Hochster–Roberts theorem

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

In other words,^[1]

If V is a rational representation of a linearly 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 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.

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