K_{0} OF INVARIANT RINGS AND NONABELIAN H^{1}

MARTIN LORENZ

Department of Mathematics Temple University Philadelphia, PA 19122-6094

                        lorenz@math.temple.edu

Abstract. We give a description of the kernel of the induction map K_{0}(R) → K_{0}(S), where S is a commutative ring and R = S^{G} is the ring of invariants of the action of a finite group G on S. The description is in terms of H^{1}(G, GL(S)).

Introduction

This article is concerned with the relationship between K_{0}(S) and K_{0}(R), where S is a commutative ring and R = S^{G} denotes the subring of invariants under the action of a finite group G on S.

Specifically, working under the assumption that the trace tr : S → R, s ↦ ∑_{g∈G} s^{g}, is surjective, we shall study the kernel of the induction map

Ind_{R}^{S} = K_{0}(f) : K_{0}(R) → K_{0}(S)

that is associated with the inclusion f : R ↪ S. We will describe an embedding of Ker(Ind_{R}^{S}) into the cohomology set H^{1}(G, GL(S)). Moreover, we will endow H^{1}(G, GL(S)) with a natural commutative monoid structure, essentially coming from the “block diagonal” maps GL_{n} × GL_{m} → GL_{n+m} ↪ GL, such that our embedding identifies Ker(Ind_{R}^{S}) with the group of units U(H^{1}(G, GL(S))). To further describe this unit group, we define S_{H}, for any subgroup H of G, to be the factor of S modulo the intersection of all maximal ideals of S whose inertia group contains H. Letting ρ_{H} : H^{1}(G, GL(S)) → H^{1}(H, GL(S_{H})) denote the map that is given by restriction from G to H and the canonical map S ↠ S_{H}, our main result reads as follows.

Theorem. Ker(Ind_{R}^{S}) ≅ U(H^{1}(G, GL(S))) = ⋂_{H} Ker ρ_{H}, where H ranges over all cyclic (or, equivalently, all) subgroups of G.

1991 Mathematics Subject Classification. 13A50, 13D15, 16E20, 16W20, 18G50, 19A49.

Key words and phrases. ring of invariants, group action, nonabelian 1-cohomology, Grothendieck group, induction map, Galois action, Hilbert’s Theorem 90.

Research supported in part by NSF Grant DMS-9618521.

The theorem follows via direct limits from a corresponding result, with GL_{n}( .) in place of GL( .), for the kernel of the maps P_{n}(f) : P_{n}(R) → P_{n}(S), where P_{n}( .) denotes the set of isomorphism classes of f.g. projective modules of constant rank n.

As applications, we present a version of Hilbert’s Theorem 90 for Galois actions on commutative rings and quickly derive the (known) structure of the Picard groups of linear and multiplicative invariants ([K], [L]). Some open problems are also discussed.

Notations and Conventions. Throughout this note, S will be a ring, assumed commutative from §2.3 onwards, G will be a finite group acting by automorphisms on S; the action will be written s ↦ s^{g}, and

R = S^{G} will denote the ring of G-invariants in S.

We make 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.

1. Nonabelian H^{1}

1.1. Definition. We recall the definition of nonabelian H^{1} following [Se] §I.5 (or [BS], [Se2] p. 123ff), except that our group actions are on the right.

Let X be a G-group, with G-action written as x ↦ x^{g}. A (1-)cocycle is a map d : G → X satisfying

d(gg′) = d(g)^{g′}d(g′) (g, g′ ∈ G) .

The set of 1-cocycles of G in X will be denoted by Z^{1}(G, X). Two cocycles d, e ∈ Z^{1}(G, X) are called cohomologous if there exists an element x ∈ X satisfying d(g) = x^{g}e(g)x^{−1} for all g ∈ G. This defines an equivalence relation on Z^{1}(G, X). The set of equivalence classes is

H^{1}(G, X) ,

a pointed set with distinguished element the class of the unit cocycle 1(g) = 1 for all g ∈ G.

Remark. H^{1}(G, X) parametrizes the conjugacy classes of complements of X in the split extension X ⋊ G, exactly as in the familiar special case where X is abelian (cf. [Ro], 11.1.2, 11.1.3).

1.2. Examples. (1) Suppose G acts trivially on X. Then Z^{1}(G, X) = Hom(G, X) and

H^{1}(G, X) = Hom(G, X)/X ,

with X acting by conjugation on Hom(G, X). The distinguished class consists of 1 alone.

K_{0} OF INVARIANT RINGS 3

(2) If G = ⟨g⟩ is cyclic of order m, then each cocycle d ∈ Z^{1}(G, X) is determined by the element x = d(g) ∈ X, and the eligible elements of X are precisely those satisfying the condition x^{g^{m−1}}x^{g^{m−2}} … x^{g}x = 1. Moreover, if d, e ∈ Z^{1}(G, X) correspond in this manner to x, y ∈ X, respectively, then d and e are cohomologous precisely if there exists z ∈ X with x = z^{g}yz^{−1}. Thus, writing ∼ for the equivalence relation on X determined by this condition, we have

H^{1}(G, X) ≅ {x ∈ X : x^{g^{m−1}}x^{g^{m−2}} … x^{g}x = 1}/ ∼ .

(3) If the order of X is finite and coprime to |G| then H^{1}(G, X) is trivial, by the uniqueness part of the Schur-Zassenhaus Theorem (cf. [Ro], 9.1.2).

(4) If X is a linear algebraic group over an algebraically closed field whose characteristic does not divide |G| and G acts algebraically on X then H^{1}(G, X) is finite. This is essentially due to A. Weil who explicitly dealt with the case of a trivial G-action on X ([We], cf. also [Sl]). The general case is an easy consequence.

1.3. Functoriality and direct limits. Suppose that we are given a homomorphism of groups α : G → G′ and a G′-group X′. Then X′ can be viewed as a G-group via α. Any map of G-groups (i.e., any group homomorphism compatible with the G-actions) f : X′ → X gives rise to a map Z^{1}(G′, X′) → Z^{1}(G, X), d ↦ f ∘ d ∘ α, and this map passes down to a map of pointed sets

(α, f)_{∗}^{1} : H^{1}(G′, X′) → H^{1}(G, X) .

In particular, we have the restriction maps H^{1}(G, X) → H^{1}(H, X) for subgroups H of G and the inflation maps H^{1}(G/N, X^{N}) → H^{1}(G, X) for normal subgroups N. Here, X^{N} denotes the N-invariants in X. We will write f_{∗}^{1} for (Id_{G}, f)_{∗}^{1}.

The following easy lemma is surely well-known, but I am not aware of a reference. (For commutative cohomology, see [Br], Prop. (4.6) on p. 195.)

Lemma. Let (X_{n}, f_{mn}) be a direct system of G-groups and let X = \underrightarrow{lim} X_{n} be the direct limit with its induced G-action (cf. [Bou3], p. A I.117). If all f_{mn} : X_{n} → X_{m} (m ≥ n) are injective then H^{1}(G, X) ≅ \underrightarrow{lim} H^{1}(G, X_{n}) (as pointed sets).

Proof. Put φ_{mn} = (f_{mn})_{∗}^{1} : H^{1}(G, X_{n}) → H^{1}(G, X_{m}) and φ_{n} = (f_{n})_{∗}^{1}, where f_{n} : X_{n} → X is the canonical map. Then the relations f_{m}∘f_{mn} = f_{n} entail φ_{m}∘φ_{mn} = φ_{n}, and so there is a unique map φ : \underrightarrow{lim} H^{1}(G, X_{n}) → H^{1}(G, X) with φ∘ψ_{n} = φ_{n}, where ψ_{n} : H^{1}(G, X_{n}) → \underrightarrow{lim} H^{1}(G, X_{n}) is the canonical map. We show that φ is bijective; the fact that φ respects distinguished elements is clear.

For surjectivity, let d ∈ Z^{1}(G, X) be given. Since G is finite, there is an n with d(G) ⊆ f_{n}(X_{n}), and so the class of d in H^{1}(G, X) belongs to Im φ_{n} ⊆ Im φ.

As to injectivity, we must show that if a, b ∈ H^{1}(G, X_{n}) satisfy φ_{n}(a) = φ_{n}(b) then there exists m ≥ n with φ_{mn}(a) = φ_{mn}(b) (cf. [Bou2], Prop. 6 on p. E III.62). Say a and b are the classes of d, e ∈ Z^{1}(G, X_{n}), respectively. Then f_{n} ∘ d and f_{n} ∘ e are cohomologous in Z^{1}(G, X), i.e., there exists x ∈ X with (f_{n} ∘d)(g) = x^{g}(f_{n} ∘e)(g)x^{−1} for all g ∈ G. Now x ∈ f_{m}(X_{m}) for some m ≥ n, say x = f_{m}(y). Then

(f_{m} ∘ f_{mn} ∘ d)(g) = f_{m}(y^{g})(f_{m} ∘ f_{mn} ∘ e)(g)f_{m}(y^{−1}) .

Since all f_{mn} are injective, so are the maps f_{n} ([Bou2], Remarque 1 on p. E III.63). Hence, (f_{mn} ∘ d)(g) = y^{g}(f_{mn} ∘ e)(g)y^{−1} holds for all g ∈ G which shows that f_{mn} ∘ d and f_{mn} ∘ e are cohomologous in Z^{1}(G, X_{m}), as required.

1.4. The monoid structure of H^{1}(G, GL(S)). Lemma 1.3 implies in particular that in H^{1}(G, GL(S)) ≅ \underrightarrow{lim} H^{1}(G, GL_{n}(S)), where G acts on GL(S) and on GL_{n}(S) via its action on S. Our goal here is to endow H^{1}(G, GL(S)) with the structure of a commutative monoid.

More generally, let X be any G-stable subgroup of GL(S) containing the matrices

s_{m,n} =_{def} (−1)^{(m+1)n} (\begin{matrix}0_{n×m} & −1_{n×n} \\ 1_{m×m} & 0_{m×n} \\ & & 1 \\ & & & . & . & . \end{matrix} ) .

Note that det s_{m,n} = 1. (In fact, it is not hard to show that the subgroup of GL(S) that is generated by the matrices s_{m,n} consists of all monomial matrices with entries ±1 and with determinant 1.) Hence s_{m,n} ∈ E(S), the group generated by the elementary matrices ([Ba], Prop. 1.6 on p. 226). Furthermore,

s_{m,n}^{−1} (\begin{matrix}a_{n×n} \\ & b_{m×m} \\ & & 1 \\ & & & . & . & . \end{matrix} ) s_{m,n} = (\begin{matrix}b_{m×m} \\ & a_{n×n} \\ & & 1 \\ & & & . & . & . \end{matrix} ) .

Thus, if x = ( \begin{matrix}x_{n×n} \\ & 1 \\ & & . & . & . \end{matrix} ) ∈ X_{n} = X ∩ GL_{n}(S) then, for any m,

x[m] =_{def} (\begin{matrix}1_{m×m} \\ & x_{n×n} \\ & & 1 \\ & & & . & . & . \end{matrix} ) = s_{m,n}^{−1}xs_{m,n} ∈ X_{m+n} .

For a, b ∈ H^{1}(G, X), define a + b ∈ H^{1}(G, X) as follows. Choose cocycles d, e ∈ Z^{1}(G, X) representing a and b, respectively. Since G is finite, we have d ∈ Z^{1}(G, X_{m}) and e ∈ Z^{1}(G, X_{n}) for suitable m and n. For g ∈ G, put

(d ⊕_{m} e)(g) =_{def} d(g) · e(g)[m] = (\begin{matrix}d(g)_{m×m} \\ & e(g)_{n×n} \\ & & 1 \\ & & & . & . & . \end{matrix} ) ∈ X .

It is trivial to verify that d ⊕_{m} e is a cocycle, and we define a + b to be its class in H^{1}(G, X). To show that this is well-defined, we first note that the class of d ⊕_{m} e is independent of m, as long as d(G) ⊆ X_{m}. Indeed, if t ≥ 0 then

(d ⊕_{m+t} e)(g) = (s_{t,n}[m])^{−1} (d ⊕_{m} e)(g)s_{t,n}[m] ,

and s_{t,n}[m] = s_{t,n}[m]^{g} ∈ X. Thus, d⊕_{m+t}e and d⊕_{m}e are cohomologous. Now suppose that a and b are also represented by the cocycles d′ ∈ Z^{1}(G, X) and e′ ∈ Z^{1}(G, X), respectively. So there are matrices x, y ∈ X with

d′(g) = x^{g}d(g)x^{−1} and e′(g) = y^{g}e(g)y^{−1} .

Fix r so that x, y, d(G), e(G) are all contained in X_{r}, and hence so are d′(G) and e′(G). By the foregoing, it suffices to show that d ⊕_{r} e and d′ ⊕_{r} e′ are cohomologous. But, putting z = x · y[r] ∈ X, we have (d′ ⊕_{r} e′)(g) = z^{g}(d ⊕_{r} e)(g)z^{−1}. Thus, a + b ∈ H^{1}(G, X) is indeed well-defined. Commutativity and associativity of + follow along similar lines, thereby making H^{1}(G, X) a commutative monoid with neutral element the class of the unit cocycle. — For a different description of the monoid structure of H^{1}(G, X) for X = GL(S), see Lemma 2.7.

1.5. Monoid maps. Let S′ be another ring that is acted on by a group G′ and suppose that we are given a group homomorphism α : G → G′ and a G-equivariant ring map ϕ : S′ → S, where G acts on S′ via α. Then we obtain a map of G-groups GL(ϕ) : GL(S′) → GL(S). Thus, if X′ ⊆ GL(S′) is a G′-stable subgroup containing the matrices s_{m,n} ∈ GL(S′) of §1.4 and if X ⊆ GL(S) is G-stable containing the image of X′ then we have a map of G-groups f : X′ → X. The map induced on cocycles Z^{1}(G′, X′) → Z^{1}(G, X) (cf. §1.3) is easily seen to respect the operations ⊕_{m} of §1.4, and hence the map (α, f)_{∗}^{1} : H^{1}(G′, X′) → H^{1}(G, X) of §1.3 is actually a monoid map. This applies in particular to restriction and induction maps (with ϕ = Id_{S}). Moreover, we have the following

Lemma. Let X ⊆ GL(S) be a G-stable subgroup with X ⊇ E(S) and let f : X → X^{ab} = X/[X, X] denote the canonical map. Then f_{∗}^{1} : H^{1}(G, X) → H^{1}(G, X^{ab}) is a monoid map, where H^{1}(G, X^{ab}) has its usual group structure.

Proof. Consider d, e ∈ Z^{1}(G, X), say d, e ∈ Z^{1}(G, X_{m}). Then, by [Ba], Prop. (1.7) on p. 226,

(d ⊕_{m} e)(g) ≡ d(g) · e(g) = (\begin{matrix}d(g)_{m×m}e(g)_{m×m} \\ & 1 \\ & & . & . & . \end{matrix} ) mod E(S) .

Inasmuch as X^{ab} = X/ E(S) (cf. [Ba], Thm. (2.1) on p. 228), our assertion follows.

1.6. Units. The kernel of Ind_{R}^{S} : K_{0}(R) → K_{0}(S) will turn out to be isomorphic to the group of units U(H^{1}(G, GL(S))) of the monoid H^{1}(G, GL(S)) (see §2.8). Here, we make some preliminary observations on the unit group

U(H^{1}(G, X)) ,

where X is any G-invariant subgroup of GL(S) containing the matrices s_{m,n}, as in §1.4.

Lemma. Let N = Ker_{G}(X) denote the kernel of the action of G on X. Then U(H^{1}(G, X)) ≅ U(H^{1}(G/N, X)) via inflation. In particular, if G acts trivially on X, then U(H^{1}(G, X)) contains only the unit class.

Proof. Use the inflation-restriction sequence H^{1}(G/N, X) → H^{1}(G, X) → H^{1}(N, X) ([Se], §I.5.8). This sequence is exact, the first map (inflation) is injective, and both maps are monoid maps. Thus it induces an exact sequence of groups

1 → U(H^{1}(G/N, X)) ⟶ U(H^{1}(G, X)) ⟶ U(H^{1}(N, X)) .

Part (ii) therefore reduces to the claim that U(H^{1}(N, X)) is trivial. To verify this, recall that H^{1}(N, X) = Hom(N, X)/X, with X acting by conjugation on Hom(N, X), and the unit class consists of the unit map 1 alone (cf. §1.2). Thus, letting ⟨ . ⟩ denote X-conjugacy classes, the equation ⟨d⟩ + ⟨e⟩ = ⟨1⟩ for d, e ∈ Hom(N, X) is equivalent with (d ⊕_{m} e)(g) = 1 for all g ∈ N, where m is chosen as above. But the latter condition says that d = e = 1.

2. The Kernel of Induction

2.1. The skew group ring. We will let

T = S∗G

denote the skew group ring that is associated with the given G-action on 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 is an R-T -bimodule with action

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

Hypothesis (*) 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 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 Rlinear 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 description of add S_{T}. From now on, S is assumed commutative. We let Max S denote the set of maximal ideals of S. For each 𝔐 ∈ Max S, we put G^{Z}(𝔐) = {g ∈ G : 𝔐^{g} = 𝔐}, the decomposition group of 𝔐, and

T (𝔐) = (S/𝔐)∗G^{Z}(𝔐) ,

the skew group ring that is associated with the action of G^{Z}(𝔐) on S/𝔐. As in §2.1, S/𝔐 is a right module over T (𝔐); this module structure can be viewed as coming from S_{T} by restriction to S∗G^{Z}(𝔐) and reduction mod 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, there is an isomorphism of T (𝔐)-modules P/P 𝔐 ≅ (S/𝔐)^{r} (r = rank 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 the homogeneous T (𝔐)-module (S/𝔐)^{n}.

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) ⊗_{T (𝔐)} T

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

P/P 𝔐^{0} ≅ (S/𝔐^{0})^{r}

as T -modules. Since S = S · I (cf. §2.1), this isomorphism implies P = P I + P 𝔐^{0}, and since 𝔐 was arbitrary, we further conclude that P = P I (cf. [Bou], Prop. 11 on p. 113). In view of Lemma 2.2(ii), this shows that P belongs to add S_{T}.

2.4. The induction triangle. For each n ≥ 0, we let P_{n}(R) denote the set of isomorphism classes of f.g. projective R-modules of constant rank n, and similarly for P_{n}(S). These are pointed sets with distinguished elements ⟨R^{n}⟩ and ⟨S^{n}⟩, respectively, where ⟨ . ⟩ denotes isomorphism classes. Furthermore, P_{S,n}(T) will denote the set of isomorphism classes of f.g. projective right T -modules having constant rank n as S-modules, with distinguished element ⟨S_{T}^{n}⟩. We have a commutative diagram of pointed sets (cf. [Ba], Prop. (7.3) on p. 130)

[figure: P_{n}(R); Φ_{n}=( .)⊗_{R}S_{T}; P_{n}(f); P_{n}(S); P_{S,n}(T); _{Res{T \atop S,n}}]

By Lemma 2.2(i), Φ_{n} is injective. The kernels of the other two maps will be described in §§2.5 and 2.6 below. Recall that the kernel of a map of pointed sets is defined to be the preimage of the distinguished element of the target set.

2.5. The kernel of Res_{S,n}^{T}. We now consider the restriction map Res_{S,n}^{T} : P_{S,n}(T) → P_{n}(S), as in §2.4.

Lemma. Ker(Res_{S,n}^{T}) ≅ H^{1}(G, GL_{n}(S)) as pointed sets.

Proof. Each cocycle d ∈ Z^{1}(G, GL_{n}(S)) gives rise to a T -module structure (S^{n})_{d} on S^{n} via

x · gs = x^{g}d(g)s (x ∈ S^{n}, g ∈ G, s ∈ S) .

This action extends the regular S-module structure on S^{n}. Conversely, if · is any right T -module operation on S^{n} extending the regular S-module structure then write, for g ∈ G,

e_{i} · g = ∑_{j=1}^{n} e_{j}d_{i,j}(g) ,

where e_{i} ∈ S^{n} is the basis element with 1 in the i-th position and 0s elsewhere and d_{i,j}(g) ∈ S. A routine verification shows that d = (d_{i,j}) is a cocycle of G in GL_{n}(S) and the given T -module structure on S^{n} is identical with (S^{n})_{d}.

Since (S^{n})_{d} for the unit cocycle d = 1 is just S_{T}^{n}, we obtain a surjective map of pointed sets

Z^{1}(G, GL_{n}(S)) ↠ Ker(Res_{S,n}^{T}), d ↦ ⟨(S^{n})_{d}⟩ .

Finally, for d, e ∈ Z^{1}(G, GL_{n}(S)), we have (S^{n})_{d} ≅ (S^{n})_{e} as T -modules precisely if there is an S-module isomorphism (S^{n})_{d} ⟶^{≅} (S^{n})_{e} that commutes with the G-actions, that is, for some matrix a ∈ GL_{n}(S), (x · g)a = (xa) · g holds for all x ∈ S^{n}, g ∈ G. The latter condition is equivalent with x^{g}d(g) = x^{g}a^{g}e(g)a^{−1} which in turn just says that d and e are cohomologous. This completes the proof of the lemma.

Remark. With T (𝔐) as in §2.3, we have a map of pointed sets P_{S,n}(T) → P_{S/𝔐,n}(T (𝔐)), P ↦ P/P 𝔐, which restricts to a map Ker(Res_{S,n}^{T}) → Ker (Res_{S/𝔐,n}^{T (𝔐)}). In terms of the identification provided by the above Lemma, the latter becomes the map

ρ′_{𝔐,n} : H^{1}(G, GL_{n}(S)) → H^{1}(G^{Z}(𝔐), GL_{n}(S/𝔐))

that is given by restriction from G to G^{Z}(𝔐) and reduction modulo 𝔐.

2.6. The kernel of P_{n}(f). For each subgroup H ≤ G, we put

J(H) = ⋂_{\begin{matrix}𝔐∈Max S \\ H⊆G^{T} (𝔐) \end{matrix}}𝔐 and S_{H} = S/J(H) ,

where G^{T} (𝔐) = {g ∈ G : s^{g} − s ∈ 𝔐 for all s ∈ S} is the inertia group of 𝔐. In addition to the maps ρ′_{𝔐,n} introduced in Remark 2.5, we will consider the analogous restriction-reduction maps

ρ_{𝔐,n} : H^{1}(G, GL_{n}(S)) → H^{1}(G^{T} (𝔐), GL_{n}(S/𝔐))

and

ρ_{H,n} : H^{1}(G, GL_{n}(S)) → H^{1}(H, GL_{n}(S_{H})) .

Since the actions of G^{T} (𝔐) on GL_{n}(S/𝔐) and of H on GL_{n}(S_{H}) are trivial, we have H^{1}(G^{T} (𝔐), GL_{n}(S/𝔐)) = Hom(G^{T} (𝔐), GL_{n}(S/𝔐))/ GL_{n}(S/𝔐) and similarly for H^{1}(H, GL_{n}(S_{H})) (cf. §1.2). We let C denote the set of cyclic subgroups of G.

Proposition. As pointed sets,

Ker P_{n}(f) ≅ ⋂_{𝔐∈Max S} Ker ρ′_{𝔐,n} = ⋂_{𝔐∈Max S} Ker ρ_{𝔐,n}
= ⋂_{C∈C} Ker ρ_{C,n}  = ⋂_{H≤G} Ker ρ_{H,n} .

Proof. In view of the induction triangle in §2.4, Ker P_{n}(f) ≅ Im Φ_{n} ∩ Ker(Res_{S,n}^{T}). Furthermore, by virtue of Lemmas 2.2 and 2.3, if ⟨P ⟩ ∈ P_{S,n}(T) then ⟨P ⟩ ∈ Im Φ_{n} iff ⟨P/P 𝔐⟩ is the distinguished element of P_{S/𝔐,n}(T (𝔐)) for all 𝔐 ∈ Max S. Therefore, by Remark 2.5,

Im Φ_{n} ∩ Ker(Res_{S,n}^{T}) ≅ ⋂_{𝔐∈Max S} Ker ρ′_{𝔐,n} ,

which establishes the isomorphism in the proposition. Now ρ_{𝔐,n} = Res_{G^{T} (𝔐)}^{G^{Z}(𝔐)} ∘ρ′_{𝔐,n}, where

Res_{G^{T} (𝔐)}^{G^{Z}(𝔐)} : H^{1}(G^{Z}(𝔐), GL_{n}(S/𝔐)) → H^{1}(G^{T} (𝔐), GL_{n}(S/𝔐))

is the restriction map. Since this map has trivial kernel, by the generalized “Theorem 90” ([Se], Lemme 1 on p. 129 and §5.8(a)), we conclude that Ker ρ′_{𝔐,n} = Ker ρ_{𝔐,n} which proves the first equality.

Consider a ∈ H^{1}(G, GL_{n}(S)), say a is the class of d ∈ Z^{1}(G, GL_{n}(S)). Then, in view of §1.2(1),

a ∈ ⋂_{𝔐∈Max S} Ker ρ_{𝔐,n} ⟺ ∀𝔐 ∈ Max S ∀g ∈ G^{T} (𝔐) : d(g) ≡ 1_{n×n} mod 𝔐

⟺ ∀g ∈ G ∀𝔐 with 𝔐 ⊇ J(⟨g⟩) : d(g) ≡ 1_{n×n} mod 𝔐 ⟺ ∀g ∈ G : d(g) ≡ 1_{n×n} mod J(⟨g⟩)

⟺ a ∈ ⋂_{C∈C} Ker ρ_{C,n} ,

proving the second equality. Finally, the proof of ⋂_{𝔐} Ker ρ_{𝔐,n} = ⋂_{H} Ker ρ_{H,n} is completely analogous, based on the observation that H ⊆ G^{T} (𝔐) if and only if 𝔐 ⊇ J(H). This completes the proof of the proposition.

2.7. Stabilization. For m ≥ n, we now consider the stabilization maps P_{n}(R) → P_{m}(R), ⟨P ⟩ ↦ ⟨P ⊕ R^{m−n}⟩, the analogous map for S, and the map P_{S,n}(T) → P_{S,m}(T), ⟨Q⟩ ↦ ⟨Q ⊕ S_{T}^{m−n}⟩. These are maps of pointed sets which are compatible with the maps Φ_{n}, Res_{S,n}^{T} and P_{n}(f) in the induction triangle (§2.4). Thus we obtain a commutative diagram

[figure: \underrightarrow{lim} P_{n}(R); φ=\underrightarrow{lim} Φ_{n}; \underrightarrow{lim} P_{n}(f); \underrightarrow{lim} P_{n}(S); \underrightarrow{lim} P_{S,n}(T); r=\underrightarrow{lim} Res_{S,n}^{T}]

with φ injective, by [Bou2], Prop. 7 on p. E III.64. Explicitly (cf. [Bou2], p. E III.61),

\underrightarrow{lim} P_{n}(R) = ⨄_{n≥0} P_{n}(R)/ ∼ ,

where ⨄ denotes the disjoint union and x ∼ y for x = ⟨P ⟩ ∈ P_{n}(R), y = ⟨Q⟩ ∈ P_{m}(R) iff P ⊕ R^{t−n} ≅ Q ⊕ R^{t−m} for some t ≥ max(m, n). In other words, ⟨P ⟩ ∼ ⟨Q⟩ iff P and Q are stably isomorphic. Now, ⨄_{n≥0} P_{n}(R) is a commutative monoid under ⊕ and the equivalence relation ∼ respects this structure. Thus, \underrightarrow{lim} P_{n}(R) becomes a commutative monoid with identity element the stable isomorphism class of {0}, that is, the f.g. free modules. Actually, \underrightarrow{lim} P_{n}(R) is a group: If ⟨P ⟩ ∈ P_{n}(R) is given then P ⊕ Q ≅ R^{r} for suitable Q and r, and hence ⟨P ⟩ ⊕ ⟨Q⟩ = ⟨R^{r}⟩ ∼ ⟨0⟩. In fact, letting K̃_{0}(R) denote the kernel of

rank : K_{0}(R) → H_{0}(R) = {continuous maps Spec R → ℤ} ,

as usual (cf. [Ba], p. 459), the map sending ⟨P ⟩ ∈ P_{n}(R) to [P] − [R^{n}] ∈ K̃_{0}(R) passes down to a homomorphism of groups

\underrightarrow{lim} P_{n}(R) → K̃_{0}(R)

which is easily seen to be an isomorphism (cf. [W], Chap. II, Lemma 2.3.1). The foregoing is valid for any commutative ring R, and so also applies to \underrightarrow{lim} P_{n}(S).

Under the identification \underrightarrow{lim} P_{n} ≅ K̃_{0}, the top map \underrightarrow{lim} P_{n}(f) of the above triangle becomes

K̃_{0}(f) = Ind_{R}^{S} |_{K̃_{0}(R)} : K̃_{0}(R) → K̃_{0}(S) .

Things are similar for \underrightarrow{lim} P_{S,n}(T): Isomorphism classes ⟨X⟩ ∈ P_{S,n}(T) and ⟨Y ⟩ ∈ P_{S,m}(T) become identified precisely if X ⊕S_{T}^{t−n} ≅ Y ⊕S_{T}^{t−m} for some t ≥ max(m, n). However, since the submonoid of (⨄_{n≥0} P_{S,n}(T), ⊕) that is generated by ⟨S_{T} ⟩ need no longer be cofinal, \underrightarrow{lim} P_{S,n}(T) is merely a commutative monoid. The maps φ and r of the above triangle are monoid maps and φ is mono, as was pointed out earlier.

Thus, summarizing, we have the following stabilized induction triangle of commutative monoids and groups

[figure: K̃_{0}(R); φ=( .)⊗_{R}S_{T}; K̃_{0}(f); K̃_{0}(S); \underrightarrow{lim} P_{S,n}(T); r]

Lemma. Ker r ≅ H^{1}(G, GL(S)) as commutative monoids.

Proof. Since direct limits commute with kernels ([Bou2], Cor. (ii) on p. E III.65),

Ker r = \underrightarrow{lim} (Ker Res_{S,n}^{T}) .

Next, we infer from Lemma 2.5 (and its proof) that the stabilization map Ker Res_{S,n}^{T} → Ker Res_{S,n}^{T} (m ≥ n) translates into the map H^{1}(G, GL_{n}(S)) → H^{1}(G, GL_{m}(S)) that is induced by GL_{n}(S) → GL_{m}(S), a ↦ ( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ). Thus, Lemma 1.3 implies that Ker r ≅ H^{1}(G, GL(S)), at least as pointed sets. The fact that this isomorphism does respect the additive structures is a consequence of the obvious isomorphism (using the notations of §§1.4,2.5)

(S^{m})_{d} ⊕ (S^{n})_{e} ≅ (S^{m+n})_{d⊕_{m}e}

for d ∈ Z^{1}(G, GL_{m}(S)) and e ∈ Z^{1}(G, GL_{n}(S)).

2.8. Proof of the main result. For each subgroup H ≤ G, let

ρ_{H} : H^{1}(G, GL(S)) → H^{1}(H, GL(S_{H})) = Hom(H, GL(S_{H}))/ GL(S_{H})

be given by restriction from G to H and reduction modulo J(H); so ρ_{H} = \underrightarrow{lim} ρ_{H,n} (cf. §2.6). Similarly, for 𝔐 ∈ Max S, put

ρ_{𝔐} = \underrightarrow{lim} ρ_{𝔐,n} : H^{1}(G, GL(S)) → H^{1}(G^{T} (𝔐), GL(S/𝔐)) .

By 1.5, these maps are monoid maps, and hence their kernels are submonoids of H^{1}(G, GL(S)). It is now a simple matter to prove the Theorem stated in the Introduction. Recall that C denotes the set of cyclic subgroups of G.

Theorem. Ker K_{0}(f) ≅ U(H^{1}(G, GL(S))), an isomorphism of groups, and

U(H^{1}(G, GL(S))) = ⋂_{C∈C} Ker ρ_{C} = ⋂_{H≤G} Ker ρ_{H} = ⋂_{𝔐∈Max S} Ker ρ_{𝔐} .

If S is Noetherian of Krull dimension d and n > d then Ker K_{0}(f) ≅ Ker P_{n}(f).

Proof. Recall that K_{0}( .) decomposes naturally as K_{0}( .) = H_{0}( .) ⊕ K̃_{0}( .), with K̃_{0}( .) ≅ \underrightarrow{lim} P_{n}( .). If the ring in question is Noetherian of Krull dimension d, then \underrightarrow{lim} P_{n}( .) ≅ P_{m}( .) holds for any m > d ([Ba2]).

Applying this to the inclusion f : R ↪ S, we obtain that K_{0}(f) = H_{0}(f) ⊕ K̃_{0}(f). Here, H_{0}(f) is injective (cf. [Ba], Lemma (3.1) on p. 459), and hence

Ker K_{0}(f) = Ker K̃_{0}(f) ≅ \underrightarrow{lim} Ker P_{n}(f)

(again using the fact that direct limits commute with kernels). Moreover, if S is Noetherian of Krull dimension d (and hence so is R, by virtue of hypothesis (*) ), then Ker K_{0}(f) ≅ Ker P_{n}(f) for n > d which establishes the last assertion of the theorem.

Writing the formula for Ker P_{n}(f) in Proposition 2.6 as

Ker P_{n}(f) ≅ Ker ρ_{n} ,

where ρ_{n} = {ρ_{C,n}} : H^{1}(G, GL_{n}(S)) → ∏_{C∈C} H^{1}(C, GL_{n}(S_{C})), the stabilization map Ker P_{n}(f) → Ker P_{m}(f) (m ≥ n) becomes the map on H^{1} that is induced by the maps GL_{n} → GL_{m}, a ↦ ( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ), for S and S_{C}. Thus, using Lemma 1.3 and the fact that direct limits commute with kernels and with finite direct products (cf. [Bou2], loc. cit. and Prop. 10/Cor. on p. E III.67/8), we deduce that

Ker K_{0}(f) ≅ Ker ρ ,

where ρ = {ρ_{C}} : H^{1}(G, GL(S)) → ∏_{C∈C} H^{1}(C, GL(S_{C})). The isomorphism is additive, by Lemma 2.7. Since Ker K_{0}(f) is a group, its image in H^{1}(G, GL(S)) must be contained in the unit group U(H^{1}(G, GL(S))). On the other hand, we infer from Lemma 1.6 that U(H^{1}(G, GL(S))) is contained in ⋂_{H≤G} Ker ρ_{H}, which in turn is clearly contained in Ker ρ = ⋂_{C∈C} Ker ρ_{C}. Hence equality must hold throughout, that is,

⋂_{C∈C} Ker ρ_{C} = ⋂_{H≤G} Ker ρ_{H} = U(H^{1}(G, GL(S))) .

Finally, the equality ⋂_{H≤G} Ker ρ_{H} = ⋂_{𝔐∈Max S} ρ_{𝔐} is proved exactly as in the proof of Proposition 2.6, thereby completing the proof of the theorem.

3. Applications and Problems

3.1. Galois actions. The G-action on S is Galois, in the sense of Auslander and Goldman [AG], if and only if G^{T} (𝔐) is trivial for all 𝔐 ∈ Max S (cf. [CHR], Theorem 1.3 on p. 4). In this case, hypothesis (*) is satisfied ([CHR], Lemma 1.6 on p. 7). Thus, by Proposition 2.6 and Theorem 2.8,

Ker P_{n}(f) ≅ H^{1}(G, GL_{n}(S)) and Ker K_{0}(f) ≅ H^{1}(G, GL(S)) .

In particular, H^{1}(G, GL(S)) is a group with the operation of 1.4. In fact:

Proposition. If the action of G on S is Galois then Ker K_{0}(f) ≅ H^{1}(G, GL(S)) is annihilated by a power of |G|. For S Noetherian of Krull dimension d, |G|^{d} will do.

Proof. By [CHR], Lemma 4.1 on p. 13, S is f.g. projective of constant rank equal to |G| as R-module. Therefore,

[S_{R}] − |G|[R] = [S_{R}] − |G|1_{K_{0}(R)} ∈ K̃_{0}(R)

and, moreover, the restriction map Res_{R}^{S} : K_{0}(S) → K_{0}(R) is defined. The composite Res_{R}^{S} ∘K_{0}(f) : K_{0}(R) → K_{0}(R) is clearly multiplication with [S_{R}]. Hence,

Ker K_{0}(f) ⊆ ann_{K_{0}(R)}([S_{R}]) .

Recall that K̃_{0}(R) is a nil ideal of K_{0}(R), and if S (or, equivalently, R) is Noetherian of Krull dimension d, then K̃_{0}(R)^{d+1} = {0} ([Ba], pp. 477 and 473). Furthermore, Ker K_{0}(f) ⊆ K̃_{0}(R) (cf. the proof of Theorem 2.8). Hence, Ker K_{0}(f)·([S_{R}]−|G|)^{t} = {0} for some t, with t = d a possible choice for S Noetherian of Krull dimension d. Consequently, Ker K_{0}(f) · |G|^{t} = {0}.

The proposition can be viewed as an extension of Hilbert’s “Theorem 90” (cf. [Se], Lemme 1 on p. 129) to commutative rings.

3.2. I don’t know if Proposition 3.1 generalizes to arbitrary G-actions:

Question. Is Ker K_{0}(f) ≅ U(H^{1}(G, GL(S))) always annihilated by |G|^{d} if S is Noetherian of Krull dimension d?

This is indeed so for d = 0, in which case K_{0}(R) = H_{0}(R) and K_{0}(f) = H_{0}(f) is mono, and for d = 1, where K_{0}(R) = H_{0}(R) ⊕ Pic(R) and Ker K_{0}(f) = Ker Pic(f) is isomorphic to a subgroup of H^{1}(G, U(S)) (cf. §3.3 below). By a routine direct limit argument, a positive answer to the above question would imply that the kernel of the induction map K_{0}(f) : K_{0}(R) → K_{0}(S) is always |G|-primary (i.e., every element is annihilated by a power of |G|), for any commutative ring S. Finally, I note that the dual statement for G_{0} is known to hold ([BrL1] or [BrL2]): The cokernel of the restriction map G_{0}(S) → G_{0}(R) is annihilated by |G|^{d+1}. The proof given in [BrL1] results from an analysis of the so-called coniveau filtration of G_{0}. The key to the above problem might very well be the Grothendieck γ-filtration ([SGA6], [FL])

K_{0}(R) = F_{γ}^{0}K_{0} ⊇ K̃_{0}(R) = F_{γ}^{1}K_{0} ⊇ … ⊇ F_{γ}^{d+1}K_{0} = 0 .

The first two slices are F_{γ}^{0}/F_{γ}^{1} = H_{0}(R) and F_{γ}^{1}/F_{γ}^{2} = Pic(R). Not much appears to be known about the higher slices.

3.3. Picard groups. For any commutative ring R, the set P_{1}(R) of isomorphism classes of f.g. projective R-modules of constant rank 1 forms a group under ⊗_{R}, with identity element the distinguished element ⟨R⟩. This group is the Picard group of R, usually denoted Pic(R) (cf. [Ba], p. 131ff).

Specializing Proposition 2.6 to the case n = 1 and letting U( .) = GL_{1}( .) denote groups of units, we obtain the following result (cf. [Kr, DMV, L]):

Proposition. There is an isomorphism of groups

Ker Pic(f) ≅ ⋂_{C∈C} Ker (H^{1}(G, U(S)) → Hom(C, U(S_{C})))
= ⋂_{H≤G} Ker (H^{1}(G, U(S)) → Hom(H, U(S_{H})))

and an exact sequence of commutative monoids

1 → Ker σ ⟶ Ker K_{0}(f) ⟶ Ker Pic(f) → 1 ,

where σ : H^{1}(G, SL(S)) → ∏_{C∈C} H^{1}(C, SL(S_{C})).

Proof. The fact that the isomorphism of Proposition 2.6 is an isomorphism of groups, not just of pointed sets, for n = 1 is a consequence of the identity of T -modules S_{d} ⊗_{S} S_{e} ≅ S_{de} for d, e ∈ Z^{1}(G, U(S)). The exact sequence is a consequence of the exact sequence 1 → SL( .) → GL( .) ⟶^{det} U( .) → 1 which is split by the canonical embedding U( .) = GL_{1}( .) ↪ GL( .). Indeed, this sequence leads to a commutative diagram of pointed sets ([Se], §§5.4, 5.5)

[figure: 1; 1; H^{1}(G, SL(S)); σ; ∏_{C∈C} H^{1}(C, SL(S_{C})); H^{1}(G, GL(S)); π_{1}; ρ; ∏_{C∈C} H^{1}(C, GL(S_{C})); ∏ π_{1,C}; H^{1}(G, U(S)); ρ_{1}; μ_{1}; ∏_{C∈C} H^{1}(C, U(S_{C})); ∏ μ_{1,C}; 1; 1]

with π_{1}∘μ_{1} = Id and π_{1,C}∘μ_{1,C} = Id. Here, ρ and ρ_{1} are the usual restriction-reduction maps, as in the proof of Theorem 2.8. The diagram yields the exact sequence

1 → Ker σ ⟶ Ker ρ ⟶ Ker ρ_{1} → 1

which is in fact a sequence of commutative monoids, by §1.5. Finally, Ker K_{0}(f) ≅ Ker ρ and Ker Pic(f) ≅ Ker ρ_{1}.

3.4. Linear actions. Here, S = S(V) is the symmetric algebra of a finite dimensional k-vector space V and G is a subgroup of GL(V) = Aut_{k}(V). The G-action on V extends uniquely to an action of G on S by k-algebra automorphisms. Hypothesis (*) amounts to the requirement that |G|^{−1} ∈ k, which will be assumed, and linear actions are never Galois. Both assertions follow from the existence of an augmentation ε : S → k which is G-invariant (i.e., ε(s^{g}) = ε(s) holds for all s ∈ S, g ∈ G); it is given by ε(V) = {0}.

3.4.1. The factors S_{H}. We now describe the factors S_{H} = S/J(H) of S that were introduced in §2.6. Fix a subgroup H of G and let V (H) denote the subspace of V that is generated by the elements v − v^{h} (v ∈ V, h ∈ H). Then H ⊆ G^{T} (𝔐) iff V (H) ⊆ 𝔐. Thus, J(H) is the intersection of all 𝔐 ∈ Max S with V (H) ⊆ 𝔐. Now V (H)S is a prime ideal of S; in fact, S/V (H)S ≅ S(V_{H}), where V_{H} = V/V (H) is the vector space of H-coinvariants of V. Moreover, since kH is semisimple, we have V = V^{H} ⊕ V (H). Therefore, J(H) = V (H)S and

S_{H} ≅ S(V_{H}) ≅ S(V^{H}) .

The canonical map S ↠ S_{H} is H-equivariantly split by S_{H} ≅ S(V^{H}) ↪ S.

3.4.2. Picard group ([K]). Here, Pic(S) = 1 and so Ker Pic(f) = Pic(R). Also, U(S) = k^{∗} and all U(S_{H}) = k^{∗}, and so the intersection in Proposition 3.3 reduces to the intersection of the kernels of the restriction maps Hom(G, k^{∗}) → Hom(H, k^{∗}) which is obviously trivial. Thus,

Pic(R) = 1 .

3.4.3. A problem of Kraft. Recall that Pic(R) is an image of K̃_{0}(R) (cf. §3.2). It is an open question, raised by Kraft ([Kr], Problem 5.1), whether in fact K̃_{0}(R) is trivial or, equivalently, Ker K_{0}(f) = {0}. A positive answer to Question 3.2 would easily entail this in characteristics > 0, and for fixed-point-free actions in characteristic 0. In the latter case, Kraft’s problem has already been settled by a different method by Holland [Ho], at least for algebraically closed base fields k. The following Proposition gives a cohomological reformulation of Holland’s result, and of a result of Gubeladze [Gu]. We put S_{+} = V S and, as usual, GL(S, S_{+}) denotes the kernel of the reduction map GL(S) → GL(S/S_{+}) = GL(k), and similarly for GL_{n}(S, S_{+}).

Proposition. Assume that k is algebraically closed of characteristic 0. If G acts fixed-point-freely on V then H^{1}(G, GL(S, S_{+})) is trivial. If, in addition, G is abelian (and hence actually cyclic) then H^{1}(G, GL_{n}(S, S_{+})) is trivial for all n.

Proof. By fixed-point-freeness, V (H) = V holds for all non-identity subgroups H of G, and hence J(H) = S_{+}. Thus, in the notation of Theorem 2.8, Ker ρ_{G} ⊆ Ker ρ_{H} and so U(H^{1}(G, GL(S))) = Ker ρ_{G}. Finally, by [Se], Prop. 38 on p. 49, the split exact sequence of G-groups 1 → GL(S, S_{+}) → GL(S) → GL(S/S_{+}) = GL(k) → 1 gives rise to an exact sequence of pointed sets

1 → H^{1}(G, GL(S, S_{+})) ⟶ H^{1}(G, GL(S)) ⟶^{ρ_{G}} H^{1}(G, GL(S/S_{+})) → 1 .

Inasmuch as Ker ρ_{G} ≅ Ker K_{0}(f) is trivial, by [Ho], triviality of H^{1}(G, GL(S, S_{+})) follows (and conversely).

If G is abelian then R is an affine normal semigroup algebra (cf. [H]), and hence all f.g. projectives over R are free, by Gubeladze’s Theorem [Gu]. In other words, all maps P_{n}(f) have trivial kernel which in turn says that H^{1}(G, GL_{n}(S, S_{+})) is trivial, exactly as above, using Proposition 2.6 instead of Theorem 2.8.

3.5. Multiplicative actions. In this case, S = kA is the group algebra of a f.g. free abelian group A and G is a subgroup of GL(A) = Aut_{ℤ}(A) acting on S by means of the unique extension of the natural GL(A)-action on A. Again, hypothesis (*) is equivalent to |G|^{−1} ∈ k and multiplicative actions are never Galois, because of the G-invariant augmentation ε : S → k given by ε(A) = {1}.

3.5.1. The factors S_{H}. Fix a subgroup H of G, let [A, H] denote the subgroup of A that is generated by the elements a^{−1}a^{h} (a ∈ A, h ∈ H), and let ω([A, H])S denote the ideal of S that is generated by the elements a − 1 with a ∈ [A, H]. So ω([A, H])S is the kernel of the canonical map of group algebras S = kA ↠ kA_{H}, where A_{H} = A/[A, H] denotes the H-coinvariants of A. Clearly, H ⊆ G^{T} (𝔐) iff ω([A, H])S ⊆ 𝔐. We claim that ω([A, H])S is a semiprime ideal of S. Since |H| is nonzero in k, this will follow if we can show that the torsion group of A_{H} is annihilated by a power of |H|. To this end, write ( .)′ = ( .)⊗_{ℤ}ℤ[1/|H|] and view A′ as a module over the group ring ℤ′H. Putting e = |H|^{−1} ∑_{h∈H} h ∈ ℤ′H, an idempotent of ℤ′H, we have A ⊆ A′ = (A′)^{e} ⊕ (A′)^{1−e} = (A′)^{H} ⊕ [A′, H]. Thus A′/[A′, H] is torsion-free, and hence so is A/(A ∩ [A′, H]). Since every element of [A′, H]/[A, H] is annihilated by a power of |H|, our claim follows. We conclude that J(H) = ω([A, H])S and

S_{H} ≅ kA_{H} .

The canonical map S ↠ S_{H} is not split, in general, as S_{H} need not be a domain.

3.5.2. Picard group ([L]). Again, Pic(S) = 1 and so Pic(R) can be determined from Proposition 3.3. Here, U(S) = k^{∗} × A and U(S_{C}) = k^{∗} × U_{1} with A_{C} ⊆ U_{1}, the group of normalized (augmentation 1) units of S_{C} = kA_{C} (a strict inclusion, in general). The maps H^{1}(G, U(S)) = Hom(G, k^{∗}) × H^{1}(G, A) → Hom(C, U(S_{C})) = Hom(C, k^{∗}) × Hom(C, U_{1}) decompose as the direct product of the restriction maps Hom(G, k^{∗}) → Hom(C, k^{∗}) times the maps H^{1}(G, A) → H^{1}(C, A) → H^{1}(C, A_{C}) = Hom(C, A_{C}) ↪ Hom(C, U_{1}). The first factor contributes nothing to the kernel, as for linear actions. Since the map H^{1}(C, A) → H^{1}(C, A_{C}) is mono for cyclic C (e.g., [B], p. 79), the contribution from the second factor is the kernel of the restriction map H^{1}(G, A) → H^{1}(C, A). Therefore,

Pic(R) = ⋂_{C∈C} Ker (Res_{C}^{G} : H^{1}(G, A) → H^{1}(C, A)) .

This group need not be trivial; the results of a systematic computer aided investigation of all cases with rank A ≤ 4 are reported in [L].

3.6. Moding out the radical. Returning to the general situation where S is an arbitrary commutative ring satisfying (*), we briefly consider the reduction maps p : GL(S) → GL(S/ rad S) and p_{n} : GL_{n}(S) → GL_{n}(S/ rad S). Here, rad S denotes the Jacobson radical of S. The following lemma is an application of Lemma 2.5.

Lemma. The maps (p_{n})_{∗}^{1} : H^{1}(G, GL_{n}(S)) → H^{1}(G, GL_{n}(S/ rad S)) have trivial kernel, and so does p_{∗}^{1} : H^{1}(G, GL(S)) → H^{1}(G, GL(S/ rad S)).

Proof. It suffices to consider the maps (p_{n})_{∗}^{1}; the case of p_{∗}^{1} then follows by taking \underrightarrow{lim}.

Using the identification Ker(Res_{S,n}^{T}) ≅ H^{1}(G, GL_{n}(S)) of Lemma 2.5 and writing \overline{S} = S/ rad S and \overline{T} = \overline{S}∗G (cf. 2.1), the map (p_{n})_{∗}^{1} becomes the map Ker(Res_{S,n}^{T}) → Ker(Res\overline{T} ; \overline{S},n) that is given by ⟨P ⟩ ↦ ⟨\overline{P} = P/P rad S⟩. Say ⟨P ⟩ belongs to the kernel of this map, that is, \overline{P} ≅ \overline{S}n ; \overline{T} or, equivalently, P/P rad S ≅ S_{T}^{n}/S_{T}^{n} rad S. Since rad S ⊆ rad T (cf. [P], Theorem 7.2.5), the Nakayama Lemma implies that P ≅ S_{T}^{n} (cf. [Ba], Prop. (2.12) on p. 90). Thus ⟨P ⟩ is the distinguished element of Ker(Res_{S,n}^{T}), as required.

References

[AG] M. Auslander and O. Goldman, The Brauer group of a commutative ring, Trans. Amer. Math. Soc. 97 (1960), 367–409.

[B] A. Babakhanian, Cohomological Methods in Group Theory, Marcel Dekker, New York, 1972.

[Ba] H. Bass, Algebraic K-theory, Benjamin, New York, 1968.

[Ba2] , K-theory and stable algebra, Publ. I.H.E.S. 22 (1964), 5–60.

[BS] A. Borel and J.-P. Serre, Théorèmes de finitude en cohomologie galoisienne, Comm. Math. Helv. 64 (1964), 111–164.

[Bou] N. Bourbaki, Algèbre commutative, chap. 1 – 2, Hermann, Paris, 1961.

[Bou2] , Théorie des Ensembles, Hermann, Paris, 1970.

[Bou3] , Algèbre, chap. 1 – 3, Hermann, Paris, 1970.

[Br] K.S. Brown, Cohomology of Groups, Springer-Verlag, New York, 1982.

[BrL1] K.A. Brown and M. Lorenz, Grothendieck groups and higher class groups of commutative invariants, Proc. of the II International Conference on Algebra in Honor of A.I. Shirshov, Contemporary Mathematics, 184 (1995), 59–74.

[BrL2] , Grothendieck groups of invariant rings and of group rings, J. Algebra 166 (1994), 423–454.

[CHR] S.U. Chase, D.K. Harrison, and A. Rosenberg, Galois Theory and Cohomology of Commutative Rings, Memoirs of the Amer. Math. Soc., No. 52, Amer. Math. Soc., Providence, 1965.

[FL] W. Fulton and S. Lang, Riemann-Roch Algebra, Springer-Verlag, New York, 1985.

[Gu] I. D. Gubeladze, Anderson’s conjecture and the maximal monoid class over which projective modules are free, Math. USSR-Sb. 63 (1989), 165–180.

[H] M. Hochster, Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes, Ann. of Math. 96 (1972), 318–337.

[Ho] M. Holland, K-theory of endomorphism rings and of rings of invariants, J. Algebra 191 (1997), 668–685.

[K] M.-C. Kang, Picard groups of some rings of invariants, J. Algebra 58 (1979), 455–461.

[Kr] H. Kraft, G-vector bundles and the linearization problem, In: Canadian Mathematical Soc. Conference Proceedings, Vol. 10, Amer. Math. Soc., Providence, R.I., 1989, pp. 111–123.

[DMV] H. Kraft, P. Slodowy, and T.A. Springer (eds.), Algebraic Transformation Groups and Invariant Theory, DMV Seminar, Band 13, Birkhäuser Verlag, Basel, 1989.

[L] M. Lorenz, Picard groups of multiplicative invariants, Comment. Math. Helv. 72 (1997), 389–399.

[P] D.S. Passman, The Algebraic Structure of Group Rings, Wiley& Sons, New York, 1977.

[Ro] D.J.S. Robinson, A Course in the Theory of Groups, 2nd ed., Springer-Verlag, New York, 1996.

[Se] J.-P. Serre, Cohomologie Galoisienne (5ème éd.), Lect. Notes in Math. No. 5, Springer-Verlag, Berlin, 1994.

[Se2] , Local Fields, Springer-Verlag, New York, 1979.

[SGA6] A. Grothendieck et al., Théorie des Intersections et Théorème de Riemann-Roch, Lect. Notes in Math. No. 225, Springer-Verlag, Heidelberg, 1977.

[Sl] P. Slodowy, Two notes on a finiteness problem in the representation theory of finite groups, .

[W] C. Weibel, Introduction to Algebraic K-Theory, preliminary version of first three chapters, Rutgers University.

[We] A. Weil, Remarks on the cohomology of groups, Ann. of Math. 80 (1964), 149–157.