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

• Boutot, Jean-François (1987), "Singularités rationnelles et quotients par les groupes réductifs", Inventiones Mathematicae 88 (1): 65–68, doi:10.1007/BF01405091, ISSN 0020-9910, MR 877006
• Hochster, Melvin; Roberts, Joel L. (1974), "Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay", Advances in Mathematics 13: 115–175, doi:10.1016/0001-8708(74)90067-X, ISSN 0001-8708, MR 0347810
• Mumford, D.; Fogarty, J.; Kirwan, F. Geometric invariant theory. Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) (Results in Mathematics and Related Areas (2)), 34. Springer-Verlag, Berlin, 1994. xiv+292 pp. MR 1304906 ISBN 3-540-56963-4