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.

References

1. Hochster, Melvin; Roberts, Joel L. (1974). "Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay". Advances in Mathematics. 13 (2): 115–175. doi:10.1016/0001-8708(74)90067-X. ISSN 0001-8708. MR 0347810.
2. Mumford, David; Fogarty, John; Kirwan, Frances (1994), Geometric invariant theory. Third edition., Ergebnisse der Mathematik und ihrer Grenzgebiete 2. Folge (Results in Mathematics and Related Areas (2)), vol. 34, Springer-Verlag, Berlin, ISBN 3-540-56963-4, MR 1304906 p. 199
3. 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.