Abstract
The theorem of Hochster and Roberts says that, for every moduleV of a linearly reductive groupG over a fieldK, the invariant ringK[V] G is Cohen-Macaulay. We prove the following converse: ifG is a reductive group andK[V] G is Cohen-Macaulay for every moduleV, thenG is linearly reductive.
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Kemper, G. A characterization of linearly reductive groups by their invariants. Transformation Groups 5, 85–92 (2000). https://doi.org/10.1007/BF01237180
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DOI: https://doi.org/10.1007/BF01237180