field $K$. Let $G\subseteq
GL_{2}(A)$ be a finite group. Let $G$ act linearly on $R=A[X,Y]$ (fixing $A$).
Let $R^{G}$ be the ring of invariants. In the equi-characteristic case we
prove $R^{G}$ is Cohen-Macaulay. In mixed characteristic case we prove that if
for all primes $p$ dividing $|G|$ the Sylow $p$-subgroup of $G$ has exponent
$p$ then $R^{G}$ is Cohen-Macaulay. We prove a similar case if for all primes
$p$ dividing $|G|$ the prime $p$ is un-ramified in $K$.

###### Key words and phrases:

invariant rings, ring of integers of global fields, Hilbert class fields,
Cohen-Macaulay rings, group cohomology

###### 1991 Mathematics Subject Classification:

Primary 13A50; Secondary 13H10

##  1\. introduction

Let $K$ be a field and let $G\subseteq GL_{n}(K)$ be a finite group acting
linearly on $R=K[X_{1},\ldots,X_{n}]$. Let $R^{G}$ be the ring of invariants.
The study of such invariant rings have a rich history both in the non-modular
case (when $|G|$ is invertible in $K$) and in the modular case (when $|G|=0$
in $K$). See [S] for a nice book on the subject.

######  1.1.

In this paper we assume $A$ is ring of integers in a global field, i.e., $A$
is one of the following two rings

  1. (1)

the ring of integers of a number field $K$ (i.e., $K$ is a finite extension of
$\mathbb{Q}$).

  2. (2)

the ring of integers of finite extension of $F_{q}(t)$ (where $F_{q}$ is a
finite field with $q$ elements).

Let $G\subseteq GL_{n}(A)$ be a finite group. Let $R=A[X_{1},\ldots,X_{n}]$
and let $G$ act linearly on $R$ (fixing $A$). In this paper we study Cohen-
Macaulay property of $R^{G}$ when $n=2$. One might wonder why should one study
such rings. As we show that there is nice interplay between algebraic number
theory, group cohomology and commutative algebra while investigating such
questions. Previously the case when $n=2,3$ was studied when $A=\mathbb{Z}$,
see [A]. In this paper it was proved that $\mathbb{Z}[X_{1},X_{2}]^{G}$ is
Cohen-Macaulay while $\mathbb{Z}[X_{1},X_{2},X_{3}]^{G}$ need not be Cohen-
Macaulay.