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,

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

• 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 0877006
• 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. MR1304906 ISBN 3-540-56963-4