Cohen–Macaulayness of invariant rings of linear actions over rings of integers in number fields

Prove that for any number field K with ring of integers A, and any finite subgroup G ⊆ GL₂(A) acting linearly on the polynomial ring R = A[X,Y] (fixing A), the invariant ring R^G is Cohen–Macaulay.

Background

The paper studies the Cohen–Macaulay property of invariant rings R^G where R = A[X,Y] and A is the ring of integers of a global field. In equi-characteristic (function field) cases, the authors prove R^G is Cohen–Macaulay (Theorem 1.3). In mixed characteristic (number field) cases, they establish Cohen–Macaulayness under additional hypotheses: when all Sylow p-subgroups of G have exponent p (Theorem 1.4), when all primes p | |G| are unramified in K (Theorem 1.5), and for cyclic groups under specific eigenvalue conditions (Theorem 1.6).

These results motivate a general expectation that no extra hypotheses are needed in the number field case. Conjecture 1.7 explicitly asserts that for any ring of integers A in a number field and any finite subgroup G ⊆ GL₂(A) acting linearly on A[X,Y], the invariant ring A[X,Y]^G is Cohen–Macaulay. The authors’ theorems provide several strong partial confirmations of this conjecture in mixed characteristic settings.