Hochster–Roberts theorem explained

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

f _{1,} … , f _{d} such that k[V] ^{G} is a free finite graded module over k[f _{1,} … , 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.

Notes and References

1. Hochster . Melvin . Melvin Hochster. Roberts . Joel L.. Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay. 10.1016/0001-8708(74)90067-X. free. 0347810. 1974. Advances in Mathematics. 0001-8708. 13. 2. 115–175.
2. p. 199
3. Boutot . Jean-François. Singularités rationnelles et quotients par les groupes réductifs. 10.1007/BF01405091. 877006. 1987. Inventiones Mathematicae. 0020-9910. 88. 1. 65–68.