algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

, the Hochster–Roberts theorem, introduced by

Melvin Hochster Melvin Hochster (born August 2, 1943) is an American mathematician working in commutative algebra. He is currently the Jack E. McLaughlin Distinguished University Professor Emeritus of Mathematics at the University of Michigan. Education Hochs ...

and Joel L. Roberts in 1974, states that rings of invariants of linearly

reductive group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation that has a finite kernel and is a ...

s acting on

regular ring In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let A be any Noetherian local ring with unique maxi ...

s are Cohen–Macaulay. In other words, if ''V'' is a rational representation of a linearly reductive group ''G'' over a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

''k'', then there exist algebraically independent invariant homogeneous polynomials

f_1, \cdots, f_d

such that

k G

is a free finite

graded module Grade most commonly refers to: * Grading in education, a measurement of a student's performance by educational assessment (e.g. A, pass, etc.) * A designation for students, classes and curricula indicating the number of the year a student has reac ...

over

k_1, \cdots, f_d /math>.

In 1987, Jean-François Boutot proved that if a variety over a field of characteristic 0 has

rational singularities In mathematics, more particularly in the field of algebraic geometry, a scheme X has rational singularities, if it is normal, of finite type over a field of characteristic zero, and there exists a proper birational map :f \colon Y \rightarrow ...

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

Theorems in algebra {{algebra-stub