Ueber die Tiefe von Invariantenringen unendlicher Gruppen

TL;DR

This paper demonstrates that for non-linearly reductive reductive groups over an algebraically closed field, there exist faithful modules with arbitrarily large Cohen-Macaulay defect in their invariant rings.

Contribution

It explicitly constructs faithful modules for such groups where the Cohen-Macaulay defect of the invariant ring grows without bound.

Findings

Existence of faithful modules with arbitrarily large Cohen-Macaulay defect.

Explicit construction of modules for non-linearly reductive groups.

Invariant rings with unbounded depth deficiency.

Abstract

Let K be an algebraically closed field. For a graded K-Algebra R, we write cmdef R:=dim R -depth R. We show that for each reductive group G (over K) which is not linearly reductive, there exists a faithful G-module V such that cmdef K[\sum_i=1^k V]^G >= k-2 for all k. We will give such a V explicitly.