PICARD GROUPS OF MULTIPLICATIVE INVARIANTS

MARTIN LORENZ

Abstract. Let S = kA denote the group algebra of a finitely generated free abelian group A over the field k and let G be a finite subgroup of GL(A). Then G acts on S by means of the unique extension of the natural GL(A)-action on A. We determine the Picard group Pic R of the algebra of invariants R = S^{G}. As an application, we produce new polycyclic group algebras with nontrivial torsion in K_{0}.

1. Introduction

It appears to be unknown whether all finitely generated projective modules over rings of invariants for rational linear actions of reductive algebraic groups on affine space 𝔸^{n} over an algebraically closed field of characteristic 0 are stably free, or even free (cf. [Kr], Problem 5.1). For finite groups, stable freeness has recently been established by M. Holland [Ho] in the special case where the underlying representation decomposes into fixed point free and trivial constituents only. Our goal here is to show that, for multiplicative actions of finite groups, the analog of Kraft’s question has a negative answer. In fact, we determine the Picard group Pic R of multiplicative invariant rings R and exhibit examples where it is nontrivial. As is well-known [K] (or see §2.6 below), the latter cannot happen for linear actions.

Specifically, let S = kA denote the group algebra of a finitely generated free abelian group A over the field k and let G be a finite subgroup of GL(A), acting on S by means of the k-linear extension of the natural action on A. Working under the assumption that the order of G is nonzero in k, we show that the Picard group Pic R of the invariant algebra R = S^{G} measures the extent to which the 1-cohomology H^{1}(G, A) fails to be detected by the cyclic subgroups of G, to wit:

Theorem. Pic R ≅ ⋂_{C} Ker (Res_{C}^{G} : H^{1}(G, A) → H^{1}(C, A)), where C runs over the cyclic subgroups of G.

A systematic investigation of the invariant algebras R arising from abelian groups A of rank at most 4, conducted with the aid of the crystallographic groups library in GAP [Sch], reveals that Pic R is rarely nontrivial, but such cases do in fact occur: 2 for rk A = 3 (out of a total of 73), and 9 (out of 710) for rk A = 4.

Two noteworthy facts about Pic R are as follows. First, Pic R survives the Cartan map K_{0}(R) → G_{0}(R) via its canonical embedding into the class group Cl R (see §3.6). Thus, our findings show that, other than in the case of linear invariants (see the Appendix), the image of the Cartan map for multiplicative invariants is not necessarily generated by the classes of free modules. Second, we show in §3.7 that Pic R is always a subquotient of the torsion subgroup of K_{0}(kΓ), where Γ denotes the semidirect product A ⋊ G. Thus our examples give rise to new polycyclic-by-finite groups Γ such that K_{0}(kΓ) has nontrivial torsion, the only instance of this kind known hitherto being a construction due to Kropholler and Moselle [KM]. The (multiplicative) invariant algebras R coming from the Kropholler-Moselle groups actually have a trivial Picard group; so Pic R is not solely responsible for torsion in K_{0}(kΓ).

1991 Mathematics Subject Classification. Primary: 13A50, 13C20, 13D15, 16E20, 16W20, 19A31; Secondary: 16S34, 20H15.

Key words and phrases. Picard group, ring of invariants, group action, Grothendieck group, polycyclic group algebra, Cartan map, class group, 1-cohomology group.

Research supported in part by NSF Grant DMS-9400643

The first half of the article, Section 2, deals quite generally with actions of finite groups on arbitrary commutative rings S, assuming only that the every invariant is a trace, and gives a description of the Picard group of the invariant ring in terms of the G-Picard group Pic_{G} S of S. To a large extent, this material, couched in geometric terms and working over an algebraically closed base field of characteristic 0, can be found in [Kr] and [DMV] for rational actions of reductive algebraic groups. However, in the light of its possible application or extension to other types of actions (e.g., Hopf algebra actions), it seemed worthwhile to give a purely ring theoretic presentation. Section 3 is then devoted entirely to multiplicative actions and contains the aforementioned results.

2. Group Actions

2.1. Throughout this section, S will be a ring and G will be a finite group acting by automorphisms on S, written s ↦ s^{g}. Furthermore,

R = S^{G} will denote the ring of G-invariants in S, and T = S ∗ G will be the skew group ring of G over S.

Thus T is an associative ring containing S as a subring, and G is a subgroup of U(T), the group of units of T. The elements of G form a free basis of T as right S-module. Multiplication in T is based on the rule ga·hb = gha^{h}b for a, b ∈ S, g, h ∈ G. The ring S becomes an R-T -bimodule via

r · a · gb = ra^{g}b (r ∈ R, a, b ∈ S, g ∈ G) .

We will work under the standing hypothesis that the trace map tr : S → R, s ↦ ∑_{g∈G} s^{g} is surjective or, equivalently:

(*) There exists an element x ∈ S with tr(x) = 1.

This entails that txt = t, where we have put t = ∑_{g∈G} g ∈ T. So e = tx is an idempotent element of T with

eT = tT = tS ≅ S_{T} .

In particular, S_{T} is projective, and the ideal I = T eT of T satisfies I^{2} = I and S_{T} · I = S_{T}.

2.2. Some module categories. Let proj-R denote the category of finitely generated projective (right) R-modules, and similarly for T, and let add(S_{T}) denote the full subcategory of proj-T whose objects are the direct summands of the modules S_{T}^{n} for n ≥ 0. The following lemma is well-known, but we include the easy proof for the reader’s convenience.

Lemma. (i) The functors E : proj- R → add (S_{T}), Q ↦ Q ⊗_{R} S_{T} and F : add (S_{T}) →

proj- R, P ↦ P^{G} yield an equivalence of categories proj- R ≈ add (S_{T}). (ii) A module P in proj-T belongs to add(S_{T}) precisely if P I = P.

Proof. (i) For Q in proj-R, let φ_{Q} : Q → (F ∘ E)(Q) = (Q ⊗_{R} S_{T})^{G} denote the R-linear map given by φ_{Q}(q) = q ⊗ 1. Then φ_{R} : R → (R ⊗_{R} S_{T})^{G} ≅ (S_{T})^{G} = R is an isomorphism, and hence so is φ_{R^{n}} for every n and φ_{Q} for every Q. Thus φ is a natural equivalence of functors Id_{proj-R} ≅ F ∘ E. Similarly, defining ψ_{P} : (E ∘ F)(P) = P^{G} ⊗_{R} S_{T} → P for P in add(S_{T}) by ψ_{P} (p ⊗ s) = ps, we obtain a natural equivalence of functors E ∘ F ≅ Id_{add(S_{T})}.

(ii) All modules P in add(S_{T}) satisfy P I = P, because S · I = S. Conversely, if P in proj-T satisfies P = P I = P eT then, for some n, (eT)^{n} ≅ S_{T}^{n} maps onto P, and so P is a direct summand of S_{T}^{n}.

2.3. Another characterization of add(S_{T}). For the remainder of this section, we assume that

S is commutative.

We let Max S denote the set of maximal ideals of S, and for each 𝔐 ∈ Max S, we let G^{Z}(𝔐) = {g ∈ G : 𝔐^{g} = 𝔐} denote the decomposition group of 𝔐. The following description of add(S_{T}) is adapted from [Kr], Proposition 3.

Lemma. A module P in proj-T belongs to add(S_{T}) if and only if, for all 𝔐 ∈ Max S, P/P 𝔐 ≅ (S/𝔐)^{r} as modules over S ∗ G^{Z}(𝔐). (The S ∗ G^{Z}(𝔐)-module structure on S/𝔐 is understood to come from S_{T}, and r = rk P_{𝔐}.)

Proof. The condition is surely necessary. For, if P is a direct summand of S_{T}^{n}, then P/P 𝔐 is a direct summand of (S/𝔐)^{n} as modules over U = S ∗ G^{Z}(𝔐).

For the converse, consider some 𝔐 ∈ Max S and put 𝔐^{0} = ⋂_{g∈G} 𝔐^{g}, a G-stable ideal of S. Then

S/𝔐^{0} ≅ ⨁_{g∈G^{Z}(𝔐)\G} S/𝔐^{g} ≅ (S/𝔐) ⊗_{U} T

as T -modules. Similarly, P/P 𝔐^{0} ≅ P ⊗_{S} S/𝔐^{0} ≅ (P/P 𝔐)⊗_{U} T as T -modules, with G acting “diagonally” on P ⊗_{S} S/𝔐^{0}. By hypothesis, P/P 𝔐 is isomorphic with (S/𝔐)^{r} as U-modules, and so

P/P 𝔐^{0} ≅ (S/𝔐^{0})^{r}

as T -modules. From S = S · I, we infer that P = P I + P 𝔐^{0}. Since 𝔐 was arbitrary, this implies P = P I (cf. [Bou], Prop. 11 on p. 113), whence P belongs to add(S_{T}), by Lemma 2.2(ii).

2.4. The G-Picard group of S. We let

Pic_{G} S

denote the set of isomorphism classes of those finitely generated T -modules P that are invertible as S-modules, that is, P_{𝔐} ≅ S_{𝔐} holds for all 𝔐 ∈ Max S (cf. [Ba], p. 132). These modules are projective over S, and hence over T as well, by virtue of (*) (e.g., [Ho], Proposition A.1). The tensor product ⊗_{S} makes Pic_{G} S an abelian group with identity element S_{T}. Here, the G-action on a tensor product X ⊗_{S} Y of T -modules is diagonal: (x ⊗ y)g = xg ⊗ yg.

2.5. Following [Kr] §7 (see also [DMV]), we now describe certain maps between Pic_{G} S and various related groups.

• Most obviously, restricting operators from T to S (forgetting the G-operation) yields a group homomorphism Pic_{G} S → Pic S whose image is clearly contained in the invariants (Pic S)^{G} of the canonical operation of G on Pic S. Thus we have

σ : Pic_{G} S → (Pic S)^{G} .

• By general principles (e.g., [J]), the cohomology group H^{1}(G, U(S)) classifies the T - module structures on S which extend the regular action of S on itself. For future reference, we outline the procedure omitting the routine verifications: Every derivation (1-cocycle) d : G → U(S) gives rise to a T -module structure S_{d} on S by a · gb = a^{g}d(g)b (a, b ∈ S, g ∈ G). One observes that S_{d} ⊗_{S} S_{d′} ≅ S_{dd′} holds for any d, d′ ∈ Der(G, U(S)), and S_{d} ≅ S_{T} = S_{1} if and only if d is an inner derivation. Finally, if · is any T -module operation on S extending the regular S-module structure, then d(g) = 1 · g (g ∈ G) defines a derivation d ∈ Der(G, U(S)) so that (S, ·) = S_{d}. Thus, summarizing, the assignment d ↦ S_{d} defines a map δ : H^{1}(G, U(S)) → Pic_{G} S which fits into an exact sequence

1 → H^{1}(G, U(S)) ⟶^{δ} Pic_{G} S ⟶^{σ} (Pic S)^{G} .

• Restricting the functor E : proj-R → add(S_{T}), Q ↦ Q ⊗_{R} S_{T} of Lemma 2.2(i) to projectives of rank 1 over R, we obtain a group monomorphism

ε : Pic R ↪ Pic_{G} S .

The foregoing proves most of the following Proposition. For each 𝔐 ∈ Max S, we let G^{T} (𝔐) = {g ∈ G : s^{g} − s ∈ 𝔐 for all s ∈ S} denote the inertia group of 𝔐 and (S/𝔐)^{∗} the nonzero elements of S/𝔐.

Proposition. There are exact sequences

1 → H^{1}(G, U(S)) ⟶^{δ} Pic_{G} S ⟶^{σ} (Pic S)^{G}

and

1 → Pic R ⟶^{ε} Pic_{G} S ⟶^{ρ} ∏_{𝔐∈Max S} Hom(G^{T} (𝔐), (S/𝔐)^{∗}) .

The composite ρ ∘ δ is the product {ρ_{𝔐}} of the maps

ρ_{𝔐} : H^{1}(G, U(S)) → H^{1}(G^{T} (𝔐), U(S)) → H^{1}(G^{T} (𝔐), (S/𝔐)^{∗}) ≅ Hom(G^{T} (𝔐), (S/𝔐)^{∗}) ,

the first map being restriction and the second coming from the canonical map U(S) → (S/𝔐)^{∗}.

Proof. Exactness of the first sequence was established above, as was injectivity of ε. It remains to construct a suitable map ρ for the second sequence. To this end, note that there is a map

τ : Pic_{G} S → ∏_{𝔐∈Max S} H^{1}(G^{Z}(𝔐), (S/𝔐)^{∗})

obtained by sending the isomorphism class of P to the element whose 𝔐-component is the isomorphism class of P/P 𝔐 as module over S ∗ G^{Z}(𝔐) via restriction from T. Inasmuch as P_{𝔐} ≅ S_{𝔐} as S_{𝔐}-modules, we have P/P 𝔐 ≅ S/𝔐 as S/𝔐-modules and so, as has been explained above, the S ∗ G^{Z}(𝔐)-module structure on P/P 𝔐 corresponds to an element of H^{1}(G^{Z}(𝔐), (S/𝔐)^{∗}), with the structure P/P 𝔐 ≅ S_{T} /S_{T} 𝔐 corresponding to the identity. Thus Lemma 2.3 coupled with Lemma 2.2(i) proves exactness of the sequence

1 → Pic R ⟶^{ε} Pic_{G} S ⟶^{τ} ∏_{𝔐∈Max S} H^{1}(G^{Z}(𝔐), (S/𝔐)^{∗}) .

Finally, for each 𝔐, the restriction map Res_{𝔐} : H^{1}(G^{Z}(𝔐), (S/𝔐)^{∗}) → H^{1}(G^{T} (𝔐), (S/𝔐)^{∗}) is injective. This follows from the exact sequence

1 → H^{1}(G^{Z}(𝔐)/G^{T} (𝔐), (S/𝔐)^{∗}) ⟶^{infl.} H^{1}(G^{Z}(𝔐), (S/𝔐)^{∗}) ⟶^{Res_{𝔐}} H^{1}(G^{T} (𝔐), (S/𝔐)^{∗}) ,

since H^{1}(G^{Z}(𝔐)/G^{T} (𝔐), (S/𝔐)^{∗}) is trivial, by Hilbert’s Theorem 90 (e.g., [B], §§35.3, 63.10). Thus, identifying H^{1}(G^{T} (𝔐), (S/𝔐)^{∗}) with Hom(G^{T} (𝔐), (S/𝔐)^{∗}), as usual, the requisite map ρ can be taken to be the composite ρ = (∏ Res_{𝔐}) ∘ τ.

2.6. In the special case where (Pic S)^{G} is trivial, we deduce from Proposition 2.5 that

Pic R ≅ ⋂_{𝔐∈Max S} Ker ρ_{𝔐} .

In particular, if G stabilizes some maximal ideal 𝔐 of S such that U(S) → (S/𝔐)^{∗} is an isomorphism, then ρ_{𝔐} is an isomorphism as well and so Pic R = 1. This applies, for example, to linear actions of G on polynomial algebras, taking 𝔐 the ideal generated by the variables ([K]).

3. Multiplicative Actions

3.1. We fix the following notations, specializing the setting of §2.1.

A will denote a (multiplicative) free abelian group of finite rank, S will be the group algebra kA of A over the field k, and G will be a finite subgroup of GL(A).

Then G acts as a group of k-algebra automorphisms on S by means of the unique extension of the natural G-action on A. Actions of this type are called multiplicative actions. The notations T, R, etc. of §2.1 remain in effect. Our hypothesis on surjectivity of the trace map is equivalent with the requirement that |G| ≠ 0 in k, which will be assumed throughout.

3.2. Following [BrL] we put, for each g ∈ G,

I(g) = (s − s^{g} : s ∈ S) ,

the ideal of S that is generated by all elements s − s^{g}. Writing [A, g] for the subgroup of A consisting of the elements [a, g] = a^{−1}a^{g}, we have the following simple

Lemma. (i) For each 𝔐 ∈ Max S, g ∈ G^{T} (𝔐) if and only if 𝔐 ⊇ I(g).

(ii) I(g) is the kernel of the canonical map S = kA ↠ kA/[A, g]. In particular, I(g) is semiprime.

Proof. (i) is evident.

(ii). The first assertion is a consequence of the fact that I(g) can clearly also be written as I(g) = (a − a^{g} : a ∈ A) = ([a, g] − 1 : a ∈ A). So S/I(g) is isomorphic with the group algebra kA/[A, g] and, in order to prove semiprimeness, we must show that the order of the torsion subgroup of A/[A, g] is nonzero in k (cf. [P], Theorems 4.2.12/13). But, letting γ denote the endomorphism a ↦ ∏_{x∈⟨g⟩} a^{x} of A, the torsion subgroup of A/[A, g] is (Ker γ) /[A, g] ≅ H^{1}(⟨g⟩, A), a finite abelian group whose exponent divides the order of g. Our assertion now follows from the fact that |G| ≠ 0 in k.

3.3. Theorem. Pic R ≅ ⋂_{C} Ker (Res_{C}^{G} : H^{1}(G, A) → H^{1}(C, A)), where C runs over the cyclic subgroups of G.

Proof. First, S is a UFD, so Pic S = 1 and §2.6 yields

Pic R ≅ ⋂_{𝔐∈Max S} Ker ρ_{𝔐} .

Next, S has only “trivial” units, U(S) = k^{∗} × A. Hence,

H^{1}(G, U(S)) = H^{1}(G, A) × Hom(G, k^{∗}) .

Moreover, for the augmentation ideal ωS = (a−1 : a ∈ A) of S, the map ρ_{ωS} : H^{1}(G, U(S)) → Hom(G^{T} (ωS), (S/ωS)^{∗}) = Hom(G, k^{∗}) is trivial on H^{1}(G, A) and the identity on Hom(G, k^{∗}). Therefore,

Pic R ≅ ⋂_{𝔐∈Max S} Ker r_{𝔐} ,

where r_{𝔐} denotes the restriction of ρ_{𝔐} to H^{1}(G, A). Writing cl(d) for the class of the derivation d ∈ Der(G, A) in H^{1}(G, A), we have

cl(d) ∈ ⋂_{𝔐∈Max S} Ker r_{𝔐} ⟺ ∀𝔐 ∈ Max S ∀g ∈ G^{T} (𝔐) : d(g) − 1 ∈ 𝔐

⟺ ∀g ∈ G : d(g) − 1 ∈ ⋂_{\begin{matrix}𝔐∈Max S \\ 𝔐⊇I(g) \end{matrix}} 𝔐

⟺ ∀g ∈ G : d(g) ∈ [A, g] .

The second and the third of the above equivalences follow from Lemma 3.2(i) and (ii), respectively, and the last statement asserts that, for all g ∈ G, the restriction of cl(d) to ⟨g⟩ is trivial in H^{1}(⟨g⟩, A), which completes the proof of the theorem.

3.4. Alternative description. The cohomology group H^{1}(G, A) can be written as follows:

H^{1}(G, A) ≅ \overline{A}^{G}/\overline{A^{G}} ,

where : A → \overline{A} = A/A^{n} is the canonical map and n = |G| is the order of G (or any multiple).

The above isomorphism follows from the cohomology sequence for 1 → A ⟶^{( .)^{n}} A ⟶ \overline{A} → 1 which starts with

1 → A^{G} → A^{G} ⟶ \overline{A}^{G} → H^{1}(G, A) ⟶^{( .)^{n}=0} H^{1}(G, A)

(e.g., [B], §15.5). Explicitly, the cohomology class that is associated with the element aA^{n} ∈ \overline{A}^{G} for a ∈ A is the class of the derivation

d_{a} : G → A, g ↦ \sqrt[n]{[a, g]} .

In these terms, the restriction map Res_{C}^{G} : H^{1}(G, A) → H^{1}(C, A) is simply the map

\overline{A}^{G}/\overline{A^{G}} ⟶ \overline{A}^{C}/\overline{A^{C}}

that is afforded by the inclusions \overline{A}^{G} ⊆ \overline{A}^{C} and \overline{A^{G}} ⊆ \overline{A^{C}}. This leads to the following easily programmeable description of Pic R:

Pic R ≅ ( ⋂_{\begin{matrix}C≤G \\ C cyclic \end{matrix}} \overline{A^{C}})/\overline{A^{G}} .

3.5. Relationship with the class group. For general reasons (e.g., [F], Cor. 18.3), Pic R embeds into the class group Cl R of R. The latter group has been determined in [L]:

Cl R ≅ Hom(G/N, k^{∗}) ⊕ H^{1}(G/D, A^{D}) ,

where N denotes the subgroup of G that is generated by all reflections in G and D the subgroup generated by the reflections that are diagonalizable over ℤ. An explicit embedding Pic R ↪ Cl R can be realized as follows. The term H^{1}(G/D, A^{D}) is canonically isomorphic with the intersection ⋂_{H} (Res_{H}^{G} : H^{1}(G, A) → H^{1}(H, A)), where H runs over the subgroups of G that are generated by a ℤ-diagonalizable reflection ([L], proof of Theorem 2.5). As all these occur among the subgroups C in Theorem 3.3, there is an obvious inclusion (strict, in general)

Pic R ⊆ H^{1}(G/D, A^{D}) .

3.6. The Cartan map. For any integrally closed commutative Noetherian domain R, the embedding Pic R ↪ Cl R that was referred to in §3.5 is part of a commutative diagram involving the Cartan map c_{R} : K_{0}(R) → G_{0}(R):

[figure: K_{0}(R)/ℤ[R]; via c_{R}; det; Pic R; G_{0}(R)/ℤ[R]; Cl R] .

The map det is described in [Ba], p. 466 and the map G_{0}(R)/ℤ[R] ↠ Cl R is as in [Bou2], p.

69. Commutativity of the diagram is a consequence of [Bou2], Exercise 12(b), p. 108. In view of this diagram, the examples of multiplicative invariants R with nontrivial Pic R

which we will exhibit in §3.8 also have the property that Im c_{R} is strictly larger than ℤ[R]. This is in contrast with the case of linear invariants R, where one always has Im c_{R} = ℤ[R]. (See the Appendix.)

3.7. Torsion in K_{0} of some polycyclic group algebras. Note that, in the setting of §3.1,

T ≅ kΓ, the group algebra of the semidirect product Γ = A ⋊ G over k.

Thus Γ is a polycyclic-by-finite group. We will use our observations about the Cartan map c_{R} in §3.6 to show:

Proposition. Pic R is a subquotient of the torsion subgroup of K_{0}(T).

Proof. Consider the following commutative diagram:

[figure: K_{0}(R); ϕ=( .)⊗_{R}S_{T}; c_{R}; G_{0}(R); K_{0}(T); c_{T}; G_{0}(T); ( .)^{G}] .

That this diagram is indeed commutative is a consequence of Lemma 2.2(i). Our assumption on |G| implies that T is regular, and so c_{T} is an isomorphism. (By [BrL], Cor. 1.3, one further knows that the fixed point map ( .)^{G} : G_{0}(T) → G_{0}(R) is surjective, but this will not be essential here.) Our goal is to establish the following

Claim. Im ϕ = ℤ[S_{T}] ⊕ F for some finite subgroup F of K_{0}(T).

Granting this for the moment, we conclude that the image of the map \overline{c_{R}} : K_{0}(R)/ℤ[R] → G_{0}(R)/ℤ[R] that was considered in §3.6 is identical with the image of F in G_{0}(R)/ℤ[R]. Thus, Im \overline{c_{R}} is a subquotient of the torsion subgroup of K_{0}(T). Moreover, from the diagram in §3.6 we infer that Pic R is a subquotient of Im \overline{c_{R}}, whence the proposition.

To prove the claim, note that K_{0}(S) = ℤ · [S]. Hence K_{0}(T) = ℤ[S_{T}] ⊕ Ker Res_{S}^{T} and Im ϕ = ℤ[S_{T}] ⊕ F, where F = Im ϕ ∩ Ker Res_{S}^{T}. To prove that F is finite, we consider the canonical map ˜ . : T = kΓ → T̃ = k[Γ/B] for a suitable normal subgroup B of Γ. Specifically, following [BrL2] §1.7, we select B ⊆ A with index [Γ : B] a divisor of some power of |G| and such that the quotient Γ̃ = Γ/B ≅ (A/B) ⋊ G is “conjugacy separating” for Γ in the sense of [BrL], Lemma 3.3. The relevant effect of this latter property is that the kernel of the map K_{0}(˜ .) : K_{0}(T) → K_{0}(T̃) is precisely the (finite) torsion subgroup of K_{0}(T) ([BrL2], §§1.6, 1.8). Thus it suffices to show that K_{0}(˜ .) annihilates F. But T̃ ≅ S̃ ∗ G, and so we have an analogous map ϕ̃ = ( .) ⊗_{R} T̃ : K_{0}(R̃) → K_{0}(T̃), where R̃ = S̃^{G}. (This is the same as the image of R under ˜ ., by surjectivity of the trace map.) Furthermore, K_{0}(˜ .) sends F = Im ϕ ∩ Ker Res_{S}^{T} to Im ϕ̃ ∩ Ker Res_{T} ; _{S}, and so we are reduced to showing

that the latter intersection is trivial. However, Res_{T} ; _{S} ∘ϕ̃ is identical with the induction map

Ind_{S} ; _{R} : K_{0}(R̃) → K_{0}(S̃), and this map is injective. Indeed, S̃ is commutative semisimple, by assumption on |G|, and hence so is R̃ = S̃^{G}. Thus the group K_{0}(R̃) is free abelian with basis

{[R̃/𝔪] : 𝔪 a maximal ideal of R̃}, and likewise for S̃. The induction map Ind_{S} ; _{R} sends a typical basis element [R̃/𝔪] to [S̃/𝔪S̃] = ∑_{𝔐}[S̃/𝔐], where 𝔐 runs over the maximal ideals of S̃ lying

over 𝔪. This shows that Ind_{S} ; _{R} is mono, thereby completing the proof of the proposition.

3.8. Computations in dimension ≤ 4. A systematic search of the crystallographic groups library of GAP, version 3.4, produced the following results.

3.8.1. Dimension 2. All multiplicative invariants R for rk A = 2 have trivial Picard group. This is easy to see directly: Since SL_{2}(ℤ) acts fixed point freely on ℤ^{2}, we have A^{C} = 0 for every nontrivial cyclic subgroup C ⊆ SL(A). Consequently, Pic R = 1 if G intersects SL(A) nontrivially. In the opposite case, G is cyclic, having order at most 2, and so Pic R = 1 again.

3.8.2. Dimension 3. Among the 73 conjugacy classes of finite subgroups of GL_{3}(ℤ), it turns out, 2 lead to an invariant algebra R with nontrivial Picard group. These classes are repre-

sented by the groups G_{1} = ⟨x, y⟩ ≅ C_{2} × C_{2} and G_{2} = ⟨x, y, z⟩ ≅ A_{4}, where x = ( \begin{matrix}0 & −1 & 1 \\ 0 & −1 & 0 \\ 1 & −1 & 0 \end{matrix} ),

y = ( \begin{matrix}0 & 1 & −1 \\ 1 & 0 & −1 \\ 0 & 0 & −1 \end{matrix} ), and z = ( \begin{matrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} ). In both cases, Pic R has order 2. Note that G_{1} is a nor-

mal subgroup of G_{2}. An explicit isomorphism Pic S^{G_{2}} ⟶^{≅} Pic S^{G_{1}} is afforded by the restriction map H^{1}(G_{2}, A) ⟶^{≅} H^{1}(G_{1}, A)^{G_{2}} (note that A^{G_{1}} = 0). We remark that Cl S^{G_{1}} = ℤ/(2) ⊕ ℤ/(2) ⊕ ℤ/(4) and Cl S^{G_{2}} = ℤ/(3) ⊕ ℤ/(4).

3.8.3. Dimension 4. The 710 conjugacy classes of finite subgroups of GL_{4}(ℤ) altogether include 9 whose invariant algebra has a nontrivial Picard group, of order 2 in all cases. The above two rank 3 groups G directly account for 4 of these: ( \begin{matrix}G \\ & 1 \end{matrix} ) has invariant algebra R[X^{±1}] and ( \begin{matrix}G \\ & ±1 \end{matrix} ) leads to R[X], both with the same Picard group (and class group) as R (cf. [Ba], Props. III.7.17 and III.7.19).

4. Appendix: The Cartan map for linear actions

The purpose of this short Appendix is to substantiate the claim, made in §3.6, that for invariants R of linear actions, the image of the Cartan map c_{R} is generated by the classes of the free R-modules. We will use the following notations:

V will denote a finite-dimensional vector space over the field k, S will be the symmetric algebra S(V), and G will be a finite subgroup of GL(V) with |G| ≠ 0 in k.

We consider the G-action by k-algebra automorphisms on S obtained by extending the natural action on V. The remaining notations R, T, etc. are as in §2.1.

Proposition. Im c_{R} = ℤ[R].

Proof. The proof is a greatly simplified version of the proof of Proposition 3.7 whose notation we employ here. It suffices to show that Im ϕ = ℤ[S_{T}] and, exactly as in the proof of Proposition 3.7, one certainly has Im ϕ = ℤ[S_{T}] ⊕ F, where F = Im ϕ ∩ Ker Res_{S}^{T}. To prove that F = 0, consider the map ˜ . : T = S ∗ G → T̃ = kG which sends V to 0. By Quillen’s induction theorem ([Q], Theorem 7), the corresponding map K_{0}(˜ .) is an isomorphism. On the other hand, a trivial variant of the argument used in the proof of Proposition 3.7, with R̃ = S̃ = k, shows that K_{0}(˜ .) annihilates F, whence F = 0.

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       anonymous ftp from math.rwth-aachen.de

Department of Mathematics, Temple University, Philadelphia, PA 19122-2585

E-mail address: lorenz@math.temple.edu