Hochster–Roberts Theorem



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What is Hochster–Roberts theorem?

In algebra, the Hochster–Roberts theorem, introduced by Hochster and Roberts , states that rings of invariants of linearly reductive groups acting on regular rings are Cohen–Macaulay. In other words, 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 . proved 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.

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Hochster-Roberts theorem

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