Hochster–Roberts theorem

In algebra, the Hochster–Roberts theorem, introduced by Hochster and Roberts (1974), states that rings of invariants of reductive groups acting on regular rings are Cohen–Macaulay.

In other words,[1]

If V is a rational representation of a reductive group G over a field k, then there exist algebraically independent invariant homogeneous polynomials f_1, \cdots, f_d such that k[V]^G is a free finite graded module over k[f_1, \cdots, f_d].

Boutot (1987) proved that if a variety has rational singularities then so does its quotient by the action of a reductive group; this implies the Hochster–Roberts theorem as rational singularities are Cohen–Macaulay.

References

  1. ↑ Mumford 1994, pg. 199