Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 43 (2002), No. 2, 551-555.

On the Intersection of Invariant Rings

Thomas Bayer

Institut für Informatik, Technische Universität München 80290 München, Germany e-mail: bayert@in.tum.de

Abstract. Based on Weitzenböck’s theorem and Nagata’s counterexample for Hilbert’s fourteenth problem we construct two finitely generated invariant rings R, S ⊂ K[x_{1}, x_{2}, …, x_{n}] s.t. the intersection R ∩ S is not finitely generated as a K-algebra.

1. Introduction

Recently the author has provided an algorithm for computing the intersection of invariant rings of finite groups and for computing K-vectorspace bases of the intersection of arbitrary graded finitely generated algebras up to a given degree, cf. [2]. One might ask if it is possible to extend the algorithm to compute the intersection of arbitrary finitely generated invariant rings. We give a negative answer by showing the existence of finitely generated invariant rings R, S ⊂ K[x_{1}, x_{2}, …, x_{n}] s.t. their intersection R ∩ S cannot be finitely generated. The example builds upon Weitzenböck’s theorem and a counterexample of Nagata for Hilbert’s fourteenth problem, which can be formulated as follows: Let K be a field and G ⊆ GL_{n}(K) be an algebraic subgroup. Is the invariant ring K[x_{1}, x_{2}, …, x_{n}]^{G} finitely generated as a Kalgebra?

In 1958 Nagata gave a negative answer by using commutative groups (cf. [6]) and in 1965 he provided invariant rings of non-commutative groups which are not finitely generated (cf. [7]). Later, these examples were greatly simplified and extended by R. Steinberg (cf. [9]). Meanwhile, based on the work of Roberts (cf. [8]), several counterexamples of invariant rings of algebraic G_{a}-actions have been found. We refer, e.g., to [3] and the references therein, and for a non-finitely generated invariant ring of a linear action of G_{a}^{12} on K^{19} we refer to [1].

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2002 Heldermann Verlag

2. Nagata’s counterexample

We present Nagata’s counterexample of 1965, given in [7]. Let G be an algebraic group and ρ : G → GL_{n}(K) be a linear representation. A polynomial f ∈ K[x_{1}, x_{2}, …, x_{n}] is invariant w.r.t. G if f(ρ(σ) · x) = f(x) for all σ ∈ G. The ring K[x_{1}, x_{2}, …, x_{n}]^{G} consisting of all invariant polynomials w.r.t. G is called the invariant ring of G (ρ will be omitted). The invariant ring is finitely generated if there exist invariants h_{1}, h_{2}, …, h_{m} s.t. the map K[y_{1}, y_{2}, …, y_{m}] → K[x_{1}, x_{2}, …, x_{n}]^{G}, sending y_{i} to h_{i}, is surjective.

For r ≥ s^{2}, where s ≥ 4, let a_{ij}, i = 1, 2, 3 and 1 ≤ j ≤ r, be algebraic independent elements over the the field k of characteristic 0 (k is the prime field Π of the algebraic curve defined in Ch. III of (loc. cit.)). Let k ⊂ K be a field extension containing the a_{ij}’s and set n = 2r. Consider the subgroup

G = \{( \begin{matrix}B_{1} & 0 & … & 0 \\ 0 & B_{2} & … & 0 \\ 0 & … & ... & … \\ 0 & … & 0 & B_{r} \end{matrix} ) : B_{i} = ( \begin{matrix}c_{i} & c_{i}b_{i} \\ 0 & c_{i} \end{matrix} )\} ⊂ GL_{n}(K)

where ∑_{j=1}^{r} a_{1j}b_{j} = ∑_{j=1}^{r} a_{2j}b_{j} = ∑_{j=1}^{r} a_{3j}b_{j} = 0 and ∏_{i=1}^{r} c_{i} = 1.

Theorem 1. (Nagata 1965) The invariant ring K[x_{1}, x_{2}, …, x_{n}]^{G} is not finitely generated. Proof. We refer to Theorem 1, Chapter III in [7]. □ Remark 1. Actually Nagata proved that the invariant ring is an ideal transform T (I, R) where I ⊂ R is an ideal and R a Noetherian integral domain. Ideal transforms are inherently non-terminating and provide counterexamples to the (generalized) Zariski problem, but there are several conditions for T (I, R) being finitely generated (cf., e.g., Chapter V of [7]). Serre proved that if R satisfies condition S_{2} then for any ideal I ⊂ R there exist f, g ∈ R s.t. T (I, R) = T (f, R) ∩ T (g, R) where T (f, R) and T (g, R) are finitely generated (cf. Section 7.1 of [11]).

3. Construction of the invariant rings

Let a_{ij}, i = 1, 2, 3 and 1 ≤ j ≤ r, be algebraic independent elements over k and k ⊂ K be a field extension containing the the a_{ij}’s (as in the previous section), let r ≥ s^{2}, s ≥ 4, and n = 2r. In order to obtain the counterexample we define two groups T and H s.t.

K[x_{1}, x_{2}, …, x_{n}]^{T} ∩ K[x_{1}, x_{2}, …, x_{n}]^{H} = K[x_{1}, x_{2}, …, x_{n}]^{G}

and show that K[x_{1}, x_{2}, …, x_{n}]^{H} is not finitely generated and that the group H contains subgroups H′, H′′ s.t. the invariant rings K[x_{1}, x_{2}, …, x_{n}]^{H′} and K[x_{1}, x_{2}, …, x_{n}]^{H′′} are finitely generated, but their intersection is not finitely generated. Consider the groups

T = \{( \begin{matrix}c_{1} & 0 & … & … & 0 \\ 0 & c_{1} & 0 & … & 0 \\ ⋮ & … & ... & … & ⋮ \\ 0 & … & 0 & c_{r} & 0 \\ 0 & … & … & 0 & c_{r} \end{matrix} ) : ∏_{i=1}^{r} c_{i} = 1\} ⊂ GL_{n}(K)

and

H_{k} = \{( \begin{matrix}B_{1} & 0 & … & 0 \\ 0 & B_{2} & … & 0 \\ ⋮ & … & ... & ⋮ \\ 0 & … & 0 & B_{k} \end{matrix} ) : B_{i} = ( \begin{matrix}1 & b_{i} \\ 0 & 1 \end{matrix} )\} ⊂ GL_{2k}(K)

where k = 4, …, r, and ∑_{j=1}^{k} a_{1j}b_{j} = ∑_{j=1}^{k} a_{2j}b_{j} = ∑_{j=1}^{r} a_{3j}b_{j} = 0. Note that both groups are closed, but only T is reductive.

Proposition 1. If T acts algebraically on an affine K-algebra R then R^{T} is finitely generated. In particular, the invariant ring K[x_{1}, x_{2}, …, x_{n}]^{T} is finitely generated.

Proof. The group T is a closed subgroup of the r-torus (K^{∗})^{r}, hence T is reductive and the invariant ring is finitely generated, cf. e.g., Chapter II.3 of [5]. □

In the sequel we define a linear action of H_{k}/H_{k−1}, (k ≥ 4), on K^{n} and we show, by using Weitzenböcks theorem (cf. [10]), that the invariant rings of H_{4} and H_{k}/H_{k−1} are finitely generated. We obtain the desired counterexample from K[x_{1}, x_{2}, …, x_{n}]^{H_{k}} = K[x_{1}, x_{2}, …, x_{n}]^{H_{k−1}} ∩ K[x_{1}, x_{2}, …, x_{n}]^{H_{k}/H_{k−1}}.

Theorem 2. (Weitzenböck 1932) Let K be a field of characteristic 0 and V be any finitedimensional rational G_{a}-module. Then the invariant ring K[V]^{G_{a}} is finitely generated.

Proof. We refer, e.g., to Theorem 10.1. in [4]. □

In the sequel we denote the nullspace of the matrix ( \begin{matrix}a_{11} & a_{12} & … & a_{1k} \\ a_{21} & a_{22} & … & a_{2k} \\ a_{31} & a_{32} & … & a_{3k} \end{matrix} ) by N_{k} and note

that N_{k} has dimension k − 3, provided that k ≥ 3. The embedding of N_{k} ↪ K^{r} by setting the additional coordinates to 0 will be omitted. The groups H_{k} can be identified with the nullspace N_{k} of A_{k} via the morphism of additive groups

ψ_{k} : N_{k} ∋ ( b_{1} ; b_{2} ; ⋮ ; b_{k} ) ↦ ( \begin{matrix}B_{1} & 0 & … & 0 \\ 0 & B_{2} & … & 0 \\ 0 & … & ... & ⋮ \\ 0 & … & 0 & B_{k} \end{matrix} ) , B_{i} = ( \begin{matrix}1 & b_{i} \\ 0 & 1 \end{matrix} ) .

We also omit the induced embedding of H_{k} ↪ GL_{n}(K) for 1 ≤ k ≤ r by the embedding of N_{k} and note that ψ_{k+1}(N_{k}) = ψ_{k}(N_{k}). In the sequel fix a basis β_{1}, β_{2}, … β_{k−3} of N_{k} for 4 ≤ k ≤ r s.t. β_{1}, β_{2}, … β_{k−4} is a basis of N_{k−1} and β_{k−3} extends the basis of N_{k−1} to a basis of N_{k}. Note that the groups N_{1} = N_{2} = N_{3} = {0} and that N_{k}/N_{k−1} is isomorphic to G_{a} for 4 ≤ k ≤ r via the mapping ∑_{i=1}^{k−4} λ_{i}β_{i} + λβ_{k−3} ↦ λ. The map is well defined since β_{1}, β_{2}, …, β_{k−4}, β_{k−3} form a basis, and bijective, hence an isomorphism of additive groups.

These isomorphisms are used to define a linear action for each H_{k}/H_{k−1} on K[x_{1}, x_{2}, …, x_{n}] in such a way that the corresponding invariant rings are finitely generated. Firstly, we define the representation ρ′_{k} of N_{k} by

ρ′_{k} : N_{k} → GL_{n}(K),
∑_{i=1}^{k−4} λ_{i}β_{i} + λβ_{k−3} ↦ ψ_{k}(λβ_{k−3}).

Note that ρ′_{k} is well defined since β_{1}, β_{2}, … β_{k−3} form a basis, that ρ′_{k} has kernel N_{k−1} and yields a linear representation of N_{k}/N_{k−1} on K^{n}. By applying Weitzenböck’s theorem we obtain the following result.

Proposition 2. For 4 ≤ k ≤ r the invariant ring K[x_{1}, x_{2}, …, x_{n}]^{H_{k}/H_{k−1}} is finitely generated.

Proof. As noted above, the group G_{a} is isomorphic to N_{k}/N_{k−1} and to H_{k}/H_{k−1} by sending λ ∈ G_{a} to [λβ_{k−3}] or to [ψ_{k}(λβ_{k−3})] respectively. Let ϕ_{k} be the inverse of the isomorphism ψ_{k} and define the linear representation ρ_{k} of H_{k}/H_{k−1} by ρ_{k}([σ]) := ρ′_{k}(ϕ_{k}(σ)). The representation is well defined because ker ρ′_{k} ∘ ϕ_{k} = H_{k−1}. Since H_{k}/H_{k−1} is isomorphic to G_{a} and acts linearly on K^{n} via ρ_{k}, the invariant ring K[x_{1}, x_{2}, …, x_{n}]^{H_{k}/H_{k−1}} is finitely generated by Weitzenböck’s theorem. □

Proposition 3. For 4 ≤ k ≤ r the invariant ring K[x_{1}, x_{2}, …, x_{n}]^{H_{k}} equals

K[x_{1}, x_{2}, …, x_{n}]^{H_{k}} = K[x_{1}, x_{2}, …, x_{n}]^{H_{k−1}} ∩ K[x_{1}, x_{2}, …, x_{n}]^{H_{k}/H_{k−1}}

Proof. If f ∈ K[x_{1}, x_{2}, …, x_{n}]^{H_{k−1}} ∩ K[x_{1}, x_{2}, …, x_{n}]^{H_{k}/H_{k−1}}, then f is invariant w.r.t. ψ_{k}(∑_{i=1}^{k−4} λ_{i}β_{i}) and ψ_{k}(λβ_{k−3}) for λ_{1}, λ_{2}, …, λ_{k−4}, λ ∈ K. In particular, f is invariant w.r.t. ψ_{k}(∑_{i=1}^{k−4} λ_{i}β_{i} + λβ_{k−3}). Since β_{1}, β_{2}, …, β_{k−3} form a basis of N_{k} and ψ_{k} is an isomorphism, the polynomial f is invariant w.r.t. the group H_{k}. The converse inclusion is obvious. □

We obtain the counterexample by showing that some K[x_{1}, x_{2}, …, x_{n}]^{H_{k}}, 5 ≤ k ≤ r, cannot be finitely generated.

Theorem 3. There exists 5 ≤ k ≤ r s.t.

K[x_{1}, x_{2}, …, x_{n}]^{H_{k−1}} ∩ K[x_{1}, x_{2}, …, x_{n}]^{H_{k}/H_{k−1}}

is not finitely generated.

Proof. Firstly, assume that K[x_{1}, x_{2}, …, x_{n}]^{H_{r}} is finitely generated. The group T acts on K[x_{1}, x_{2}, …, x_{n}]^{H_{r}} since it is the normalizer of H_{r}. By Proposition 1, (K[x_{1}, x_{2}, …, x_{n}]^{H_{r}})^{T} is finitely generated and, since

(K[x_{1}, x_{2}, …, x_{n}]^{H_{r}})^{T} = K[x_{1}, x_{2}, …, x_{n}]^{T} ∩ K[x_{1}, x_{2}, …, x_{n}]^{H_{r}}
= K[x_{1}, x_{2}, …, x_{n}]^{G}

is a contradiction to Nagata’s theorem, the ring K[x_{1}, x_{2}, …, x_{n}]^{H_{r}} cannot be finitely generated. Therefore let k ≤ r be the minimal index s.t. K[x_{1}, x_{2}, …, x_{n}]^{H_{k}} is not finitely generated. By Proposition 2, k > 4 and K[x_{1}, x_{2}, …, x_{n}]^{H_{k}/H_{k−1}} is finitely generated. By assumption, the ring K[x_{1}, x_{2}, …, x_{n}]^{H_{k−1}} is finitely generated, but the intersection K[x_{1}, x_{2}, …, x_{n}]^{H_{k}} = K[x_{1}, x_{2}, …, x_{n}]^{H_{k−1}} ∩ K[x_{1}, x_{2}, …, x_{n}]^{H_{k}/H_{k−1}} is not finitely generated. □

Acknowledgments

The author is grateful to the anonymous referee for the useful comments.

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Received March 12, 2001