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 [math]\displaystyle{ f_1, \cdots, f_d }[/math] such that [math]\displaystyle{ k[V]^G }[/math] is a free finite graded module over [math]\displaystyle{ k[f_1, \cdots, f_d] }[/math].
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
- ↑ "Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay". Advances in Mathematics 13 (2): 115–175. 1974. doi:10.1016/0001-8708(74)90067-X. ISSN 0001-8708.
- ↑ 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)), 34, Springer-Verlag, Berlin, ISBN 3-540-56963-4 p. 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.
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