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Graded invariants of a finitely generated rational G-algebra are finitely generated

Statement

Assume AC inherited from the invariant-theory suppliers. Let $G$ be a complex reductive affine algebraic group and let $A = ⨁_{n \geq 0} A_{n}$ be a finitely generated graded commutative $C$ -algebra with $A_{0} = C$ , equipped with a rational action of $G$ by graded algebra automorphisms (Classical complex affine algebraic actions and rational modules). Then the graded invariant subalgebra $A^{G} = ⨁_{n \geq 0} A_{n}^{G}$ is a finitely generated $C$ -algebra.

Facts & Assumptions

Given: A complex reductive affine algebraic group $G$ , a finitely generated graded $C$ -algebra $A$ with $A_{0} = C$ and a rational action of $G$ on $A$ by graded algebra automorphisms.

[F1]

Local finiteness. Every element of a rational $G$ -module lies in a finite-dimensional $G$ -stable subspace on which $G$ acts by a morphism; sums of finitely many such subspaces are again finite-dimensional and $G$ -stable. (Classical complex affine algebraic actions and rational modules)

[F2]

Finite homogeneous generation. There are finitely many homogeneous elements $a_{1}, \dots, a_{r}$ generating $A$ as a $C$ -algebra, with $a_{i} \in A_{d_{i}}$ , $d_{i} \geq 1$ because $A_{0} = C$ ; the invariant subalgebra is graded, $A^{G} = ⨁_{n} A_{n}^{G}$ . (Subalgebra generated by a subset, algebras of finite type, and module-finite algebras, Nonnegatively graded rings and modules, homogeneous elements, and twists)

[F3]

Surjectivity of invariants. If $φ : B \to A$ is a surjective $G$ -equivariant homomorphism of rational $G$ -algebras, then $φ (B^{G}) = A^{G}$ ; the conclusion holds also for the graded subalgebra of invariants. (The Reynolds operator and the ideal theory of the invariant subring (c), (d))

[F4]

Invariants of a finite-dimensional module. For a finite-dimensional rational $G$ -module $W$ the invariant algebra $C [W^{*}]^{G}$ is a finitely generated $C$ -algebra, and $C [W^{*}] = Sym (W)$ as a graded algebra. (Invariants of a finite-dimensional module are finitely generated)

Proof

technique · direct

1.1F1F2algebra

A finite-dimensional generating module. By [F2] choose homogeneous generators $a_{1}, \dots, a_{r}$ of $A$ . By [F1] each $a_{i}$ lies in a finite-dimensional $G$ -stable subspace $W_{i}$ ; since $A = ⨁_{n} A_{n}$ and the action is graded, the homogeneous components of the elements of $W_{i}$ span a finite-dimensional graded $G$ -stable space containing $a_{i}$ , so we may take each $W_{i}$ graded. Then $W = W_{1} + \dots + W_{r}$ is a finite-dimensional graded $G$ -stable subspace of $A$ whose elements contain the generators $a_{i}$ , hence generate $A$ as a $C$ -algebra.

2.1F4step 1.1construct

The symmetric algebra surjection. The universal property of the symmetric algebra of the finite-dimensional graded vector space $W$ gives a graded $C$ -algebra surjection $φ : Sym (W) \to A$ sending $W$ identically onto its image in $A$ ; it is $G$ -equivariant because $W$ is $G$ -stable and the identification $Sym (W) = C [W^{*}]$ carries the induced action to the action on polynomial functions.

3.1F3F4step 2.1

By [F3] the induced map on invariants $Sym (W)^{G} \to A^{G}$ is surjective, and $Sym (W)^{G} = C [W^{*}]^{G}$ is a finitely generated $C$ -algebra by [F4].

4.1step 3.1algebra∎

A quotient of a finitely generated $C$ -algebra is finitely generated, so $A^{G}$ is finitely generated, as claimed; the argument is the graded form of Nagata's theorem used by Brion and Hoskins.

Remarks

Noetherianity of $A$ . The hypothesis that $A$ is finitely generated over $C$ is what makes $A$ Noetherian and the quotient argument in step 3.1 available; no Hilbert-basis input beyond finite generation is used.
Gradings. The proof keeps the $Z_{\geq 0}$ -grading throughout: the generators are homogeneous, the module $W$ is chosen graded, and the surjection of step 2.1 is a graded map, so the finite generating set produced for $A^{G}$ consists of homogeneous invariants.

Depends on

Used by

Dependency tree · two levels

21 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Victoria Hoskins, Moduli Problems and Geometric Invariant Theory, FU Berlin lecture notes (2015/16) (standard reference, not scraped)
Michel Brion, Introduction to actions of algebraic groups, Les cours du CIRM 1 (2010), no. 1, 1-22 (standard reference, not scraped)