Hochster–Roberts theorem

From Wikipedia, the free encyclopedia

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


This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.