Hochster–Roberts theorem
{{short description|Theorem in ring theory}}
In algebra, the Hochster–Roberts theorem, introduced by Melvin Hochster and Joel L. Roberts in 1974,
{{Cite journal
| last1=Hochster | first1=Melvin | author1-link=Melvin Hochster
| last2=Roberts | first2=Joel L.
| title=Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay
| doi=10.1016/0001-8708(74)90067-X
| doi-access=free
| mr=0347810
| year=1974
| journal=Advances in Mathematics
| issn=0001-8708
| volume=13
| issue=2
| pages=115–175}} states that rings of invariants of linearly reductive groups acting on regular rings are Cohen–Macaulay.
In other words,{{citation
| first1=David | last1=Mumford
| first2=John | last2=Fogarty
| first3=Frances | last3=Kirwan
| title=Geometric invariant theory. Third edition.
| series= Ergebnisse der Mathematik und ihrer Grenzgebiete 2. Folge (Results in Mathematics and Related Areas (2))
| volume=34
| publisher=Springer-Verlag, Berlin
| year=1994
| mr=1304906
| isbn=3-540-56963-4}} p. 199 if V is a rational representation of a linearly reductive group G over a field k, then there exist algebraically independent invariant homogeneous polynomials such that is a free finite graded module over .
In 1987, Jean-François Boutot proved{{Cite journal
| last1=Boutot | first1=Jean-François
| title=Singularités rationnelles et quotients par les groupes réductifs
| doi=10.1007/BF01405091
| mr=877006
| year=1987
| journal=Inventiones Mathematicae
| issn=0020-9910
| volume=88
| issue=1
| pages=65–68}} 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.