RELATIVE INVARIANTS OF GROUPS ACTING

ON KRULL DOMAINS AND APPLICATIONS

Haruhisa Nakajima

Abstract. In this paper, we will study on relative invariants of a group G consisting of automorphims of a Krull domain R and give a criterion for R_{χ} to be a free R^{G}-module of rank one for a 1-cocycle χ of G in the unit group U(R), in terms of local 1-cocycles which is similar to one in [N1, S1, S2]. On the other hand, suppose that G is equal to the centralizer of an algebraic torus G^{0}. Consider an affine factorial G-variety X with trivial units, over an algebraically closed field K of characteristic zero. Let V be the K-dual of a finite dimensional generating G-submodule V^{∨} of O(X) having a K-basis Ω consisting of weight vectors of G^{0} such that Ω does not degenerate under V \{0} → ℙ(V). Suppose that the action (X, G) is stable (cf. [K1, P]) and consider the G-submodule W of V such that G is diagonal on V/W and W ∋ x ↦ x+w ∈ V induces W//G_{w} ≅ V//G, , for some w ∈ V. As an application of the result on modules of relative invariants mentioned above, we show that, for a minimal W, Cl(X//G) ↪ 𝔛(G_{w}/ℜ_{W} (G_{w})) whose cokernel is finite. Here 𝔛(G_{w}/ℜ_{W} (G_{w})) is a group of rational characters of G_{w} modulo its largest pseudo-reflection subgroup on W. Some related results are discussed.

1. Introduction.

Let Q(R), U(R) and Ht_{1}(R) denote the total quotient ring of a commutative ring R, the unit group of R and the set consisting of all prime ideals 𝔭 of R of height one, respectively. For a ring extension S ↪ R, put

Ht_{1}(R, S) := {𝔭 ∈ Ht_{1}(R) | 𝔭 ∩ S ∈ Ht_{1}(S)}

and, moreover for any 𝔔 ∈ Ht_{1}(S), put

X_{𝔔}(R) := {𝔭 ∈ Ht_{1}(R) | 𝔭 ∩ S = 𝔔}.

In the case where R is a Krull domain, let v_{R,𝔭} be the discrete valuation of R defined by 𝔭 ∈ Ht_{1}(R) and, for a subset Ω of Q(R), let div_{R}(Ω) be the divisor on R associated to the divisorialization { \atop Ω · R} of Ω·R, if it is a fractional ideal of R. Moreover, supp_{R}(Ω) stands for the set

{𝔭 ∈ Ht_{1}(R) | v_{R,𝔭}(Ω · R) ≠ 0}.

2000 Mathematics Subject Classification. 13A50, 13B15, 14L30, 20G05,. Key words and phrases. relative invariant; inertia group; ramification index; Krull domain; divisor

class groups, pseudo-reflection.

The author is partially supported by the Grant-in-Aid for Scientific Research (C); the Ministry of Education, Science, Sports and Culture, Japan; Japan Society for the Promotion of Science.

This is an expository article on some results of the author which shall be published in the forthcoming papers with detailed proofs.

If R is a Krull domain, then a subring S of R satisfying the condition that Q(S)∩R is so and we can define the reduced ramification index e(𝔭, 𝔭 ∩ S) of 𝔭 ∈ Ht_{1}(R, S) over 𝔭 ∩ S (cf. [N4]). In this case, the ring R is said to be unramified over S at a subset Γ ⊆ Ht_{1}(R, S), if R is unramified over S at each 𝔭 ∈ Γ (cf. [N4]). Let us consider an action of a group G on R as ring automorphisms. For a 1-cocycle χ of G in the unit group U(R) of R, we denote by R_{χ} the R^{G}-module of χ-invariants or invariants relative to χ, where R^{G} denotes the subring of R of invariants of G.

(1.1) Suppose that X is an affine normal variety over an algebraically closed field K of characteristic p ≥ 0, whose structure sheaf is denoted to O, with a regular action of an affine algebraic K-group G. Suppose that Q(O(X)^{G}) = Q(O(X))^{G} and that O(X)^{R_{u}(G)} is noetherian, where R_{u}(G) is the unipotent radical of G. Regard Z^{1}(G, U(O(X))), the group of 1-cocyles of G, as an additive group and also 𝔛(G), the rational character group of G, as an additive subgroup. We can choose a finite subset Γ of Ht_{1}(O(X), O(X)^{G}) in such a way that

Γ ⊇supp_{O(X)}(O(X)_{χ}) ∩ Ht_{1}(O(X), O(X)^{G})
(∀χ ∈ Z^{1}(G, U(O(X))) with O(X)_{χ} ≠ {0})

and X_{𝔭∩O(X)^{G}}(O(X)) ⊆ Γ for all 𝔭 ∈ Γ. For any 𝔭 ∈ Γ, let I_{G}(𝔭 ∩ O(X)^{G^{0}}) denote the inertia group of G at 𝔭 ∩ O(X)^{G^{0}} (cf. [N4]). Set U = ∩_{𝔭∈Γ}(O(X)\𝔭) and let π_{𝔭} ∈ 𝔭 (𝔭 ∈ Γ) be a relative invariant of G^{0} whose associated 1-cocycle is denoted to δ_{π_{𝔭}} ∈ Z^{1}(G^{0}, U(U^{−1}O(X))) such that π_{𝔭} generates 𝔭U^{−1}O(X). Let Δ_{𝔭} be the 1-cocycle in Z^{1}(I_{G}(𝔭 ∩ O(X)^{G^{0}}), U(U^{−1}R)) defined by ∏_{𝔮∈I_{G}(𝔭∩O(X)^{G^{0}} )𝔭} π_{𝔮}. For

any χ ∈ Z^{1}(G, U(O(X))) with O(X)_{χ} ≠ {0}, s_{𝔭}(χ) stands for the smallest a ∈ ℤ_{0} satisfying

χ|_{I_{G}(𝔭∩O(X)^{G^{0}} )} ≡ a · Δ_{𝔭} mod B^{1}(I_{G}(𝔭 ∩ O(X)^{G^{0}}), U(U^{−1}O(X))).

We can choose a nonnegative integer b_{𝔭}(χ) (𝔭 ∈ Γ) such that

χ|_{G^{0}} − ∑_{𝔭∈Γ} s_{𝔭}(χ) · δ_{π_{𝔭}} ≡ ∑_{𝔭∈Γ} b_{𝔭}(χ) · δ_{π_{𝔭}} mod B^{1}(G^{0}, U(U^{−1}O(X)))

and, for any minimal subset Δ of Γ satisfying

{∑_{𝔭∈Δ} ℕ · δ_{π_{𝔭}} } ∩ B^{1}(G^{0}, U(U^{−1}R)) ≠ ∅,

{b_{𝔭}(χ) | 𝔭 ∈ Δ} contains 0.

As an application of our main result in Sect. 2, we obtain

Theorem 1.2. Under the circumstances as in (1.1), for χ ∈ Z^{1}(G, U(O(X))) such that O(X)_{χ} ≠ {0}, the following conditions are equivalent:

(1) O(X)_{χ} ≅ O(X)^{G} as O(X)^{G}-modules.

(2) dim_{Q(O(X)^{G})}(O(X)_{χ} ⊗_{O(X)^{G}} Q(O(X)^{G})) = 1, and the intersection of

∑_{𝔭∈Γ}(b_{𝔭}(χ) + s_{𝔭}(χ)) · div_{R}(𝔭) + ∑_{𝔭∈Ht_{1}(O(X))\Ht_{1}(O(X),O(X)^{G})} ℤ · div_{O(X)}(𝔭)

and {div_{O(X)}(f) | f ∈ O(X)_{χ}\{0}} is nonempty.

If these conditions are satisfied, then O(X)_{χ} is generated by the element f as an O(X)^{G}-module such that {div_{O(X)}(f)} is equal to the non-empty set of divisors stated in (2).

This can be applied to establishing a reduction of invariants of certain reductive algebraic groups to ones of their stabilizers, which is similar to slice method.

(1.3) Furthermore suppose that G^{0} is an algebraic torus. Let O(X)_{st} be the largest G-invariant affine K-subalgebra of O(X) such that the action of G on Spec(O(X)_{st}) is stable (e.g. [K1, P]) and X_{st} the affine variety defined by O(X_{st}) = O(X)_{st} (cf. [W1, N3]). Consider a finite dimensional rational G-submodule V^{∨} of O(X) which generates O(X) as a K-algebra. Let V denote the dual space of V^{∨} on which G acts naturally. We have a canonical G-equivariant closed embedding X ↪ V and identify the dual of V with V^{∨}. A pair (W, w) is defined to be a paralleled linear hull of (X, G) through V^{∨} or, simply, of (V, G), if W is a G-submodule of V_{st} such that G is diagonalizable on V_{st}/W, w is a nonzero vector of V_{st} such that W ∩ < G·w >_{K}= {0} and the morphism

(• + w) : W ∋ x ↦ x + w ∈ V_{st}

induces the isomorphism

π_{V_{st}//G_{w},V//G} ∘ (• + w)//G_{w} : W//G_{w} ⟶^{∼} V_{st}//G.

Here (• + w)//G_{w} : W//G_{w} ⟶ V_{st}//G_{w} is the algebraic quotient of (• + w) and

π_{V_{st}//G_{w},V//G} : V_{st}//G_{w} ⟶ V_{st}//G

is induced by O(V_{st})^{G} ↪ O(V_{st})^{G_{w}}. Under the assumption that G = G^{0}, the pair (W, w) seems to be initially defined and used by H. P. Kraft and D. H. Wehlau (cf. §1 of [K2] and [W1, W2]). A paralleled linear hull (W_{0}, w_{o}) of (V, G) is said to be minimal, if W_{0} is minimal in the subspaces W which admit paralleled linear hulls (W, w) of (V, G) for some w ’s. A element of σ ∈ G is said to be a pseudo-reflection on V (resp. a generalized-reflection on O(X)), if dim(σ−1)(V) = ht((σ−1)(V)·O(V)∩O(V)^{G}) = 1 (resp. ht((σ−1)(O(X))·O(X)) = ht((σ−1)(O(X))·O(X)∩O(X)^{G}) = 1). We denote by R_{V} (G) (resp. by R_{O}(X)(G)) the subgroup of G generated by all pseudo-reflections (resp. generalized-reflections) of G on V (resp on O(X)).

Theorem 1.4. Under the circumstances as in (1.3), moreover suppose that p = 0 and X is an affine conical factorial variety with a regular conical action of G. Let V^{∨} be the rational homogeneous G-submodule of O(X) minimally generating O(X) as a K-algebra. Suppose that Z_{G}(G^{0}) = G. Let (W, w) be a minimal paralleled linear hull of (V, G) through V^{∨}, where V denotes the K-dual (V^{∨})^{∨} of V^{∨}. Then

(1) The quotient morphism π_{W,G_{w}} (resp. π_{X_{st},G_{w}}) is no-blowing-up of codimension one, and for any 𝔭 ∈ Ht_{1}(O(X_{st})^{G_{w}}) and 𝔮 ∈ Ht_{1}(O(W)^{G_{w}}), the actions of G_{w} on X_{𝔭}(O(X_{st})) and X_{𝔮}(O(W)) are transitive.

(2) Cl(O(X)^{G}) ≅ Cl(O(X_{st})^{G_{w}}) ≅ Cl(O(V)^{G}) ≅ 𝔛(G_{w}/ℜ_{O(X_{st})}(G_{w})).

(3) If X_{st}//G_{w} is an affine space, then so is V//G.

(4) Suppose that G_{w} is solvable. Then V//G is an affine space if and only if O(X)^{G} is factorial.

The last assertion can be regarded as a generalization of [W2] and Proposition 3.6 of [N1]. Similarly by the use of paralleled linear hulls, we can determine representations of diagonalizable groups with the algebra of invariants which are hypersurfaces. Further applications to invariants of non-semisimple reductive groups shall be given by the author elsewhere.

The notations ℕ and ℤ are standard and ℤ_{0} stands for the set of all nonnegative integers.

2. Modules of relative invariants.

From now on to the end of Proposition 2.10, suppose that R is a Krull domain with an action of a group G through the automorphism group Aut(R) of R. The group of cocycles of degree 1 including character groups are represented as additive groups.

Proposition 2.1. For χ ∈ Z^{1}(G, U(R)), R_{χ} ≅ R^{G} as R^{G}-modules if and only if

dim_{Q(R^{G})}(Q(R^{G}) ⊗_{R^{G}} R_{χ}) = 1 and \frac{1}{f} R ∩ Q(R^{G}) = R^{G} for some f ∈ R_{χ}.

We derive the next theorem from Proposition 2.1.

Theorem 2.2. For χ ∈ Z^{1}(G, U(R)), the R^{G}-module R_{χ} is free and of rank one if and only if the following two conditions are satisfied:

(1) dim_{Q(R^{G})}(Q(R^{G}) ⊗_{R^{G}} R_{χ}) = 1

(2) There exists a nonzero element f satisfying

(2.2.1)

∀𝔮 ∈ Ht_{1}(R^{G}) ⇒
∃𝔓 ∈ X_{𝔮}(R) such that v_{R,𝔓}(f) < e(𝔓, 𝔮)..

If these conditions are satisfied, then R_{χ} = R^{G} · f for any nonzero f ∈ R_{χ} such that (3.3.1) holds for f. □

The following two propositions play an important role in defining the local datum which shall be introduced in (2.5).

Proposition 2.3. Let χ ∈ Z^{1}(G, U(R)) and let H be a normal subgroup of G of a finite index. Suppose that R^{H} is unramified over R^{G} at supp_{R}(R_{χ}) ∩ Ht_{1}(R, R^{G}). Then the following conditions are equivalent:

(1) R_{χ} is an R^{G}-free module of rank one and

dim_{Q(R^{H})}(Q(R^{H}) ⊗_{R^{H}} R_{χ|_{H}}) = 1.

(2) There are nonzero g ∈ R_{χ|_{H}} and u ∈ U(R^{H}) such that R_{χ|_{H}} = R^{H} · g and R_{χ} ∋ g · u.

If these conditions are satisfied, then R_{χ} = R^{G} ·gu for the elements g and u in (2).

Proposition 2.4. Suppose that Q(R^{G}) = Q(R)^{G} and let χ ∈ Z^{1}(G, U(R)) such that R_{χ} ≠ {0}. Let U be a multiplicative system of R invariant under the action of G such that

𝔭 ∩ U = ∅ (∀𝔭 ∈ ⋃_{𝔭∈supp(R_{χ})∩Ht_{1}(R,R^{G})} X_{𝔭∩R^{G}}(R)),
𝔭 ∩ U ≠ ∅ (∀𝔭 ∈ supp(R_{χ}) satisfying ht(𝔭 ∩ R^{G}) ≥ 2).

Then we have

(1) For any 𝔮 ∈ supp(R_{χ}) ∩ Ht_{1}(R, R^{G}) and 𝔭 ∈ X_{𝔮∩R^{G}}(R), the following congruence holds;

v_{R,𝔭}(R_{χ}) ≡ v_{U^{−1}R,𝔭U^{−1}R}((U^{−1}R)_{χ}) mod e(𝔭, 𝔮 ∩ R^{G}).

(2) If R_{χ} ≅ R^{G} as R^{G}-modules, (U^{−1}R)_{χ} ≅ (U^{−1}R)^{G} as (U^{−1}R)^{G}-modules and (U^{−1}R)_{χ} = R_{χ} · (U^{−1}R)^{G}.

(2.5) We explain our notations and circumstances which shall be considered as follows. Suppose that Q(R^{G}) = Q(R)^{G}. Let χ ∈ Z^{1}(G, U(R)) be a 1-cocycle such that R_{χ} ≠ {0} and Γ a finite subset of Ht_{1}(R, R^{G}) such that

Γ ⊇ supp_{R}(R_{χ}) ∩ Ht_{1}(R, R^{G})

and Γ ⊇ X_{𝔭∩R^{G}}(R) for any 𝔭 ∈ Γ. Let H be a normal subgroup of G of a finite index stabilizing each 𝔭 in Γ. Put U = ∩_{𝔭∈Γ}(R\𝔭). Let π_{𝔭} be an element which generates 𝔭U^{−1}R and, for any subgroup N of G, set

π_{N𝔭} = ∏_{𝔮∈N𝔭} π_{𝔮}.

Let Δ_{Nπ_{𝔭}} ∈ Z^{1}(N, U(U^{−1}R)) be the cocycle defined by

N ∋ τ ⟼ Δ_{Nπ_{𝔭}}(τ) = \frac{τ(π_{N𝔭})}{π_{N𝔭}} ∈ U(U^{−1}R)

and δ_{π_{𝔭}} ∈ Z^{1}(H, U(U^{−1}R)) the cocycle defined by

H ∋ σ ⟼ δ_{π_{𝔭}}(σ) = \frac{σ(π_{𝔭})}{π_{𝔭}} ∈ U(U^{−1}R).

Moreover we donote by s_{I_{G}(𝔭∩R^{H})𝔭}(χ) the infimum of

{a ∈ ℤ_{0} | χ|_{I_{G}(𝔭∩R^{H})} ≡ a · Δ_{I_{G}(𝔭∩R^{H})𝔭}
mod B^{1}(I_{G}(𝔭 ∩ R^{H}), U(U^{−1}R))}

and denote by χ^{H} the cocycle

χ|_{H} − ∑_{𝔭∈Γ} s_{I_{G}(𝔭∩R^{H})𝔭}(χ) · δ_{𝔭} ∈ Z^{1}(H, U(U^{−1}R)).

The independence of the number s_{I_{G}(𝔭∩R^{H})𝔭}(χ) on the choice of the subgroup H is guaranteed in the next lemma.

Lemma 2.6. Under the same circumstances as in (2.5), the following properties hold for each 𝔭 ∈ Γ:

(1) e(𝔮, 𝔮 ∩ R^{G}) = e(𝔭, 𝔭 ∩ R^{G}) for all 𝔮 ∈ I_{G}(𝔭 ∩ R^{H}).

(2) ord(Δ_{I_{G}(𝔭∩R^{H})𝔭} mod B^{1}(I_{G}(𝔭 ∩ R^{H}), U(U^{−1}R))) = e(𝔭, 𝔭 ∩ R^{G}).

(3) 0 ≤ s_{I_{G}(𝔭∩R^{H})𝔭}(χ) < e(𝔭, 𝔭 ∩ R^{G}) and

s_{I_{G}(𝔭∩R^{H})𝔭}(χ) ≡ v_{R,𝔭}(R_{χ}) mod e(𝔭, 𝔭 ∩ R^{G}).

In order to define the characteristic divisor D_{χ}, we need

Lemma 2.7. Under the same circumstances as in (2.5), there exist non-negative integers a_{𝔭}(χ) (∀𝔭 ∈ Γ) such that {a_{𝔭}(χ) | 𝔭 ∈ X_{𝔔}(R)} contains 0, for any restriction 𝔔 ∈ {𝔭 ∩ R^{H} | 𝔭 ∈ Γ} and the following congruence holds;

χ^{H} ≡ ∑_{𝔭∈Γ} a_{𝔭}(χ)e(𝔭, 𝔭 ∩ R^{G}) · δ_{π_{𝔭}} mod B^{1}(H, U(U^{−1}R)).

Furthermore:

(1) (a_{𝔭}(χ) | 𝔭 ∈ Γ) is uniquely determined by χ.

(2) Extending a_{𝔭}(χ) = 0 for any 𝔭 ∈ Ht_{1}(R, R^{G})\Γ, we see that (a_{𝔭}(χ) | 𝔭 ∈ Ht_{1}(R, R^{G})) is independent on the choice of Γ.

(3) There is an element u ∈ U(U^{−1}R) such that the (U^{−1}R)^{G}-module (U^{−1}R)_{χ} is generated by the element

u · ∏_{𝔭∈Γ} π_{𝔭}^{a_{𝔭}(χ)e(𝔭,𝔭∩R^{G})+s_{I_{G}(𝔭∩R^{H})𝔭}(χ)}.

(2.8) Under the same circumstances as in (2.5) and Lemma 2.7, let D_{χ} denote the divisor

∑_{𝔭∈supp_{R}(R_{χ})∩Ht_{1}(R,R^{G})}{a_{𝔭}(χ)e(𝔭, 𝔭 ∩ R^{G}) + s_{I_{G}(𝔭∩R^{H})𝔭}(χ)} · div_{R}(𝔭)

in Div(R) and we call D_{χ} the characteristic divisor of χ. Let D_{G} denote the subgroup

∑_{𝔭∈Ht_{1}(R)\Ht_{1}(R,R^{G})} ℤ · div_{R}(𝔭)

of Div(R).

Using the residue class of the characteristic divisor, we obtain the main result of this section which can be regarded as a generalization of the Stanley criterion (cf. [S1, S2]) for R_{χ} to be a free R^{G}-module of rank one:

Theorem 2.9. Under the circumstances as in (2.5) and Lemma 2.7, R_{χ} ≅ R^{G} as R^{G}-modules if and only if dim_{Q(R^{G})}(R_{χ} ⊗_{R^{G}} Q(R^{G})) = 1 and

(D_{χ} + D_{G}) ∩ {div_{R}(f) | f ∈ R_{χ}\{0}} ≠ ∅.

In the case where these equivalent conditions are satisfied,

(D_{χ} + D_{G}) ∩ {div_{R}(f) | f ∈ R_{χ}\{0}} = {div_{R}(f_{χ})}

for some f_{χ} ∈ R_{χ} and R_{χ} = R^{G} · f_{χ}.

The criterion in the case of algebraic group actions is reduced to Theorem 2.9, by the aid of the following two results.

Proposition 2.10. Suppose that Q(R^{G}) = Q(R)^{G}. Let H be a normal subgroup of G of a finite index and χ ∈ Z^{1}(G, U(R)). If R_{χ} ≅ R^{G} as R^{G}-modules, then

supp_{R}(R_{χ}) ∩ Ht_{1}(R, R^{G}) ⊆ { ⋂_{f∈R_{χ|_{H}} \{0}} supp_{R}(f)}
∪ {𝔭 ∈ Ht_{1}(R, R^{G}) | I_{G}(𝔭 ∩ R^{H})|_{R^{H}} ≠ {1}}.

Corollary 2.11. Suppose that G is an affine algebraic group over an algebraically closed field K. Suppose that R is an affine normal domain over K on which G acts Krationally as K-algebra automorphisms. Suppose that Q(R^{G}) = Q(R)^{G} and R^{R_{u}(G^{0})} is noetherian. Then

{ ⋃_{χ∈{χ∈Z^{1}(G,U(R))|R_{χ}≅R^{G}}} supp_{R}(R_{χ})} ∩ Ht_{1}(R, R^{G})

is a finite set.

(2.12) Proof of Theorem 1.2. By Corollary 2.11, we see that the finite set Γ in (1.1) exists. If Δ is a minimal subset of Γ such that there exist a_{𝔭} ∈ ℕ (𝔭 ∈ Δ) and a unit u of U^{−1}O(X), satisfying

u · ∏_{𝔭∈Δ} π_{𝔭}^{a_{𝔭}} ∈ (U^{−1}O(X))^{G^{0}},

then, for 𝔭 ∈ Δ,

(2.12.1)  {𝔭U^{−1}O(X) | 𝔭 ∈ Δ} = X_{𝔭U^{−1}O(X)∩(U^{−1}O(X))^{G^{0}}} (U^{−1}O(X))

For any subset Δ of Γ, the equality (2.12.1) holds for a prime ideal 𝔭 ∈ Δ if and only if Δ is minimal in the subsets Δ’s satisfying

{∑_{𝔭∈Δ} ℕ · δ_{π_{𝔭}} } ∩ B^{1}(G^{0}, U(U^{−1}R)) ≠ ∅.

Hence, by (2.5), such a subset Δ as in (2.11.1) is identical with X_{𝔭∩O(X)^{G^{0}}} ( O(X)) for a prime ideal 𝔭 ∈ Δ. Consequently the assertion of Theorem 1.2 follows from Theorem 2.9. □

3. Applications.

In this section, suppose that R is an affine factorial domain with trivial units and G^{0} is an algebraic torus over an algebraically closed field K of characteristic zero. A subset {f_{1}, …, f_{n}} of n elements in R is said to be (R, G)-basic, if each f_{i} is a relative invariant of G which is prime in R and R·f_{i} ≠ R·f_{j} for any 1 ≤ i < j ≤ n. Moreover, a subset {f_{1}, …, f_{n}} of n elements is said to be (R, G)-free, if it is (R, G)-basic and, for any a_{i} ∈ ℤ_{0}, there exists a character χ ∈ 𝔛(G) such that

R_{χ} = R^{G} · ∏_{i=1}^{n} f_{i}^{a_{i}}.

We further say that {f_{1}, …, f_{n}} of n elements is (R, G)-afforded, if it is (R, G^{0})-basic, the K-subspace < f_{1}, …, f_{n} >_{K} of R is G-invariant and {f_{1}, …, f_{n}} forms a K-basis of this space. There exists a G-afforded generating system {g_{1}, …, g_{n}}, which is denoted to Ω for a convenience sake, of R of n elements as a K-algebra. We denote by (K^{×} ∘ Ω)_{R,st}^{G} the set consisting of all g_{i}’s such that R · g_{i} ∩ R^{G} ≠ {0} and g_{i} are relative invariants of G. The subring R_{st} is an affine factorial K-domain on which G acts naturally (cf. Sect.1 and [N3]) and (K^{×} ∘ Ω)_{R,st}^{G} ⊆ R_{st}. Let Ω_{R,G} denote the set

{g_{i} ∈ (K^{×} ∘ Ω)_{R,st}^{G} | ht(R · g_{i} ∩ R^{G}) = 1 and 𝔛_{R·g_{i}∩R^{G}}(R)^{G} = 𝔛_{R·g_{i}∩R^{G}}(R)}.

For any pair g_{i_{1}}, g_{i_{2}} of elements in Ω_{R,G}, define the relation g_{i_{1}} ≡_{G} g_{i_{2}}, if R·g_{i_{1}} ∩R^{G} = R · g_{i_{2}} ∩ R^{G}. Then we denote by Ω_{R,G}^{≡_{G}} the set consisting of all complete systems of representatives of Ω_{R,G} modulo ≡_{G}.

As an application of Theorem 2.9, we have

Theorem 3.1. Let {f_{1}, …, f_{m}} be a subset of R_{st} of m elements. Then {f_{1}, …, f_{m}} is (R, G)-free if and only if there exists a system Υ ∈ Ω_{R,G}^{≡_{G}} such that

(K^{×} ∘ Ω)_{R,st}^{G}\Υ ∩ K^{×} ∘ f_{i} ≠ ∅ (1 ≤ i ≤ m).

Let V be the dual space of < Ω >_{K} and denote by ρ the canonical K-morphism O(V) → R defined by the inclusion Ω ⊆ R. We easily see that (K^{×} ∘ Ω)_{R,st}^{G} = (K^{×} ∘ Ω)_{O(V),st}^{G}.

Corollary 3.2. The restriction of the mapping

2^{O(V_{st})} ∋ Γ ⟼ ρ(Γ) ∈ 2^{R_{st}}

is a one-to-one correspondence between the set consisting of all finite (O(V), G)-free subsets of O(V_{st}) and the set consisting of all finite (R, G)-free subsets of R_{st}. Moreover the equality Ω_{R,G}^{≡_{G}} = Ω_{O(V),G}^{≡_{G}} holds.

Proof. Thanks to Theorem 3.1, the proof of the first assertion is reduced to show that, for a subset Γ of Ω, the condition that Γ is (O(V), G^{0})-free is equivalent to the one that Γ is (R, G^{0})-free. The last equality follows easily from Theorem 3.1 and the first assertion. □

Proposition 3.3. Suppose that {f_{1}, …, f_{m}} is a maximal (R, G)-free subset of R of m elements. Then there is not a non-empty (R, ∩_{i=1}^{m}G_{f_{i}})-free subset consisting relative invariants of G. Especially in the case where G is connected, we have ht(R · f ∩ R^{∩_{i=1}^{m}G_{f_{i}}}) = 1 and ♯(X_{R·f∩R^{(∩{m \atop i=1}G_{f_{i}})^{0}}} (R)) = 1, for any prime element f ∈ R such

that R · f ∩ R^{∩_{i=1}^{m}G_{f_{i}}} ≠ {0}.

Proposition 3.4. For a subspace W of V, the following conditions are equivalent: (1) (W, w) is a paralleled linear hull of (V, G) for a nonzero w ∈ V_{st}.

(2) There exists a system Υ ∈ Ω_{O(V),G}^{≡G} and a subset F_{W} of (K^{×} ∘ Ω)_{O(V),st}^{G}\Υ

such that

W = {x ∈ V_{st} | F (x) = 0 (∀F ∈ F_{W})},

where Ω is regarded as a subset of O(V).

Proof. (2) ⇒ (1) : By Theorem 3.1, we see that the set F_{W} is (O(V_{st}), G)-free. For each z ∈ F_{W}, let χ_{z} ∈ 𝔛(G) to satisfy O(V_{st})_{χ_{z}} ∋ z. Since (V_{st}^{∨})_{χ_{z}} = K · z, for each z ∈ F_{W}, we can choose a unique vector z^{∨} from (V_{st})_{−χ_{z}} in such a way that z^{∨}(z) = 1. Putting w = ∑_{z∈F_{W}} z^{∨}, we infer that (W, w) is a paralleled linear hull of (V, G).

(1) ⇒ (2) : There are weight vectors v_{i} (1 ≤ i ≤ m) of G in V_{st} whose reduction modulo W form a K-basis of V_{st}/W such that w = ∑_{i=1}^{m} v_{i}. Let {v_{1}^{∨}, … v_{m}^{∨}} be a basis of W^{⊥} = (V_{st}/W)^{∨} dual to {v_{1} mod W, …, v_{m} mod W }. Then, since

π_{V_{st}//G_{w},V//G} ∘ (• + w)//G_{w} : W//G_{w} ⟶^{∼} V_{st}//G,

the set {v_{1}^{∨}, … v_{m}^{∨}} is (O(X), G)-free. Thus the assertion follows from Theorem 3.1. □

Consequently (W, w) is a minimal paralleled linear hull of (V, G) for some w ∈ V if and only if

W = {x ∈ V_{st} | F (x) = 0 (∀F ∈ (K^{×} ∘ Ω)_{O(V),st}^{G}\Υ)},

for some system Υ ∈ Ω_{O(V),G}^{≡G}.

(3.5) Suppose that g ∈ R_{χ} a fixed relative invariant of G such that {g} is (R, G)-free and, for a convenience sake, put H = Kerχ and S = R^{H}. Furthermore, suppose that the action (Spec(R), G) is stable and Z_{G}(G^{0}) = G. As in [M] (e.g, [N2]), we similarly introduce the following subgroups of Div(S);

E^{∗}(G, S) = ⨁_{𝔭∈Ht_{1}(S^{G})} ℤ· ( ∑_{𝔮∈X_{𝔭}(S)} e(𝔮, 𝔭) · div_{S}(𝔮))

⊕ ⨁_{𝔮∈Ht_{1}(S),ht(𝔮∩S^{G})≥2} ℤ · div_{S}(𝔮),

Δ_{S·g}(G, S) = ⨁_{𝔮∈Ht_{1}(S)\Y_{S·g}(S),𝔮∩S^{G}≠{0}} ℤ · div_{S}(𝔮),

where Y_{S·g}(S) = X_{S·g∩S^{G}}(S), if ht(S · g ∩ S^{G}) = 1, and Y_{S·g}(S) = {S · g}, otherwise. Let Φ^{∗} : E^{∗}(G, S) → Div(S^{G}) be a homomorphism defined by

∑_{𝔭∈Ht_{1}(S^{G})} a_{𝔭} · D_{𝔭} + ∑_{𝔮∈Ht_{1}(S),ht(𝔮∩S^{G})≥2}b_{𝔮} · div_{S}(𝔮)

⟼ ∑_{𝔭∈Ht_{1}(S^{G})} a_{𝔭} · div_{S^{G}}(𝔭) (a_{𝔭}, 𝔟_{𝔮} ∈ ℤ),

where D_{𝔭} = ∑_{𝔮∈X_{𝔭}(S)} e(𝔮, 𝔭) · div_{S}(𝔮).

Under the same circumstances as in (3.5), we have Lemma 3.6, Proposition 3.6 and Proposition 3.7 as follows:

Lemma 3.6. The canonical homomorphism Div(S)^{G} → Cl(S) is an epimorphism.

Proof. For any 𝔭 ∈ Ht_{1}(S), choose a prime f from R in such a way that R·f ∩R^{H} = 𝔭. Then, expressing H · Rf = {R · f_{1}, …, R · f_{♯(H·Rf)}}, we see

a := ∏_{1≤i≤♯(H·Rf)} f_{i}

is an invariant relative to some ψ ∈ 𝔛(H) such that R · a ∩ R^{H} = R · a · R_{−ψ}. As Z_{G}(G^{0}) = G, there is a character δ ∈ 𝔛(G^{0} · H) such that δ|_{H} = δ and δ(Ker((G^{0} · H) → Aut(R))) is trivial. Since δ is regarded as a character of the group G^{0} · H|_{R},

the module R_{δ} contains a nonzero element b. So R^{H} · \frac{b}{a} · (R · a)^{H} = (R · b)^{H}. By the

stability of the action of H on R (cf. [P, N3]), we see \frac{b}{a} ∈ Q(R)^{H} = Q(R^{H}), which implies that

div_{R^{H}} ((R · b) ∩ R^{H}) ≡ div_{R^{H}} (𝔭) mod Prin(R^{H}).

From this, we infer that the homomorphism

Div(R^{H})^{G^{0}·H} ↪ Div(R^{H}) ⟶ Cl(R^{H})

is surjective. □

The next two propositions are based on Lemma 3.6, in which the assumption that G is a central extension of G^{0} is essential.

Proposition 3.7. Suppose that ht((S · g ∩ S^{G}) ≥ 2. Then the sequence

0 → Prin(S) ∩ Δ_{S·g}(G, S) → Δ_{S·g}(G, S) → Cl(S) → 0

is exact and

Φ^{∗}(Prin(S) ∩ Δ_{S·g}(G, S)) = Prin(S^{G}).

Moreover, Φ^{∗} induces the isomorphism Cl(S) ≅ Cl(S^{G}).

Proposition 3.8. Suppose that ht(S · g ∩ S^{G}) = 1. Let 𝔮_{0} ∈ Ht_{1}(S) satisfy the equality X_{S·g∩S^{G}}(S)\{S · g} = {𝔮_{0}}. Set

∇(G, S) := Δ_{S·g}(G, S) ⊕ ℤ · div_{S}(𝔮_{0}) ⊆ Div(S)

and let Θ : E^{∗}(G, S) → ∇(G, S) be the composite

E^{∗}(G, S) ↪ Div(S) ⟶^{pr} ∇(G, S).

Then the square

Cl(S^{G}) ⟶^{∼} Cl(S)
can.↑  ↑^{can.}
Div(S^{G}) ⟶_{Θ∘Ψ_{S^{G},S}}^{∼} ∇(G, S)

is a commutative diagram with horizontal isomorphisms.

Applying Proposition 3.7 and Proposition 3.8 inductively to a maximal (R, G)-free set, by the aid of Theorem 3.1, we now establish

Theorem 3.9. Suppose that Z_{G}(G^{0}) = G. Let (W, w) be a minimal paralleled linear hull of (V, G) through < Ω >_{K}. Then

(1) The inclusions R_{st}^{G_{w}} ↪ R_{st} and O(W)^{G_{w}} ↪ O(W) are no-blowing-up of codimension one and, for each 𝔭 ∈ Ht_{1}(R_{st}^{G_{w}}) and 𝔮 ∈ Ht_{1}(O(W)^{G_{w}}), the actions of G_{w} on X_{𝔭}(R_{st}) and X_{𝔮}(O(W)) are, respectively, transitive.

(2) We have the monomorphism

Cl(R^{G}) ≅ Cl(R_{st}^{G_{w}}) ≅ 𝔛(G_{w}/R_{R_{st}}(G_{w}))
⟶^{φ}𝔛(G_{w}/R_{W} (G_{w})) ≅ Cl(O(V)^{G})

with its cokernel, isomorphic to a subgroup of the abelialization of the finite group R_{R_{st}}(G_{w})/R_{W} (G_{w})|_{R_{st}}.

Proof of Theorem 1.4. The first and second assertions follow from Theorem 3.9. Suppose that Cl(R^{G}) is finite. Then, the assertion (2) of Theorem 3.9 implies that G_{w} is also finite. Hence the third assertion follows from [C, R1] and, in the case where G_{w} is solvable, applying Proposition 3.6 of [N1] to the G_{w}-module W, we see that O(V)^{G} is a polynomial ring over K if and only if it is factorial. Thus the last assertion follows from this observation and Theorem 3.9. □

As in [D] and [N1], it shall be expected that Theorem 2.9 and Theorem 3.9 imply the detailed description of Cl(O(X)^{G}) in terms of group theory. Finally, we point out that these theorems seem somewhat useful in studying invariants of non-semisimple reductive algebraic groups which are treated in [P] for semisimple groups.

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Department of Mathematics Faculty of Science Josai University Keyakidai, Sakado 350-0295, Japan