Hochster–Roberts theorem

In algebra, the Hochster–Roberts theorem, introduced by Melvin Hochster and Joel L. Roberts in 1974,^[1] states that rings of invariants of linearly reductive groups acting on regular rings are Cohen–Macaulay.

In other words,^[2] if V is a rational representation of a linearly reductive group G over a field k, then there exist algebraically independent invariant homogeneous polynomials ${\displaystyle f_{1},\cdots ,f_{d}}$ such that ${\displaystyle k[V]^{G}}$ is a free finite graded module over ${\displaystyle k[f_{1},\cdots ,f_{d}]}$.

In 1987, Jean-François Boutot proved^[3] that if a variety over a field of characteristic 0 has rational singularities then so does its quotient by the action of a reductive group; this implies the Hochster–Roberts theorem in characteristic 0 as rational singularities are Cohen–Macaulay.

In characteristic p>0, there are examples of groups that are reductive (or even finite) acting on polynomial rings whose rings of invariants are not Cohen–Macaulay.

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