pacific journal of mathematics Vol. 181, No. 3, 1997

BLOCKS OF FULLY GRADED RINGS

Everett C. Dade

To the memory of Olga Taussky-Todd

We develop Clifford Theory, through the Fong-Reynolds Theorem, for blocks of a ring ℜ fully graded by a finite group G, with respect to an arbitrary G-invariant subring 𝔖 of ℜ. Our only assumption is that ℜ be a finite sum of indecomposable subrings.

The writing of this paper was precipitated by a curious observation relating the blocks of a G-ring 𝔗, for some finite group G, to the blocks of the fixed subring 𝔗^{G} of G acting on 𝔗. We shall say the an arbitrary ring “has finite block theory” if it is a finite direct sum of indecomposable subrings. The curious observation is that a G-ring 𝔗 has finite block theory whenever its fixed subring 𝔗^{G} has finite block theory. This is easily proved in Theorem 2.2 below. Incidently the converse statement can be false, as Example 2.4 below shows.

A fully G-graded ring ℜ, as defined in [U] or §4 below, is a G-graded ring whose σ-components form a group under multiplication. The centralizer of the identity component ℜ_{1} in any such ℜ is naturally a G-ring ℭ. The fixed subring ℭ^{G} of G in ℭ is exactly the center Z(ℜ) of ℜ (see [U] or §5 below). This allows us to apply the above observation to ℭ, concluding in Theorem 5.6 that ℜ has finite block theory if and only if ℭ does. We can even obtain a one to one correspondence between blocks of ℜ and Gconjugacy classes of blocks of ℭ, whether or not ℜ has finite block theory (see Theorem 5.8 below).

When the fully G-graded ring ℜ has finite block theory, we can reproduce Clifford theory for its blocks, as far as the Fong-Reynolds Theorem (Theorem 8.12 below). Following the observation of Ellers [E] that ordinary Clifford theory works just as well over arbitrary G-invariant subalgebras of group algebras, we develop this theory over an arbitary “G-invariant” subring 𝔖 of ℜ (see §8 below). One intermediate result in an otherwise straightforward development is surprisingly delicate to prove. It is Theorem 6.6 below, which says that an idempotent e of ℭ which is orthogonal to all its G-conjugates e^{σ} ≠ e must lie in the restriction ℜ[H] of ℜ to its stabilizer H in G. Once this is out of the way, the rest of Clifford theory is easy.

Perhaps the most interesting part of this paper is to be found in §10 below, where we speculate about some possible extensions of the theories discussed here. The question of whether our curious observation about G-rings has a counterpart for G-graded rings is particularly intriguing. We wish we could answer it.

This paper, like the entire journal issue to which it belongs, is dedicated to the memory of the late Olga Taussky-Todd. My first hesitant steps in mathematical research, along with those of so many other young mathematicians, were guided and encouraged by her. I was privileged to coauthor a few papers [DT1], [DT2], [DRTW] with her and others, notably a joint effort [DTZ] with the late Hans Zassenhaus. But she also read over and improved many of my other works. More than forty years ago she introduced me to Max Deuring’s Algebren, thus sparking a life-long interest in rings and algebras. The present paper is only the latest consequence of that long ago act.

1. Finite block theory.

When we speak of a ring ℜ, we mean an associative ring with identity element 1 = 1_{ℜ}. Any subring 𝔖 of ℜ must have an identity element 1_{𝔖}, but this identity element need not coincide with 1_{ℜ}. When 1_{𝔖} is equal to 1_{ℜ} we say that 𝔖 is a unitary subring of ℜ. Similary, a homomorphism γ : ℜ → 𝔗 of rings need not send 1_{ℜ} to 1_{𝔗}. If it does, we say that γ is identity-preserving.

Any ℜ-module 𝔐, whether right, left or two-sided, is understood to be unitary, in the sense that multiplication by 1_{ℜ} is the identity map of 𝔐 onto itself. We write Mod(ℜ) for the abelian category of all right ℜ-modules and their ℜ-homomorphisms. If 𝔐, 𝔑 ∈ Mod(ℜ) (i.e., if 𝔐 and 𝔑 are right ℜ-modules), then Hom_{ℜ}(𝔐, 𝔑) denotes the additive group of all ℜhomomorphisms of 𝔐 into 𝔑.

We denote by Z(ℜ) the center of a ring ℜ, and by ZI(ℜ) the set of all idempotents in Z(ℜ). The set ZI(ℜ) is naturally a Boolean algebra, with Boolean operations

e ∧ f = ef = fe, e ∨ f = e + f − ef and e′ = 1 − e (1.1)

for any e, f ∈ ZI(ℜ). The zero element 0 = 0_{ℜ} and identity element 1 = 1_{ℜ} of the ring ℜ are also the zero and identity elements of the Boolean algebra ZI(ℜ).

The fact that ZI(ℜ) is a Boolean algebra gives us two sets of names for most basic properties of idempotents in the ring Z(ℜ). Thus two idempotents e, f in Z(ℜ) are orthogonal, in the sense that ef = fe = 0, if and only if they are disjoint in the Boolean algebra ZI(ℜ), in the sense that e∧f = f ∧e = 0. Furthermore an idempotent e is primitive in Z(ℜ), i.e., is non-zero but not the sum of two non-zero orthogonal idempotents in Z(ℜ), if and only if it is an atom in the Boolean algebra ZI(ℜ), i.e., is non-zero but not the join of two disjoint non-zero elements in ZI(ℜ). Of course distinct primitive idempotents in Z(ℜ) are always orthogonal, just as distinct atoms in ZI(ℜ) are always disjoint.

We define a block B of a ring ℜ to be an ordered pair (ℜ, e) consisting of ℜ and a primitive idempotent e of Z(ℜ). We denote by Blk(ℜ) the (possibly empty) set of all such blocks of ℜ, and by 1_{B} the primitive idempotent of Z(ℜ) lying in a given block B = (ℜ, 1_{B}) of ℜ. Thus the 1_{B}, for B ∈ Blk(ℜ), are both the distinct primitive idempotents in the ring Z(ℜ) and the distinct atoms in the Boolean algebra ZI(ℜ). Note that this definition of blocks ensures that two blocks B and B′ are equal if and only if they are blocks of the same ring ℜ corresponding to the same primitive idempotent 1_{B} = 1_{B′} in Z(ℜ).

Definition 1.2. A ring ℜ has finite block theory if the set Blk(ℜ) is finite and 1_{ℜ} is the finite sum

1_{ℜ} = ∑_{B∈Blk(ℜ)} 1_{B} (1.3)

of pairwise orthogonal idempotents in Z(ℜ).

There a couple of different ways of expressing this property of ℜ.

Proposition 1.4. A ring ℜ has finite block theory if and only if the set ZI(ℜ) of all idempotents in Z(ℜ) is finite. This happens if and only if Blk(ℜ) is finite and ℜ is the finite direct sum

ℜ = ∑_{B∈Blk(ℜ)}^{.} ℜ1_{B} (1.5)

of indecomposable subrings.

Proof. If ℜ has finite block theory, then ZI(ℜ) has a finite number of atoms 1_{B}, for B ∈ Blk(ℜ). Since these atoms are pairwise disjoint elements in the Boolean algebra ZI(ℜ), their sum in ℜ is also their join

⋁_{B∈Blk(ℜ)} 1_{B} = ∑_{B∈Blk(ℜ)} 1_{B} = 1_{ℜ}

in ZI(ℜ). So the Boolean algebra ZI(ℜ) has a finite number n ≥ 0 of distinct atoms, and its identity element is the join of those atoms. This implies that ZI(ℜ) is isomorphic to the Boolean algebra of all subsets of its set of atoms, and hence has finite order 2^{n} (see [B, Chapt. III, Th. 4]).

Conversely, if the Boolean algebra ZI(ℜ) is finite, then it has a finite number n ≥ 0 of distinct atoms 1_{B}, for B ∈ Blk(ℜ), and is isomorphic to the Boolean algebra of all subsets of its set of atoms. Hence its identity element is the finite join of its atoms. As above, this is equivalent to (1.3). Therefore ℜ has finite block theory if and only if ZI(ℜ) is finite.

Since the idempotents 1_{B}, for B ∈ Blk(ℜ), are pairwise orthogonal and central in ℜ, the decomposition (1.3) of 1_{ℜ} is equivalent to the decomposition (1.5) of ℜ. The primitivity in Z(ℜ) of each idempotent 1_{B} in the former decomposition is equivalent to the indecomposability of each subring ℜ1_{B} in the latter decomposition. So the remaining statement of the proposition holds.

Corollary 1.6. If a ring ℜ has finite block theory, then the distinct idempotents in Z(ℜ) are the finite sums

1_{B} = ∑_{B∈B} 1_{B}, (1.7)

where B runs over all subsets of the finite set Blk(ℜ).

Proof. We follow the convention that empty sums or joins are always zero. So the idempotent 1_{B} in (1.7) is zero when the subset B of Blk(ℜ) is empty.

We saw in the above proof that the Boolean algebra ZI(ℜ) is generated by its finite number of atoms 1_{B} for B ∈ Blk(ℜ). It follows that the distinct elements in ZI(ℜ) are the joins

1_{B} = ⋁_{B∈B} 1_{B},

where B runs over all subsets of the finite set Blk(ℜ). But this join 1_{B} is equal to the sum on the right side of (1.7) by (1.1), because the atoms 1_{B}, for B ∈ B, are pairwise disjoint. Hence the corollary holds.

One final remark about rings with finite block theory is

Proposition 1.8. A ring ℜ has finite block theory if and only if its center Z(ℜ) has finite block theory.

Proof. The set ZI(ℜ) of all idempotents in Z(ℜ) is also the set ZI(Z(ℜ)) of all idempotents in Z(Z(ℜ)) = Z(ℜ). So the former set is finite if and only if the latter is. By Proposition 1.4 this implies the present proposition.

2. G-rings.

Let G be any multiplicative group. We denote by 1 = 1_{G} the identity element of G. We use exponential notation for the conjugation action of G on itself. So conjugation by any τ ∈ G sends any element σ ∈ G to σ^{τ} = τ^{−1}στ ∈ G, and any subset H ⊆ G to H^{τ} = τ^{−1}Hτ ⊆ G.

A G-ring 𝔗 is a ring, also called 𝔗, together with an action of the group G as automorphisms of the ring 𝔗. We write the action of G on 𝔗 exponentially, so that any σ ∈ G sends any t ∈ 𝔗 to t^{σ} ∈ 𝔗. We also use exponential notation for the fixed subring

𝔗^{H} = { t ∈ 𝔗 | t^{σ} = t, for all σ ∈ H } (2.1)

in 𝔗 of any subgroup H of G.

A G-subring 𝔖 of a G-ring 𝔗 is any subring 𝔖 of 𝔗 such that 𝔖 is invariant under the action of G on 𝔗. That action then restricts to one of G on 𝔖, making 𝔖 a G-ring in its own right. Evidently the center Z(𝔗) is always a unitary G-subring of 𝔗.

The following curious result will be our major tool for proving that certain rings have finite block theory.

Theorem 2.2. Let G be a finite multiplicative group, and 𝔗 be a G-ring whose fixed subring 𝔗^{G} has finite block theory. Then 𝔗 has finite block theory.

Proof. The fixed subring Z(𝔗)^{G} of the G-subring Z(𝔗) is clearly a unitary subring of Z(𝔗^{G}). Since 𝔗^{G} has finite block theory, its center Z(𝔗^{G}) has only a finite number of distinct idempotents by Proposition 1.4. Hence there are only a finite number n of distinct idempotents in the subring Z(𝔗)^{G} of Z(𝔗^{G}).

As usual, we write |S| for the cardinality of any set S. So |G| is the order of the finite group G. Suppose that 𝔗 does not have finite block theory. Then Proposition 1.4 tells us that the Boolean algebra ZI(𝔗) of all idempotents in Z(𝔗) is infinite. In particular, there is some finite subset E of ZI(𝔗) such that

2^{n|G|} < |E|.

The definition (1.1) of the operations in ZI(𝔗) implies that the action of group G on the ring 𝔗 restricts to one of G as automorphisms of the Boolean algebra ZI(𝔗). The union

E′ = ⋃_{σ∈G} E^{σ}

of the translates of the finite set E by elements of the finite group G is a finite, G-invariant subset of ZI(𝔗) satisfying

2^{n|G|} < |E| ≤ |E′|.

The Boolean subalgebra 𝔅 of ZI(𝔗) generated by E′ is both finite and Ginvariant. It follows (see [B, Chapt. III, Th. 4]) that it is the full Boolean algebra on its atoms, which form a G-invariant subset F of some finite cardinality m = |F|. Hence 𝔅 has order 2^{m}. The inclusion E′ ⊆ 𝔅 and the above inequalities imply that

2^{n|G|} < |E′| ≤ |𝔅| = 2^{m}.

So we have

n|G| < m = |F|.

Let F_{1}, F_{2}, …, F_{h} be the distinct G-orbits in the finite G-set F. Each orbit F_{i} satisfies

0 < |F_{i}| ≤ |G|.

Since F is the disjoint union of the F_{i}, we conclude that

n|G| < |F| = ∑_{i=1}^{h} |F_{i}| ≤ ∑_{i=1}^{h} |G| = h|G|.

Therefore

n < h.

The members of F are atoms in the Boolean subalgebra 𝔅 of ZI(𝔗). So they are pairwise disjoint non-zero elements in ZI(𝔗). This implies that the joins

g_{i} = ⋁_{f∈F_{i}} f = ∑_{f∈F_{i}} f,

for i = 1, 2, …, h, are non-zero, pairwise disjoint elements of ZI(ℜ). Furthermore, each g_{i} is fixed by G, and hence lies in Z(𝔗)^{G}. So g_{1}, g_{2}, … g_{h} are h distinct idempotents in Z(𝔗)^{G}. This is impossible, because h > n and Z(𝔗)^{G} has exactly n distinct idempotents (see the first paragraph of this proof). Thus the theorem must hold.

The following example shows that the equivalent of the above theorem need not be valid for infinite G.

Example 2.3. Let ℤ be the ring of all ordinary integers. We form a ring 𝔗 whose additive group is a free ℤ-module with a basis consisting of the identity element 1 = 1_{𝔗} and an infinite number of other elements e_{i}, one for each i ∈ ℤ. Multiplication in 𝔗 is determined by the rule that

e_{i}e_{j} = \begin{cases}e_{i} & if i = j, \\ 0 & if i ≠ j, \end{cases}

for any i, j ∈ ℤ, and the fact that 1 is the two-sided identity element of 𝔗. It is straightforward to verify that 𝔗 is a commutative ring with an infinite number of distinct blocks (𝔗, e_{i}) for i ∈ ℤ. So 𝔗 does not have finite block theory.

Let G be the infinite cyclic group ⟨σ⟩ on one generator σ. We make G act as automorphisms of the ring 𝔗 so that

(e_{i})^{σ^{j}} = e_{i+j} and 1^{σ^{j}} = 1

for all i, j ∈ ℤ. Then 𝔗 becomes a G-ring whose fixed subring 𝔗^{G} just consists of all multiples n1_{𝔗} of 1_{𝔗} for n ∈ ℤ. So 𝔗^{G} ≃ ℤ has finite block theory with just one block (𝔗^{G}, 1), even though 𝔗 does not have finite block theory.

Our next example shows that the converse to Theorem 2.2 need not hold, even when the group G is finite.

Example 2.4. Let 𝔗 and the e_{i}, for i ∈ ℤ, be as in Example 2.3. We form a two-sided unitary 𝔗-module 𝔐. The additive group of 𝔐 is a free ℤmodule with a basis consisting of one element m_{i,j} for each pair of elements i, j ∈ ℤ such that i < j. The module multiplication in 𝔐 is determined by ℤ-bilinearity, the fact that right or left multiplication by 1_{𝔗} must be the identity map of 𝔐 onto itself, and the rule that

e_{i}m_{i′,j′}e_{j} = \begin{cases}m_{i′,j′} & if i = i′ and j = j′, \\ 0 & otherwise, \end{cases}

for any i, i′, j′, j ∈ ℤ with i′ < j′.

We make the direct sum 𝔗 ⊕ 𝔐 of the additive groups 𝔗 and 𝔐 into a ring with the multiplication given by

(t ⊕ m)(t′ ⊕ m′) = (tt′) ⊕ (tm′ + mt′)

for any t, t′ ∈ 𝔗 and m, m′ ∈ 𝔐. Here, of course, tt′ is computed in the ring 𝔗, and tm′ + mt′ in the two-sided 𝔗-module 𝔐. It is straightforward to verify that

Z(𝔗 ⊕ 𝔐) = ℤ(1_{𝔗} ⊕ 0_{𝔐}) = ℤ1_{𝔗⊕𝔐} ≃ ℤ.

So 𝔗 ⊕ 𝔐 has finite block theory, with (𝔗 ⊕ 𝔐, 1) as its only block.

Let G be the cyclic group ⟨τ⟩ generated by an element τ of order two. We make G act as automorphisms of the ring 𝔗 ⊕ 𝔐 so that

(t ⊕ m)^{τ} = t ⊕ (−m)

for any t ∈ 𝔗 and m ∈ 𝔐. Then the fixed subring (𝔗 ⊕ 𝔐)^{G} in the G-ring 𝔗 ⊕ 𝔐 is 𝔗 ⊕ 0, which is isomorphic to 𝔗 as a ring. We know from the discussion in Example 2.3 that the ring 𝔗 has an infinite number of blocks. Hence the isomorphic ring (𝔗 ⊕ 𝔐)^{G} does not have finite block theory, even though 𝔗 ⊕ 𝔐 has finite block theory.

3. G-graded rings.

Let G be any multiplicative group. A G-graded ring ℜ is a ring, also denoted by ℜ, together with a decomposition

(3.1 a)  ℜ = ∑_{σ∈G}^{.} ℜ_{σ}

of ℜ as a direct sum of additive subgroups ℜ_{σ}, whose products in the ring ℜ satisfy

(3.1 b)  ℜ_{σ}ℜ_{τ} ⊆ ℜ_{στ}

for all σ, τ ∈ G. The decomposition (3.1a) is the G-grading of ℜ, and the additive subgroup ℜ_{σ} is the σ-component of ℜ for any σ ∈ G. We always have

1_{ℜ} ∈ ℜ_{1}. (3.2)

So the identity component ℜ_{1} is a unitary subring of ℜ, and each σ-compo-nent ℜ_{σ} is a two-sided ℜ_{1}-submodule of ℜ.

For the rest of this section we fix a multiplicative group G and a G-graded ring ℜ. Given a subset H ⊆ G, we denote by ℜ[H] the additive subgroup

(3.3 a)  ℜ[H] = ∑_{σ∈H}^{.} ℜ_{σ}

of ℜ. By convention ℜ[H] is zero when H is empty. If H is a subgroup of G, then ℜ[H] is a unitary subring of ℜ. In that case the H-grading (3.3a) turns ℜ[H] into an H-graded ring with the σ-component

(3.3 b)  ℜ[H]_{σ} = ℜ_{σ}

for any σ ∈ H. We call this H-graded ring ℜ[H] the restriction of ℜ to H.

Let H be a subgroup of G. An H-graded subring 𝔖 of ℜ is any subring 𝔖 of the ring ℜ such that

(3.4 a)  𝔖 = ∑_{σ∈H}^{.}(𝔖 ∩ ℜ_{σ}).

Such an 𝔖 is itself an H-graded ring with the σ-component

(3.4 b)  𝔖_{σ} = 𝔖 ∩ ℜ_{σ}

for any σ ∈ H. Clearly ℜ[H] is the unique largest H-graded subring of ℜ. For our purposes the most important subring of ℜ is the centralizer

ℭ = C_{ℜ}(ℜ_{1}) = { c ∈ ℜ | cr_{1} = r_{1}c for all r_{1} ∈ ℜ_{1} } (3.5)

of the identity component ℜ_{1}. It follows easily from (3.1) that:

(3.6) The centralizer ℭ is a unitary G-graded subring of ℜ.

For any σ ∈ G the σ-component of ℭ is the centralizer ℭ_{σ} = C(ℜ_{1} in ℜ_{σ}) = { c ∈ ℜ_{σ} | cr_{1} = r_{1}c for all r_{1} ∈ ℜ_{1} }

of ℜ_{1} in ℜ_{σ}.

In particular, the identity component of ℭ is the center

ℭ_{1} = C(ℜ_{1} in ℜ_{1}) = Z(ℜ_{1}) (3.7)

of ℜ_{1}. Since ℭ_{1} ⊆ ℜ_{1} centralizes ℭ = C_{ℜ}(ℜ_{1}), and contains the common identity element 1_{ℜ} of ℭ and ℜ_{1}, we have:

(3.8) The identity component ℭ_{1} is a unitary central subring of both ℜ_{1} and ℭ.

If H is any subgroup of G, then H\G (which we pronounce as “H under G”) will denote the set of all left cosets Hτ of H in G. The group G acts on the set H\G, with any σ ∈ G sending any T ∈ H\G to T σ ∈ H\G. An H\G-graded ℜ-module 𝔐 is a right ℜ-module, also called 𝔐, together with a decomposition

(3.9 a)  𝔐 = ∑_{T ∈H\G}^{.} 𝔐_{T}

of 𝔐 as a direct sum of additive subgroups 𝔐_{T}, whose products with the σ-components of ℜ satisfy

(3.9 b)  𝔐_{T} ℜ_{σ} ⊆ 𝔐_{T σ}

for any T ∈ H\G and σ ∈ G. The decomposition (3.9a) is the H\G-grading of 𝔐, and the additive subgroup 𝔐_{T} is the T -component of 𝔐 for any T ∈ H\G.

An H\G-graded ℜ-submodule 𝔑 of an H\G-graded ℜ-module 𝔐 is any ℜ-submodule 𝔑 of the ℜ-module 𝔐 such that

(3.10 a)  𝔑 = ∑_{T ∈H\G}^{.}(𝔐_{T} ∩ 𝔑).

Any such 𝔑 is itself an H\G-graded ℜ-module, with the T -component

(3.10 b)  𝔑_{T} = 𝔐_{T} ∩ 𝔑

for any T ∈ H\G. Furthermore, the factor ℜ-module 𝔐/𝔑 is naturally H\G-graded, with the T -component

(𝔐/𝔑)_{T} = (𝔐_{T} + 𝔑)/𝔑 ≃ 𝔐_{T} /𝔑_{T} (3.11)

for any T ∈ H\G.

We form a category GrMod(H\G, ℜ), whose objects are the H\G-graded ℜ-modules, and whose morphisms ϕ : 𝔐 → 𝔑 are those homomorphisms of the underlying ℜ-modules which preserve H\G-gradings, in the sense that

ϕ(𝔐_{T}) ⊆ 𝔑_{T} (3.12)

for any T ∈ H\G. For fixed objects 𝔐, 𝔑 ∈ GrMod(H\G, ℜ), the morphisms ϕ : 𝔐 → 𝔑 in GrMod(H\G, ℜ) form a group GrHom_{H\G,ℜ}(𝔐, 𝔑) under addition of module homomorphisms. It follows that GrMod(H\G, ℜ) is an additive category, with composition of maps as multiplication.

We indicate that ϕ : 𝔐 → 𝔑 is a morphism in GrMod(H\G, ℜ) by saying that it is a homomorphism of H\G-graded ℜ-modules. In that case the kernel

ker(ϕ) = { m ∈ 𝔐 | ϕ(m) = 0 }

of ϕ is an H\G-graded ℜ-submodule of 𝔐, and the image ϕ(𝔐) of ϕ is an H\G-graded ℜ-submodule of 𝔑. It follows that the injection of ker(ϕ) into 𝔐 and the projection of 𝔑 onto 𝔑/ϕ(𝔐) are a kernel and cokernel, respectively, for the morphism ϕ in the category GrMod(H\G, ℜ). With this observation it is easy to see that GrMod(H\G, ℜ) is an abelian category, in which monomorphisms, epimorphisms and isomorphisms are just those morphisms which are monomorphisms, epimorphisms or isomorphisms, respectively, of the underlying ℜ-modules.

The inclusions (3.9b) imply that the H-component of any H\G-graded ℜ-module 𝔐 is an ℜ[H]-submodule 𝔐_{H} of 𝔐. Furthermore, any homomorphism ϕ : 𝔐 → 𝔑 of H\G-graded ℜ-modules restricts to a homomorphism ϕ_{H} : 𝔐_{H} → 𝔑_{H} of right ℜ[H]-modules. It follows that restriction to Hcomponents is an additive functor (·)_{H} from GrMod(H\G, ℜ) to Mod(ℜ[H]).

Since G is the disjoint union of the cosets T ∈ H\G, the G-graded ring ℜ is the direct sum

(3.13 a)  ℜ = ∑_{T ∈H\G}^{.} ℜ[T]

of additive subgroups. It follows easily from (3.1b) and (3.3a) that

(3.13 b)  ℜ[T]ℜ_{σ} ⊆ ℜ[T σ]

for any T ∈ H\G and σ ∈ G. So the H\G-grading (3.13a) turns the regular right ℜ-module ℜ into an H\G-graded ℜ-module.

Any right ℜ[H]-module 𝔎 induces a right ℜ-module 𝔎 ⊗ ℜ = 𝔎 ⊗_{ℜ[H]} ℜ. Each coset T = Hτ ∈ H\G is closed under left multiplication by elements of H. Hence the corresponding summand ℜ[T] in (3.13a) is a left ℜ[H]- submodule of ℜ. It follows that 𝔎 ⊗ ℜ is, after natural identifications, the direct sum

(3.14 a)  𝔎 ⊗ ℜ = ∑_{T ∈H\G}^{.} 𝔎 ⊗ ℜ[T]

of additive subgroups. In view of (3.13b) we have

(3.14 b)  (𝔎 ⊗ ℜ[T])ℜ_{σ} ⊆ 𝔎 ⊗ ℜ[T σ]

for any T ∈ H\G and σ ∈ G. Hence the H\G-grading (3.14a) turns 𝔎 ⊗ ℜ into an H\G-graded ℜ-module. Any homomorphism ψ : 𝔎 → 𝔏 of right ℜ[H]-modules induces a homomorphism ψ ⊗ ℜ : 𝔎 ⊗ ℜ → 𝔏 ⊗ ℜ of H\Ggraded ℜ-modules, sending k ⊗ r to ψ(k) ⊗ r for any k ∈ 𝔎 and r ∈ ℜ. In this way induction from ℜ[H] to ℜ becomes an additive functor · ⊗ ℜ from Mod(ℜ[H]) to GrMod(H\G, ℜ).

The composite functor (· ⊗ ℜ)_{H} : Mod(ℜ[H]) → Mod(ℜ[H]) sends any right ℜ[H]-module 𝔎 to the H-component 𝔎 ⊗ ℜ[H] of the induced H\Ggraded ℜ-module 𝔎 ⊗ ℜ. Since these tensor products are over ℜ[H], the natural map

(3.15 a)  γ_{𝔎} : k ↦ k ⊗ 1_{ℜ}

is an isomorphism of 𝔎 onto 𝔎 ⊗ ℜ[H] as right ℜ[H]-modules. It is straightforward to verify that:

The above isomorphisms γ_{𝔎} form a natural equivalence γ (3.15b)

between the identity functor on the category Mod(ℜ[H])

and the composite functor (· ⊗ ℜ)_{H} : Mod(ℜ[H]) → Mod(ℜ[H]).

4. Fully graded rings.

By a fully G-graded ring ℜ we mean a G-graded ring ℜ in which the inclusion (3.1b) is equality

ℜ_{σ}ℜ_{τ} = ℜ_{στ} (4.1)

for any σ, τ ∈ G. Such rings are also called strongly G-graded (see [D]). Notice that the product ℜ_{σ}ℜ_{τ} in (4.1), like all our products of additive subgroups of ℜ, is the additive subgroup of ℜ generated by the products r_{σ}r′_{τ} ∈ ℜ of elements r_{σ} ∈ ℜ_{σ} and r′_{τ} ∈ ℜ_{τ}, and not just the set of those products.

For the rest of this paper we assume that:

Hypothesis 4.2. G is an arbitrary multiplicative group, and ℜ is a fully G-graded ring.

Our main interest is in the case of finite G. But enough results hold for arbitrary G to make it undesirable to restrict ourselves just to finite groups.

The following consequence of (4.1) has many uses.

Lemma 4.3. Given any τ ∈ G, there exist a finite number n ≥ 1 of elements s_{1}, s_{2}, …, s_{n} ∈ ℜ_{τ^{−1}} and t_{1}, t_{2}, …, t_{n} ∈ ℜ_{τ} such that

1_{ℜ} = s_{1}t_{1} + s_{2}t_{2} + ⋯ + s_{n}t_{n}.

Proof. Equation (4.1) for σ = τ^{−1} is

ℜ_{1} = ℜ_{τ^{−1}}ℜ_{τ}.

Since 1_{ℜ} lies in ℜ_{1} by (3.2), this implies the lemma.

We first apply the above lemma in the study of the structure of ℜ as a module over its restriction to a subgroup H of G. Specifically, we use it to prove:

Lemma 4.4. Suppose that elements s_{1}, s_{2}, …, s_{n} and t_{1}, t_{2}, …, t_{n} satisfy the conditions in Lemma 4.3 for some τ ∈ G. For any subgroup H of G, the left ℜ[H]-submodule ℜ[Hτ] of ℜ is projective and generated by t_{1}, t_{2}, …, t_{n}. Similarly, the right ℜ[H]-submodule ℜ[τ^{−1}H] of ℜ is projective and generated by s_{1}, s_{2}, …, s_{n}.

Proof. By left-right symmetry we only need prove the lemma for ℜ[Hτ]. The fact that the subset Hτ of G is closed under left multiplication by elements of H implies that ℜ[Hτ] is a left ℜ[H]-submodule of ℜ. Since s_{i} ∈ ℜ_{τ^{−1}} for each i = 1, 2, …, n, we have

rs_{i} ∈ ℜ[Hτ]ℜ_{τ^{−1}} ⊆ ℜ[Hττ^{−1}] = ℜ[H]

for any r ∈ ℜ[Hτ]. It follows that the map

Φ : r ↦ (rs_{1}, rs_{2}, …, rs_{n})

is a homomorphism of the left ℜ[H]-module ℜ[Hτ] into the external direct sum

ℜ[H]^{(n)} = ℜ[H] ⊕ ℜ[H] ⊕ ⋯ ⊕ ℜ[H]

of n copies of the regular left ℜ[H]-module ℜ[H]. Similarly, the product ut_{i} lies in ℜ[H]ℜ_{τ} ⊆ ℜ[Hτ] for any u ∈ ℜ[H] and i = 1, 2, …, n. Hence the map

Ψ : (u_{1}, u_{2}, …, u_{n}) ↦ u_{1}t_{1} + u_{2}t_{2} + ⋯ + u_{n}t_{n}

is a homomorphism of ℜ[H]^{(n)} into ℜ[Hτ] as left ℜ[H]-modules. The composite homomorphism ΨΦ : ℜ[Hτ] → ℜ[Hτ] is the identity map, because

Ψ(Φ(r)) = rs_{1}t_{1} + rs_{2}t_{2} + ⋯ + rs_{n}t_{n} = r(s_{1}t_{1} + s_{2}t_{2} + ⋯ + s_{n}t_{n}) = r1 = r

for any r ∈ ℜ[Hτ]. We conclude that ℜ[Hτ] is generated by t_{1}, t_{2}, …, t_{n} as a left ℜ[H]-module, and is isomorphic to a direct summand of the free left ℜ[H]-module ℜ[H]^{(n)}. So it is a projective left ℜ[H]-module, and the lemma is proved.

What we really need is the following consequence of the preceding lemma.

Proposition 4.5. If Hypothesis 4.2 holds, then ℜ is projective as both a left and a right module over its restriction ℜ[H] to any subgroup H of G.

Proof. By left-right symmetry we only need prove the proposition for the left ℜ[H]-module ℜ. Because G is the disjoint union of the left cosets T = Hτ ∈ H\G, the left ℜ[H]-module ℜ is the direct sum

ℜ = ∑_{T ∈H\G}^{.} ℜ[T]

of left ℜ[H]-submodules. Lemma 4.4 tells us that each summand ℜ[T] = ℜ[Hτ] in this decomposition is a projective left ℜ[H]-module. Hence so is their direct sum ℜ. Thus the proposition holds.

There is an equality similar to (4.1) for components of H\G-graded ℜmodules.

Lemma 4.6. If 𝔐 is an H\G-graded ℜ-module, then

𝔐_{T} ℜ_{σ} = 𝔐_{T σ}

for any T ∈ H\G and σ ∈ G.

Proof. We know from (3.9b) that

𝔐_{T} ℜ_{σ} ⊆ 𝔐_{T σ}.

Since

1_{ℜ} ∈ ℜ_{1} = ℜ_{σ^{−1}}ℜ_{σ}

by (3.2) and (4.1), we have

𝔐_{T σ} = (𝔐_{T σ})1_{ℜ} ⊆ 𝔐_{T σ}ℜ_{σ^{−1}}ℜ_{σ} ⊆ 𝔐_{T σσ^{−1}}ℜ_{σ} = 𝔐_{T} ℜ_{σ}.

Hence the lemma holds.

We apply the above lemma to prove:

Proposition 4.7. If Hypothesis 4.2 holds and H is any subgroup of G, then a homomorphism ϕ : 𝔐 → 𝔑 of H\G-graded ℜ-modules is a monomorphism, epimorphism or isomorphism if and only if its restriction ϕ_{H} : 𝔐_{H} → 𝔑_{H} to H-components is a monomorphism, epimorphism or isomorphism, respectively.

Proof. The kernel of ϕ is an H\G-graded ℜ-submodule 𝔎 of 𝔐. Its Hcomponent 𝔎_{H} is the kernel of the restriction ϕ_{H} of ϕ to 𝔐_{H}. Clearly 𝔎_{H} = 0 when 𝔎 = 0. Conversely, if 𝔎_{H} = 0, then Lemma 4.6 for the H\G-graded ℜ-module 𝔎 implies that

𝔎_{T} = 𝔎_{Hσ} = 𝔎_{H}ℜ_{σ} = 0

for any coset T = Hσ ∈ H\G. Hence

𝔎 = ∑_{T ∈H\G}^{.} 𝔎_{T} = 0.

Thus we have shown that 𝔎 = 0 if and only if 𝔎_{H} = 0, i.e., that ϕ is a monomorphism if and only if ϕ_{H} is a monomorphism.

The image of ϕ is an H\G-graded ℜ-submodule 𝔏 of 𝔑. Its H-component 𝔏_{H} is the image of ϕ_{H}. The H\G-graded factor ℜ-module 𝔑/𝔏 has an H-component (𝔑/𝔏)_{H} isomorphic to 𝔑_{H}/𝔏_{H}. So 𝔑_{H}/𝔏_{H} = 0 whenever 𝔑/𝔏 = 0. Conversely, if 𝔑_{H}/𝔏_{H} = 0, then (𝔑/𝔏)_{H} = 0. As above, this and Lemma 4.6 for the H\G-graded ℜ-module 𝔑/𝔏 imply that 𝔑/𝔏 = 0. Thus we have shown that 𝔑/𝔏 = 0 if and only if 𝔑_{H}/𝔏_{H} = 0, i.e., that ϕ is an epimorphism if and only if ϕ_{H} is an epimorphism.

The homomorphism ϕ is an isomorphism if and only if it is both a monomorphism and an epimorphism. We have seen above that this happens if and only if ϕ_{H} is both a monomorphism and an epimorphism, i.e., if and only if ϕ_{H} is an isomorphism. So the proposition is proved.

Any H\G-graded ℜ-module 𝔐 restricts to a right ℜ[H]-module 𝔐_{H}, from which we can form the induced H\G-graded ℜ-module (𝔐_{H}) ⊗ ℜ as in (3.14). There is a natural homomorphism δ_{𝔐} of the ℜ-module (𝔐_{H}) ⊗ ℜ into 𝔐, sending m ⊗ r to

δ_{𝔐}(m ⊗ r) = mr ∈ 𝔐 (4.8)

for any m ∈ 𝔐_{H} and r ∈ ℜ. We use these homomorphisms in the following explicit version of [NRvO, 3.12].

Proposition 4.9. If Hypothesis 4.2 holds and H is any subgroup of G, then the map δ_{𝔐} : (𝔐_{H}) ⊗ ℜ → 𝔐 is an isomorphism of H\G-graded ℜmodules for any 𝔐 ∈ GrMod(H\G, ℜ) . These isomorphisms form a natural equivalence δ between the composite functor

(·)_{H} ⊗ ℜ : GrMod(H\G, ℜ) → GrMod(H\G, ℜ)

and the identity functor on GrMod(H\G, ℜ). Hence the two functors

(·)_{H} : GrMod(H\G, ℜ) → Mod(ℜ[H])

and

· ⊗ ℜ : Mod(ℜ[H]) → GrMod(H\G, ℜ)

form an equivalence between the abelian categories GrMod(H\G, ℜ) and Mod(ℜ[H]).

Proof. The ℜ-homomorphism δ_{𝔐} sends the T -component (𝔐_{H}) ⊗ ℜ[T] of (𝔐_{H}) ⊗ ℜ onto the additive subgroup 𝔐_{H}ℜ[T] of 𝔐 for any T ∈ H\G. It follows from (3.9b) that

𝔐_{H}ℜ[T] = ∑_{σ∈T} 𝔐_{H}ℜ_{σ} ⊆ ∑_{σ∈T} 𝔐_{Hσ}.

Since Hσ = T for any σ in the left coset T of H, this last sum is just 𝔐_{T}. Hence δ_{𝔐} sends (𝔐_{H}) ⊗ ℜ[T] into 𝔐_{T}. So δ_{𝔐} : (𝔐_{H}) ⊗ ℜ → 𝔐 is a homomorphism of H\G-graded ℜ-modules. The restriction of δ_{𝔐} to Hcomponents is the natural isomorphism of (𝔐_{H})⊗ℜ[H] = (𝔐_{H})⊗_{ℜ[H]} ℜ[H] onto the right ℜ[H]-module 𝔐_{H}. We conclude from this and Proposition 4.7 that δ_{𝔐} is an isomorphism. Thus the first statement of the proposition holds.

It is straightforward to verify that the δ_{𝔐} form a natural transformation δ of the composite functor (·)_{H} ⊗ ℜ into the identity functor on GrMod( H\G, ℜ). Because each δ_{𝔐} is an isomorphism, this natural transformation is a natural equivalence. We know from (3.15b) that the other composite functor (· ⊗ ℜ)_{H} is naturally equivalent to the identity functor on Mod(ℜ[H]). Therefore the remaining statements of the proposition hold.

5. Centralizers of identity components.

As in (3.5) we define ℭ to be the unitary G-graded subring C_{ℜ}(ℜ_{1}) of our fully G-graded ring ℜ. There is a natural action of the group G as automorphisms of the ring ℭ (see [U]). Since we’re going to need various properties of this action, we discuss its definition in detail.

Lemma 5.1. If c ∈ ℭ and τ ∈ G, then there is a unique element c^{τ} ∈ ℜ such that

(5.2 a)  r_{τ}c^{τ} = cr_{τ}

for all r_{τ} ∈ ℜ_{τ}. This element c^{τ} is given by

(5.2 b)  c^{τ} = ∑_{i=1}^{n} s_{i}ct_{i} ∈ ℜ

whenever s_{1}, s_{2}, …, s_{n} and t_{1}, t_{2}, …, t_{n} satisfy the conditions in Lemma 4.3.

Proof. Suppose that c^{τ} ∈ ℜ satisfies (5.2a) for all r_{τ} ∈ ℜ_{τ}. If s_{1}, s_{2}, …, s_{n} and t_{1}, t_{2}, …, t_{n} satisfy the conditions in Lemma 4.3, then each t_{i} lies in ℜ_{τ}. So ct_{i} = t_{i}c^{τ} by (5.2a). This implies that

∑_{i=1}^{n} s_{i}ct_{i} = ∑_{i=1}^{n} s_{i}t_{i}c^{τ} = 1c^{τ} = c^{τ}.

Hence (5.2b) holds if (5.2a) does. This proves that there can be at most one c^{τ} ∈ ℜ satisfying (5.2a).

There are always some elements s_{1}, s_{2}, …, s_{n} and t_{1}, t_{2}, …, t_{n} satisfying the conditions in Lemma 4.3. We can use these elements to define c^{τ} ∈ ℜ by (5.2b). Since each s_{i} lies in ℜ_{τ^{−1}}, its product r_{τ}s_{i} with any r_{τ} ∈ ℜ_{τ} lies in ℜ_{τ}ℜ_{τ^{−1}} = ℜ_{1}. Hence r_{τ}s_{i} commutes with c ∈ ℭ = C_{ℜ}(ℜ_{1}). So

r_{τ}c^{τ} = ∑_{i=1}^{n} r_{τ}s_{i}ct_{i} = ∑_{i=1}^{n} cr_{τ}s_{i}t_{i} = cr_{τ}1 = cr_{τ}.

Therefore c^{τ} satisfies (5.2a), and the lemma is proved.

We call the element c^{τ} defined in the above lemma the conjugate of c ∈ ℭ by τ ∈ G. Of course conjugation is an action of G on ℭ.

Proposition 5.3. If Hypothesis 4.2 holds, then c^{τ} lies in ℭ for any c ∈ ℭ = C_{ℜ}(ℜ_{1}) and any τ ∈ G. The map c, τ ↦ c^{τ} is then an action of the group G as automorphisms of the ring ℭ. So it makes ℭ a G-ring.

Proof. Given τ ∈ G we may fix elements s_{1}, s_{2}, …, s_{n} and t_{1}, t_{2}, …, t_{n} satisfying the conditions in Lemma 4.3. If r_{1} ∈ ℜ_{1}, then t_{i}r_{1} lies in ℜ_{τ}ℜ_{1} = ℜ_{τ} for all i = 1, 2, …, n. This and (5.2a) imply that

ct_{i}r_{1} = t_{i}r_{1}c^{τ}

for any c ∈ ℭ and i = 1, …, n. In view of (5.2b) it follows that

c^{τ}r_{1} = ∑_{i=1}^{n} s_{i}ct_{i}r_{1} = ∑_{i=1}^{n} s_{i}t_{i}r_{1}c^{τ} = 1r_{1}c^{τ} = r_{1}c^{τ}.

So c^{τ} ∈ ℜ commutes with any r_{1} ∈ ℜ_{1}, and hence lies in ℭ = C_{ℜ}(ℜ_{1}).

Suppose that σ is also an element of G. Then (5.2a) says that r′_{σ}c^{σ} = cr′_{σ} for any c ∈ ℭ and r′_{σ} ∈ ℜ_{σ}. Since c^{σ} lies in ℭ by the above arguments, the element (c^{σ})^{τ} is defined and satisfies r_{τ}(c^{σ})^{τ} = c^{σ}r_{τ} for any r_{τ} ∈ ℜ_{τ}. It follows that

r′_{σ}r_{τ}(c^{σ})^{τ} = r′_{σ}c^{σ}r_{τ} = cr′_{σ}r_{τ}

for any such r′_{σ} and r_{τ}. By (4.1) the products r′_{σ}r_{τ} generate ℜ_{στ} as an additive group. So (c^{σ})^{τ} satisfies r′′_{στ}(c^{σ})^{τ} = cr′′_{στ} for all r′′_{στ} ∈ ℜ_{στ}. Hence (c^{σ})^{τ} = c^{στ} by Lemma 5.1. Any element c ∈ ℭ = C_{ℜ}(ℜ_{1}) satisfies r_{1}c = cr_{1} for all r_{1} ∈ ℜ_{1}. So c^{1} = c by Lemma 5.1. Therefore the map c, τ ↦ c^{τ} is an action of the group G on the set ℭ.

If c, d ∈ ℭ and τ ∈ G, then (5.2a) tells us that

r_{τ}(c^{τ} + d^{τ}) = r_{τ}c^{τ} + r_{τ}d^{τ} = cr_{τ} + dr_{τ} = (c + d)r_{τ}

and

r_{τ}(c^{τ}d^{τ}) = cr_{τ}d^{τ} = cdr_{τ}

for any r_{τ} ∈ ℜ_{τ}. Applying Lemma 5.1, we conclude that (c + d)^{τ} = c^{τ} + d^{τ} and (cd)^{τ} = c^{τ}d^{τ}. Therefore the bijection c ↦ c^{τ} is an automorphism of the ring ℭ, and the proposition is proved.

Conjugation restricts properly to subgroups of G.

Proposition 5.4. If Hypothesis 4.2 holds, then the restriction of ℜ to any subgroup H of G is a fully H-graded ring ℜ[H]. The centralizer C_{ℜ[H]}(ℜ[H]_{1}) of the identity component in ℜ[H] is equal as an H-graded ring to the restriction ℭ[H] of the centralizer ℭ = C_{ℜ}(ℜ_{1}) to H. The conjugation action of H on ℭ[H], as defined by Lemma 5.1 for the fully H-graded ring ℜ[H], is the restriction of the conjugation action of G on ℭ, as defined by Lemma 5.1 for the fully G-graded ring ℜ.

Proof. It follows from (3.3b) and (4.1) that

ℜ[H]_{σ}ℜ[H]_{τ} = ℜ_{σ}ℜ_{τ} = ℜ_{στ} = ℜ[H]_{στ}

for any σ, τ ∈ H. Hence ℜ[H] is fully H-graded.

By (3.6) the σ-component of C_{ℜ[H]}(ℜ[H]_{1}) is

C(ℜ[H]_{1} in ℜ[H]_{σ}) = C(ℜ_{1} in ℜ_{σ}) = ℭ_{σ}

for any σ ∈ H. It follows that

C_{ℜ[H]}(ℜ[H]_{1}) = ∑_{σ∈H}^{.} C(ℜ[H]_{1} in ℜ[H]_{σ}) = ∑_{σ∈H}^{.} ℭ_{σ} = ℭ[H]

as H-graded rings.

If c ∈ C_{ℜ[H]}(ℜ[H]_{1}) and τ ∈ H, then the τ-conjugate of c, as defined by Lemma 5.1 for the fully H-graded ring ℜ[H], is the unique element c^{τ} ∈ ℜ[H] such that r_{τ}c^{τ} = cr_{τ} for all r_{τ} ∈ ℜ[H]_{τ}. Since ℜ[H]_{τ} is equal to ℜ_{τ}, this c^{τ} is also the unique element in ℜ such that r_{τ}c^{τ} = cr_{τ} for all r_{τ} ∈ ℜ_{τ}. So it is the τ-conjugate of c ∈ ℭ, as defined by Lemma 5.1 for the fully G-graded ring ℜ. Thus the proposition holds.

The fixed subring under conjugation is easy to compute.

Proposition 5.5. If Hypothesis 4.2 holds, then the fixed subring ℭ^{G} of G in ℭ = C_{ℜ}(ℜ_{1}) satisfies

ℭ^{G} = Z(ℜ) = Z(ℭ)^{G} ⊆ Z(ℭ).

Proof. By (5.2a) an element c ∈ ℭ lies in ℭ^{G} if and only if it centralizes ℜ_{τ} for every τ ∈ G. Clearly this happens if and only if c centralizes ℜ = ∑_{τ∈G} ℜ_{τ}. Hence ℭ^{G} is the intersection Z(ℜ) ∩ ℭ. Since the centralizer ℭ of ℜ_{1} must contain Z(ℜ), this implies that ℭ^{G} = Z(ℜ). It follows that ℭ^{G} is contained in Z(ℭ), and so must equal Z(ℭ)^{G}. Thus the proposition holds.

One elementary consequence of the above proposition and Theorem 2.2 is:

Theorem 5.6. If G is a finite group, then a fully G-graded ring ℜ has finite block theory if and only if its subring ℭ = C_{ℜ}(ℜ_{1}) has finite block theory.

Proof. Suppose that ℜ has finite block theory. Then Z(ℜ) has finite block theory by Proposition 1.8. Since ℭ^{G} is equal to Z(ℜ) by Proposition 5.5, it also has finite block theory. Then ℭ has finite block theory by Theorem 2.2.

Conversely, suppose that ℭ has finite block theory. Then Proposition 1.4 tells us that Z(ℭ) has only a finite number of distinct idempotents. We know from Proposition 5.5 that Z(ℜ) is a subring of Z(ℭ). Hence Z(ℜ) has only a finite number of distinct idempotents. So ℜ has finite block theory by Proposition 1.4, and the theorem is proved.

In fact, we can always describe the blocks of ℜ in terms of those of ℭ when G is finite, whether or not ℜ has finite block theory. The conjugation action of the group G on the ring ℭ must permute among themselves the distinct primitive idempotents 1_{C}, for C ∈ Blk(ℭ), of Z(ℭ). So there is an induced conjugation action of G on the set Blk(ℭ), with any τ ∈ G sending any block C ∈ Blk(ℭ) to the unique conjugate block C^{τ} of ℭ such that

1_{C^{τ}} = (1_{C})^{τ}. (5.7)

We write Blk_{G}(ℭ) for the set of all G-orbits C in Blk(ℭ) under this action.

Theorem 5.8. If G is a finite group, then there is a one to one correspondence between all blocks B of any fully G-graded ring ℜ and all G-orbits C ∈ Blk_{G}(ℭ) of blocks of the subring ℭ = C_{ℜ}(ℜ_{1}) of ℜ. Two such B and C correspond if and only if

1_{B} = 1_{C} = ∑_{C∈C} 1_{C} ∈ Z(ℭ)^{G} = Z(ℜ). (5.9)

Proof. Let B be any block of ℜ. Then 1_{B} is a primitive idempotent in Z(ℜ) = ℭ^{G}. It follows that ℭ is the direct sum

ℭ = ℭ1_{B} ∔ ℭ(1 − 1_{B}) (5.10) of G-subrings. The fixed subring of the first summand in this decomposition is the indecomposable subring (ℭ1_{B})^{G} = ℭ^{G}1_{B} = Z(ℜ)1_{B}. Hence (ℭ1_{B})^{G} has finite block theory. Because G is finite, Theorem 2.2 tells us that the G-ring ℭ1_{B} has finite block theory.

The decomposition (5.10) implies that the distinct primitive idempotents in Z(ℭ1_{B}) are the 1_{C}, where C runs over the set C of all blocks C ∈ Blk(ℭ) such that 1_{C} lies in ℭ1_{B}. Since the G-ring ℭ1_{B} has finite block theory, Corollary 1.6 tells us that the distinct idempotents in Z(ℭ1_{B}) are just the finite sums

1_{C′} = ∑_{C′∈C′} 1_{C′}, (5.11)

where C′ runs over all subsets of C. It is clear from (5.7) that such a sum 1_{C′} is fixed by G if and only if the subset C′ is invariant under conjugation by G. Because the ring (ℭ1_{B})^{G} = Z(ℜ)1_{B} is indecomposable, its only non-zero idempotent is its identity element 1_{B}. It follows that C must have only one non-empty G-invariant subset C′, and that 1_{C′} is equal to 1_{B}. This forces C to be a G-orbit in Blk_{G}(ℭ) such that (5.9) holds.

Conversely, suppose that C is any G-orbit in Blk_{G}(ℭ). Then C is a finite set, and the second equation in (5.9) defines an idempotent 1_{C} in Z(ℭ)^{G}. The 1_{C}, for C ∈ C, are the distinct primitive idempotents in the G-subring

Z(ℭ)1_{C} = ∑_{C∈C}^{.} Z(ℭ)1_{C}

of Z(ℭ). It follows that the distinct idempotents in Z(ℭ)1_{C} are the finite sums 1_{C′} in (5.11), where C′ runs over all subsets of C. Because C is a G-orbit, the only such 1_{C′} which is non-zero and G-invariant is 1_{C}. Hence the unique non-zero idempotent in (Z(ℭ)1_{C})^{G} is the identity element 1_{C} of that subring. But Proposition 5.5 implies that

(Z(ℭ)1_{C})^{G} = Z(ℭ)^{G}1_{C} = Z(ℜ)1_{C}.

Therefore 1_{C} is a primitive idempotent in Z(ℜ), and there is a unique block B of ℜ such that (5.9) holds.

6. Orthogonal conjugate idempotents.

We continue to discuss the centralizer ℭ = C_{ℜ}(ℜ_{1}) of the identity component in the fully G-graded ring ℜ of Hypothesis 4.2. The conjugation action of G must permute among themselves the idempotents of the G-ring ℭ. Let e be one of those idempotents, and H be its stabilizer

H = { τ ∈ G | e^{τ} = e }

in G. Each left coset T = Hσ of H in G determines a unique T -conjugate e^{T} of e such that

e^{T} = e^{τ} ∈ ℭ (6.1)

for all τ ∈ T. The resulting idempotents e^{T}, for T ∈ H\G, are the distinct G-conjugates of e.

We’re going to investigate the case where the G-conjugates of e are pairwise orthogonal, i.e., where

e^{S}e^{T} = \begin{cases}e^{S} & if S = T, \\ 0 & if S ≠ T, \end{cases} (6.2)

for any S, T ∈ H\G. So all our present assumptions are collected in:

Hypothesis 6.3. G is a multiplicative group, ℜ is a fully G-graded ring, ℭ is the G-ring C_{ℜ}(ℜ_{1}), and H is the stabilizer in G of some idempotent e ∈ ℭ satisfying (6.2).

We begin by discussing the right ideal eℜ of ℜ.

Lemma 6.4. The right ideal eℜ is projective as both a right ℜ-module and a right ℜ[H]-module. It is also an H\G-graded ℜ-module, with the T -component

(eℜ)_{T} = eℜ[T] = eℜe^{T} (6.5)

for any T ∈ H\G.

Proof. Since eℜ is a direct summand of the regular right ℜ-module ℜ = eℜ∔ (1−e)ℜ, it is projective as a right ℜ-module. We know from Proposition 4.5 that ℜ is projective as a right ℜ[H]-module. Hence so is its direct summand eℜ.

If S ∈ H\G and σ ∈ S, then (6.1) and (5.2a) imply that

ℜ_{σ}e^{S} = ℜ_{σ}e^{σ} = eℜ_{σ}.

Hence

ℜ[S]e^{S} = ∑_{σ∈S} ℜ_{σ}e^{S} = ∑_{σ∈S} eℜ_{σ} = eℜ[S].

This and (6.2) tell us that

eℜ[S]e^{T} = ℜ[S]e^{S}e^{T} = \begin{cases}ℜ[S]e^{S} = eℜ[S] & if S = T, \\ 0 & if S ≠ T, \end{cases}

for any S, T ∈ H\G. It follows that

eℜe^{T} = ∑_{S∈H\G} eℜ[S]e^{T} = eℜ[T]

for any T ∈ H\G, and that

eℜ = ∑_{T ∈H\G} eℜ[T] = ∑_{T ∈H\G} eℜe^{T} .

Because the idempotents e^{T}, for T ∈ H\G, are pairwise orthogonal, we conclude that eℜ is the direct sum of its additive subgroups (eℜ)_{T} defined by (6.5) for any T ∈ H\G. Since

eℜ[T]ℜ_{σ} ⊆ eℜ[T σ]

for any T ∈ H\G and σ ∈ G, this direct sum decomposition is an H\Ggrading making eℜ an H\G-graded ℜ-module. So the lemma is proved.

Our goal in this section is the surprisingly delicate proof of:

Theorem 6.6. If Hypothesis 6.3 holds, then the idempotent e must lie in Z(ℜ[H]) = ℭ[H]^{H}.

Proof. By Lemma 6.4 the right ideal eℜ is the direct sum

eℜ = ∑_{T ∈H\G}^{.} eℜ[T]

of its T -components (eℜ)_{T} = eℜ[T] for T ∈ H\G. Since right multiplication by any σ ∈ H fixes the coset H ∈ H\G, and permutes among themselves all the other cosets T ∈ H\G, it follows that eℜ is the direct sum

eℜ = eℜ[H] ∔ eℜ[G − H]

of its ℜ[H]-submodules eℜ[H] and

eℜ[G − H] = ∑_{\begin{matrix}T ∈H\G \\ T ≠H \end{matrix}}^{.} eℜ[T].

Because eℜ is projective as a right ℜ[H]-module (see Lemma 6.4), so is its direct summand eℜ[H].

The map ϕ : r ↦ er is an epimorphism of ℜ[H] onto eℜ[H] as right ℜ[H]- modules. Since eℜ[H] is projective, there is some right ideal ℑ in ℜ[H] such that

ℜ[H] = ℑ ∔ ker(ϕ).

The projection of ℜ[H] onto ℑ in this decomposition sends 1 = 1_{ℜ} = 1_{ℜ[H]} to an idempotent f ∈ ℜ[H] such that

ℑ = fℜ[H] and ker(ϕ) = (1 − f)ℜ[H].

It follows that the epimorphism ϕ restricts to an isomorphism ϕ′ : r ↦ er of fℜ[H] onto eℜ[H] as right ℜ[H]-modules. Furthermore, this isomorphism sends f to

ef = ϕ′(f) = ϕ(1) = e1 = e. (6.7)

If T ∈ H\G, then

fℜ[T] ⊆ ℜ[H]ℜ[T] ⊆ ℜ[HT] = ℜ[T].

Because ℜ is the direct sum of its additive subgroups ℜ[T], for T ∈ H\G, it follows that its right ideal fℜ is the direct sum

fℜ = ∑_{T ∈H\G}^{.} fℜ[T]

of additive subgroups. This H\G-grading makes fℜ an H\G-graded ℜmodule, since

fℜ[T]ℜ_{σ} ⊆ fℜ[T σ]

for any T ∈ H\G and σ ∈ G.

In view of (6.7) we have

efℜ[T] = eℜ[T]

for each T ∈ H\G. It follows that left multiplication by e is an epimorphism Φ′ : r ↦ er of fℜ onto eℜ as H\G-graded ℜ-modules. This epimorphism restricts to the isomorphism ϕ′ of the H-component fℜ[H] of fℜ onto the H-component eℜ[H] of eℜ. So it must be an isomorphism of fℜ onto eℜ by Proposition 4.7.

Since the H-component eℜ[H] of eℜ is equal to eℜe^{H} = eℜe by (6.5), the H-component fℜ[H] of the isomorphic H\G-graded module fℜ must satisfy

fℜ[H] = fℜe.

In particular, the element fe = f1e must lie in fℜ[H] ⊆ ℜ[H]. But (5.2a) implies that e commutes with every element of ℜ_{σ} for each σ in its stabilizer H. Hence e centralizes the direct sum ℜ[H] of all those ℜ_{σ}. Because f lies in ℜ[H], we conclude from this and (6.7) that e = ef = fe lies in the center of ℜ[H]. That center is equal to ℭ[H]^{H} by Proposition 5.5, applied to the fully H-graded ring ℜ[H] in Proposition 5.4. So the theorem is proved.

7. Morita equivalent subrings.

Now we’re going to assume that the stabilizer H of the idempotent e in Hypothesis 6.3 has finite index in G. So e has only a finite number of distinct G-conjugates e^{T}, for T ∈ H\G. Since these conjugates are pairwise orthogonal by (6.2), their sum

(7.1 a)  E = ∑_{T ∈H\G} e^{T}

is an idempotent in ℭ^{G} satisfying

(7.1 b)  eE = Ee = e.

The fixed subring ℭ^{G} is equal to Z(ℜ) by Proposition 5.5. Hence its idempotent E determines a decomposition of the ring ℜ as the direct sum

ℜ = ℜE ∔ ℜ(1 − E) (7.2)

of subrings. This decomposition allows us to consider Mod(ℜE) as the full additive subcategory of Mod(ℜ) having as objects all right ℜ-modules 𝔐 such that 𝔐E = 𝔐.

The idempotent e lies in Z(ℜ[H]) by Theorem 6.6. So it determines a decomposition of the ring ℜ[H] as the direct sum

ℜ[H] = ℜ[H]e ∔ ℜ[H](1 − e) (7.3)

of subrings. As above, this allows us to consider Mod(ℜ[H]e) to be the full additive subcategory of Mod(ℜ[H]) having as objects all right ℜ[H]-modules 𝔎 satisfying 𝔎e = 𝔎.

Proposition 7.4. The idempotent e lies in the direct summand ℜE of ℜ. The direct summand ℜ[H]e = eℜe of ℜ[H] is the subring e(ℜE)e of ℜE. Furthermore, the two-sided ideal (ℜE)e(ℜE) = ℜeℜ of ℜE is equal to ℜE.

Proof. It follows immediately from (7.1b) that e = eE lies in ℜE. We know from (6.5) for T = H that ℜ[H]e = eℜ[H] is the subring eℜe of ℜ. In view of (7.1b) this subring is equal to e(ℜE)e. Similarly, the two-sided ideal (ℜE)e(ℜE) of ℜE is equal to the two-sided ideal ℜeℜ of ℜ. If T ∈ H\G and τ ∈ T, then

e^{T} = e^{τ} ∈ ℜ_{τ^{−1}}eℜ_{τ} ⊆ ℜeℜ = (ℜE)e(ℜE)

by (6.1) and (5.2b). Hence the ideal (ℜE)e(ℜE) of ℜE contains the sum E in (7.1a). Since E is the identity element of ℜE, we conclude that (ℜE)e(ℜE) = ℜE. Thus the proposition is proved.

Because e is an idempotent generating ℜE as a two-sided ideal, the two rings ℜ[H]e = e(ℜE)e and ℜE are Morita equivalent. We’re going to describe the resulting equivalence between the categories Mod(ℜ[H]e) and Mod(ℜE) in a way that emphasizes its relation with the equivalence between the categories Mod(ℜ[H]) and GrMod(H\G, ℜ) in Proposition 4.9. All the work is done in two lemmas.

Lemma 7.5. Any right ℜE-module 𝔐 is also an H\G-graded ℜ-module, with the T -component 𝔐e^{T} for any T ∈ H\G. Any homomorphism ϕ : 𝔐 → 𝔑 of right ℜE-modules is also a homomorphism of H\G-graded ℜ-modules. Thus Mod(ℜE) is naturally a full additive subcategory of GrMod(H\G, ℜ). The H-component functor (·)_{H} : GrMod(H\G, ℜ) → Mod(ℜ[H]) sends any object 𝔐 in this subcategory to the ℜ[H]-submodule 𝔐e of 𝔐. It also sends any morphism ϕ : 𝔐 → 𝔑 in this subcategory to its restriction to a morphism ϕe : 𝔐e → 𝔑e in Mod(ℜ[H]). Hence (·)_{H} restricts to an additive functor (·)e sending Mod(ℜE) into the subcategory Mod(ℜ[H]e) of Mod(ℜ[H]).

Proof. The decomposition (7.1a) of E as a finite sum of the pairwise orthogonal idempotents e^{T} gives a decomposition of 𝔐 = 𝔐E as the finite direct sum

𝔐 = ∑_{T ∈H\G}^{.} 𝔐e^{T}

of additive subgroups. It follows from (6.1) and (5.2a) that

(𝔐e^{T})ℜ_{σ} = 𝔐e^{τ}ℜ_{σ} = 𝔐ℜ_{σ}e^{τσ} = 𝔐e^{T σ}

for any T ∈ H\G, any τ ∈ T, and any σ ∈ G. Hence the first statement of the lemma holds.

Clearly any homomorphism ϕ : 𝔐 → 𝔑 of right ℜE-modules sends 𝔐e^{T} into 𝔑e^{T} for any T ∈ H\G. So ϕ is also a homomorphism of H\G-graded ℜmodules. Thus the second statement of the lemma holds. This immediately implies the third statement.

The H-component functor (·)_{H} : GrMod(H\G, ℜ) → Mod(ℜ[H]) discussed in §3 sends any object 𝔐 in the subcategory Mod(ℜE) to its Hcomponent 𝔐e^{H} = 𝔐e, which certainly lies in Mod(ℜ[H]e). It also sends any morphism ϕ : 𝔐 → 𝔑 in Mod(ℜE) to its restriction to a homomorphism ϕ_{H} = ϕe of 𝔐_{H} = 𝔐e into 𝔑_{H} = 𝔑e as right ℜ[H]-modules. So the remaining statements of the lemma hold.

Lemma 7.6. If 𝔎 is any right ℜ[H]e-module, then the H\G-graded ℜmodule 𝔎 ⊗ ℜ = 𝔎 ⊗_{ℜ[H]} ℜ is equal to (𝔎 ⊗ ℜ)E, and hence is also a right ℜE-module. The T -component 𝔎 ⊗ ℜ[T] of 𝔎 ⊗ ℜ is equal to (𝔎 ⊗ ℜ)e^{T} for any T ∈ H\G. So the induction functor · ⊗ ℜ : Mod(ℜ[H]) → GrMod(H\G, ℜ) sends the subcategory Mod(ℜ[H]e) of Mod(ℜ[H]) into the subcategory Mod(ℜE) of GrMod(H\G, ℜ).

Proof. Of course we regard 𝔎 as a right ℜ[H]-module using the decomposition (7.3) of ℜ[H]. Since the idempotent e lies in ℜ[H], we have

𝔎 ⊗ ℜ[T] = (𝔎e) ⊗ ℜ[T] = 𝔎 ⊗ (eℜ[T])

for any T ∈ H\G. It view of (6.5) this last expression is equal to

𝔎 ⊗ (eℜe^{T}) = (𝔎e) ⊗ (ℜe^{T}) = (𝔎 ⊗ ℜ)e^{T} ,

where all tensor products are over ℜ[H]. It follows that

𝔎 ⊗ ℜ = ∑_{T ∈H\G}^{.}(𝔎 ⊗ ℜ[T]) = ∑_{T ∈H\G}^{.}(𝔎 ⊗ ℜ)e^{T} = (𝔎 ⊗ ℜ)E.

So the lemma holds.

We can now state the equivalent of the Fong-Reynolds Theorem [F, V.2.5] for our present situation as:

Theorem 7.7. If Hypothesis 6.3 holds with H of finite index in G, and if E is the central idempotent (7.1 a) of ℜ, then the H-component functor and the induction functor restrict to additive functors

(·)e : Mod(ℜE) → Mod(ℜ[H]e) and · ⊗ℜ : Mod(ℜ[H]e) → Mod(ℜE),

which form an equivalence between the abelian categories Mod(ℜE) and Mod(ℜ[H]e).

Proof. This is an immediate consequence of the preceding two lemmas and Proposition 4.9.

Of course the above equivalence between the categories Mod(ℜE) and Mod(ℜ[H]e) is a Morita equivalence between the rings ℜE and ℜ[H]e. So it induces an isomorphism between the centers Z(ℜE) and Z(ℜ[H]e) of those rings. This isomorphism can also be described in a more convenient form. We start with the observation that

Proposition 7.8. The center Z(ℜE)= Z(ℜ)E is the fixed subring (ℭE)^{G} = ℭ^{G}E in the G-subring ℭE of ℭ. The center Z(ℜ[H]e) = Z(ℜ[H])e is the fixed subring (ℭ[H]e)^{H} = ℭ[H]^{H}e in the H-subring ℭ[H]e of ℭ[H].

Proof. Because the idempotent E lies in ℭ^{G}, which is contained in Z(ℭ) by Proposition 5.5, the G-ring ℭ is the direct sum

ℭ = ℭE ∔ ℭ(1 − E)

of G-subrings. Hence ℭ^{G} is the direct sum

ℭ^{G} = (ℭE)^{G} ∔ (ℭ(1 − E))^{G}

of its subrings (ℭE)^{G} = ℭ^{G}E and (ℭ(1−E))^{G} = ℭ^{G}(1−E). Since ℭ^{G} = Z(ℜ) by Proposition 5.5, this last decomposition coincides with the decomposition

Z(ℜ) = Z(ℜ)E ∔ Z(ℜ)(1 − E)

of Z(ℜ) as a direct sum of its subrings Z(ℜ)E = Z(ℜE) and Z(ℜ)(1 − E) = Z(ℜ(1 − E)). So the first statement of the proposition is proved. The second statement is proved similarly, using the fully H-graded ring ℜ[H], the centralizer ℭ[H] of its identity component, and the idempotent e ∈ Z(ℜ[H]) in place of ℜ, ℭ and E, respectively (see Proposition 5.4).

Suppose that c ∈ ℭ^{H} and T ∈ H\G. As in (6.1), we may define the T -conjugate c^{T} ∈ ℭ of c to be the common value of the τ-conjugates c^{τ} for τ ∈ T. Then the trace map tr_{H}^{G} : ℭ^{H} → ℭ^{G} is the homomorphism of additive groups sending c ∈ ℭ^{H} to the finite sum

tr_{H}^{G}(c) = ∑_{T ∈H\G} c^{T} , (7.9)

computed in ℭ. Comparing this with (7.1a), we see that E is the trace tr_{H}^{G}(e) of e ∈ ℭ[H]^{H} ⊆ ℭ^{H}. We use the trace map in the following description of the isomorphism between Z(ℜE) and Z(ℜ[H]e).

Theorem 7.10. If Hypothesis 6.3 holds with H of finite index in G, and if E is the central idempotent tr_{H}^{G}(e) of ℜ, then multiplication by e is an isomorphism

λ_{e} : z ↦ ze = ez (7.11) of the ring Z(ℜE) = ℭ^{G}E onto the ring Z(ℜ[H]e) = ℭ[H]^{H}e. The inverse isomorphism is the restriction of the trace map tr_{H}^{G} : ℭ^{H} → ℭ^{G} = Z(ℜ) to the subring ℭ[H]^{H}e of ℭ^{H}.

Proof. The idempotent e in the subring Z(ℜ[H]) of ℜ certainly commutes with any element z ∈ Z(ℜ). It follows that multiplication by e is a homomorphism of Z(ℜ) into the center of the subring eℜe of ℜ. We know from Proposition 7.4 that this subring is equal to ℜ[H]e. In view of (7.1b) this homomorphism sends E to Ee = e. Thus its restriction to Z(ℜE) = Z(ℜ)E is an identity-preserving homomorphism λ_{e} of that ring into Z(ℜ[H]e).

By Proposition 7.8 the center Z(ℜE) is the subring ℭ^{G}E of ℭ^{G}. Hence any element z ∈ Z(ℜE) lies in (ℭE)^{H} and is fixed under conjugation by any element of either G or H\G. Its image λ_{e}(z) ∈ Z(ℜ[H]e) lies in the subring ℭ[H]^{H}e of ℭ^{H} by Proposition 7.8. So the trace tr_{H}^{G}(λ_{e}(z)) is defined. It follows from (7.9) and (7.1a) that

tr_{H}^{G}(λ_{e}(z)) = ∑_{T ∈H\G}(ze)^{T} = ∑_{T ∈H\G} z^{T} e^{T} = ∑_{T ∈H\G} ze^{T} = zE = z.

Thus the restriction of tr_{H}^{G} is a left inverse to λ_{e} : Z(ℜE) → Z(ℜ[H]e). If w ∈ Z(ℜ[H]e) = ℭ[H]^{H}e, then tr_{H}^{G}(w) lies in ℭ^{G} and satisfies

tr_{H}^{G}(w)E = tr_{H}^{G}(w) tr_{H}^{G}(e) = ∑_{S,T ∈H\G} w^{S}e^{T} .

Because w is equal to we, it follows from (6.2) that

w^{S}e^{T} = (we)^{S}e^{T} = w^{S}e^{S}e^{T} = \begin{cases}w^{S}e^{S} = w^{S} & if S = T, \\ 0 & if S ≠ T, \end{cases}

for any S, T ∈ H\G. Hence

tr_{H}^{G}(w)E = ∑_{S∈H\G} w^{S} = tr_{H}^{G}(w).

So tr_{H}^{G}(w) lies in ℭ^{G}E = Z(ℜE), and λ_{e}(tr_{H}^{G}(w)) is defined.

We compute that

λ_{e}(tr_{H}^{G}(w)) = tr_{H}^{G}(w)e = ∑_{T ∈H\G} w^{T} e.

But w is equal to we, and e^{T} is orthogonal to the idempotent e = e^{H} whenever T ∈ H\G is different from H. Hence

tr_{H}^{G}(w)e = ∑_{T ∈H\G} w^{T} e^{T} e = w^{H}e^{H}e = we = w.

Therefore tr_{H}^{G} : Z(ℜ[H]e) → Z(ℜE) is a right inverse to λ_{e} : Z(ℜE) → Z(ℜ[H]e), as well as a left inverse. So the theorem is proved.

It remains to be seen that the isomorphism λ_{e} of Z(ℜE) onto Z(ℜ[H]e) in the above theorem is actually the isomorphism ι associated with the equivalence of Mod(ℜE) with Mod(ℜ[H]e) in Theorem 7.7. The latter isomorphism can be described as sending any z ∈ Z(ℜE) to the unique element ι(z) ∈ Z(ℜ[H]e) such that the functor (·)e sends the ℜE-endomorphism m ↦ mz of any right ℜE-module 𝔐 to the ℜ[H]e-endomorphism k ↦ kι(z) of the right ℜ[H]e-module 𝔐e. Since (·)e is just restriction to 𝔐e, the following observation shows that λ_{e} is indeed ι.

Proposition 7.12. If 𝔐 is any right ℜE-module, and z is any element of Z(ℜE), then the ℜ-endomorphism m ↦ mz of 𝔐 restricts to the ℜ[H]- endomorphism k ↦ kλ_{e}(z) of 𝔐e.

Proof. Clearly any element k ∈ 𝔐e satisfies k = ke. It follows from this and (7.11) that

kz = kez = kλ_{e}(z).

So the proposition holds.

8. G-invariant subrings.

The most natural idempotents e ∈ ℭ satisfying (6.2) are the block idempotents of G-invariant subrings of our fully G-graded ring ℜ. By a G-invariant subring of ℜ we mean a subring 𝔖 of ℜ such that

ℜ_{τ^{−1}}𝔖ℜ_{τ} = 𝔖 (8.1)

for every τ ∈ G. It follows immediately from (4.1) and (3.3a) that:

(8.2) The restriction of ℜ to any normal subgroup N of G is a unitary, G-invariant subring ℜ[N] of ℜ.

So such subrings always exist.

From now on we fix an arbitrary G-invariant subring 𝔖 of ℜ. For our purposes the most important property of 𝔖 is:

Proposition 8.3. The center Z(𝔖) of 𝔖 is a G-subring of ℭ = C_{ℜ}(ℜ_{1}).

Proof. We first show the the identity element 1_{𝔖} of 𝔖 lies in ℭ, i.e., that (1_{𝔖})r_{1} = r_{1}1_{𝔖} for any r_{1} ∈ ℜ_{1}. Since 1_{ℜ} lies in ℜ_{1} by (3.2), it follows from (8.1) for τ = 1 that

(1_{𝔖})r_{1} = (1_{ℜ}1_{𝔖})r_{1} ∈ ℜ_{1}𝔖ℜ_{1} = 𝔖.

So

(1_{𝔖})r_{1} = ((1_{𝔖})r_{1})1_{𝔖} = (1_{𝔖})r_{1}1_{𝔖}.

By left-right symmetry, r_{1}1_{𝔖} is also equal to (1_{𝔖})r_{1}1_{𝔖}, and hence to (1_{𝔖})r_{1}. Thus 1_{𝔖} lies in ℭ.

The above argument shows that the product (1_{𝔖})r_{1} = r_{1}1_{𝔖} lies in 𝔖 for each r_{1} ∈ ℜ_{1}. Hence this product commutes with any z ∈ Z(𝔖). It follows that

zr_{1} = (z1_{𝔖})r_{1} = z((1_{𝔖})r_{1}) = (r_{1}1_{𝔖})z = r_{1}((1_{𝔖})z) = r_{1}z.

Therefore Z(𝔖) is a subring of ℭ.

If z ∈ Z(𝔖) and τ ∈ G, then (5.2b) and (8.1) imply that

z^{τ} = ∑_{i=1}^{n} s_{i}zt_{i} ∈ ℜ_{τ^{−1}}𝔖ℜ_{τ} = 𝔖,

for some s_{1}, s_{2}, …, s_{n} ∈ ℜ_{τ^{−1}} and t_{1}, t_{2}, …, t_{n} ∈ ℜ_{τ}. Furthermore, (5.2a) tells us that

z^{τ}r′_{τ^{−1}}sr_{τ} = r′_{τ^{−1}}(z^{τ})^{τ^{−1}}sr_{τ} = r′_{τ^{−1}}zsr_{τ} = r′_{τ^{−1}}szr_{τ} = r′_{τ^{−1}}sr_{τ}z^{τ}

for any r′_{τ^{−1}} ∈ ℜ_{τ^{−1}}, any s ∈ 𝔖, and any r_{τ} ∈ ℜ_{τ}. The resulting products r′_{τ^{−1}}sr_{τ} generate 𝔖 as an additive group by (8.1). Hence z^{τ} both lies in 𝔖 and centralizes 𝔖. So z^{τ} lies in Z(𝔖), and the proposition is proved.

The conjugation action of G on the G-subring Z(𝔖) of ℭ must permute among themselves the primitive idempotents 1_{C}, for C ∈ Blk(𝔖), of that subring. As in (5.7), we conclude that there is a natural conjugation action of the group G on the set Blk(𝔖), with any τ ∈ G sending any block C of 𝔖 to the unique conjugate block C^{τ} ∈ Blk(𝔖) such that

1_{C^{τ}} = (1_{C})^{τ} ∈ Z(𝔖). (8.4)

Now we fix a block C ∈ Blk(𝔖) and its stabilizer G_{C} in G. So all our current assumptions are gathered in:

Hypothesis 8.5. G is a multiplicative group, ℜ is a fully G-graded ring, 𝔖 is a G-invariant subring of ℜ, and C is a block of 𝔖 with stabilizer G_{C} in G.

The idempotent 1_{C} of Z(𝔖) lies in ℭ by Proposition 8.3. Its distinct Gconjugates (1_{C})^{τ} are distinct primitive idempotents 1_{C^{τ}} in Z(𝔖), and hence are pairwise orthogonal. So Hypothesis 6.3 holds with e = 1_{C} and H = G_{C}. This allows us to reproduce the above relations between G, ℜ, 𝔖 and C on a smaller scale.

Proposition 8.6. If Hypothesis 8.5 holds, then the restriction ℜ[G_{C}] is a fully G_{C}-graded ring. The intersection 𝔖_{C} = ℜ[G_{C}] ∩ 𝔖 is a unitary G_{C}- invariant subring of ℜ[G_{C}]. The idempotent 1_{C} lies in Z(𝔖_{C}). Furthermore, the direct summand 𝔖_{C}1_{C} of 𝔖_{C} is equal to the direct summand 𝔖1_{C} of 𝔖. Hence there is a unique G_{C}-invariant block C_{C} of 𝔖_{C} such that 1_{C_{C}} = 1_{C}. So Hypothesis 8.5 holds with G_{C}, ℜ[G_{C}], 𝔖_{C}, and C_{C} in place of G, ℜ, 𝔖 and C, respectively.

Proof. The first statement of the proposition holds by Proposition 5.4. Since ℜ[G_{C}] and 𝔖 are unitary subrings of ℜ, their intersection 𝔖_{C} is a unitary subring of each of them. If τ ∈ G_{C}, then (3.3a), (4.1) and (8.1) imply that

ℜ_{τ^{−1}}𝔖_{C}ℜ_{τ} = ℜ_{τ^{−1}}(ℜ[G_{C}] ∩ 𝔖)ℜ_{τ} ⊆
ℜ_{τ^{−1}}ℜ[G_{C}]ℜ_{τ} ∩ ℜ_{τ^{−1}}𝔖ℜ_{τ} = ℜ[G_{C}] ∩ 𝔖 = 𝔖_{C}.

Similarly we have

ℜ_{τ}𝔖_{C}ℜ_{τ^{−1}} ⊆ 𝔖_{C}.

Since 1 = 1_{ℜ} lies in ℜ_{1} = ℜ_{τ^{−1}}ℜ_{τ} by (3.2) and (4.1), we conclude that

𝔖_{C} = 1(𝔖_{C})1 ⊆ ℜ_{τ^{−1}}ℜ_{τ}𝔖_{C}ℜ_{τ^{−1}}ℜ_{τ} ⊆ ℜ_{τ^{−1}}𝔖_{C}ℜ_{τ}.

Therefore ℜ_{τ^{−1}}𝔖_{C}ℜ_{τ} is equal to 𝔖_{C} for any τ ∈ G_{C}, and the second statement of the proposition is proved.

Theorem 6.6 for e = 1_{C} and H = G_{C} tells us that 1_{C} lies in Z(ℜ[G_{C}]). Since 1_{C} already lies in Z(𝔖), this implies that it lies in Z(𝔖_{C}). The direct summand 𝔖1_{C} = 1_{C}𝔖 of 𝔖 is contained in 1_{C}ℜ1_{C}, which is equal to the direct summand ℜ[G_{C}]1_{C} = 1_{C}ℜ[G_{C}] of ℜ[G_{C}] by (6.5) for e = 1_{C} and T = H = G_{C}. Hence 𝔖1_{C} is contained in 𝔖_{C} = ℜ[G_{C}]∩𝔖. This forces 𝔖1_{C} to equal 𝔖_{C}1_{C}. Thus the third and fourth statements of the proposition hold.

Because 1_{C} is a primitive idempotent in Z(𝔖), the ring 𝔖1_{C} is indecomposable. Hence so is the equal ring 𝔖_{C}1_{C}. Therefore 1_{C} is a primitive idempotent in Z(𝔖_{C}). So there is a unique block C_{C} of 𝔖_{C} such that 1_{C_{C}} = 1_{C}. The G_{C}-invariance of 1_{C} implies that of both 1_{C_{C}} and C_{C}. Thus the fifth statement of the proposition holds. The remaining statement follows from the preceding ones.

In order to apply the results in §7 we must assume that the stabilizer G_{C} has finite index in G, i.e., that C belongs to a finite G-orbit C of blocks of 𝔖. Then the equivalent of the idempotent E in (7.1a) is the finite sum

1_{C} = ∑_{C′∈C} 1_{C′} (8.7)

of pairwise orthogonal idempotents 1_{C′} in Z(𝔖). The decompositions (7.2) and (7.3) now become the decompositions

(8.8 a)  ℜ = ℜ1_{C} ∔ ℜ(1 − 1_{C})

and

(8.8 b)  ℜ[G_{C}] = ℜ[G_{C}]1_{C_{C}} ∔ ℜ[G_{C}](1 − 1_{C_{C}})

of ℜ and ℜ[G_{C}] as direct sums of subrings.

We say that a block B of ℜ lies over the block C of 𝔖 if the idempotent 1_{B}1_{C} = 1_{C}1_{B} in ℭ is non-zero. In that case we also say that C lies under B. We denote by Blk( ℜ | C) the set of all blocks B of ℜ lying over C. Similarly, we say that a block B′ of ℜ[G_{C}] lies over the block C_{C} of 𝔖_{C}, or that C_{C} lies under B′, if the idempotent 1_{B′}1_{C_{C}} = 1_{C_{C}}1_{B′} of ℭ[G_{C}] is non-zero. We denote by Blk( ℜ[G_{C}] | C_{C} ) the set of all blocks B′ of ℜ[G_{C}] lying over C_{C}.

Proposition 8.9. If C belongs to a finite G-orbit C of blocks of 𝔖, then the 1_{B}, for B ∈ Blk( ℜ | C), are the distinct primitive idempotents in the center Z(ℜ1_{C}) of the summand ℜ1_{C} in (8.8 a). Furthermore, the 1_{B′}, for B′ ∈ Blk( ℜ[G_{C}] | C_{C} ) are the distinct primitive idempotents in the center Z(ℜ[G_{C}]1_{C_{C}}) of the summand ℜ[G_{C}]1_{C_{C}} in (8.8 b).

Proof. Fix a block B ∈ Blk(ℜ). The primitive idempotent 1_{B} of Z(ℜ) must lie in exactly one of the summands ℜ1_{C} and ℜ(1 − 1_{C}) in (8.8a). If 1_{B} lies in ℜ(1 − 1_{C}), then 1_{B}1_{C} = 0. Since 1_{C}1_{C} = 1_{C} by (8.7), this implies that 1_{B}1_{C} = 1_{B}1_{C}1_{C} = 0. Therefore B does not lie over C when 1_{B} lies in ℜ(1 − 1_{C}).

On the other hand, if 1_{B} lies in ℜ1_{C}, then

0 ≠ 1_{B} = 1_{B}1_{C} = ∑_{C′∈C} 1_{B}1_{C′}.

So B lies over some block C′ ∈ C. Since C′ is equal to C^{τ} for some τ ∈ G, and 1_{B} ∈ Z(ℜ) = ℭ^{G} is fixed by τ, we have

(1_{B}1_{C})^{τ} = (1_{B})^{τ}(1_{C})^{τ} = 1_{B}1_{C′} ≠ 0.

Hence 1_{B}1_{C} ≠ 0, and B lies over C when 1_{B} lies in ℜ1_{C}.

Because the primitive idempotents of Z(ℜ1_{C}) are exactly those primitive idempotents of Z(ℜ) which lie in the summand ℜ1_{C} in (8.8a), the above arguments prove the first statement in the proposition. The second statement is proved similarly, using G_{C}, ℜ[G_{C}], 𝔖_{C} and C_{C} in place of G, ℜ, 𝔖 and C, respectively, as in Proposition 8.6.

Now we can apply Theorem 7.10 to obtain:

Theorem 8.10. If Hypothesis 8.5 holds and C belongs to a finite G-orbit C in Blk(𝔖), then multiplication by 1_{C} is an isomorphism λ_{C} of the ring Z(ℜ1_{C}) onto the ring Z(ℜ[G_{C}]1_{C_{C}}). The inverse isomorphism is a restriction of the trace map from G_{C} to G. These isomorphisms induce a bijection of Blk( ℜ | C) onto Blk( ℜ[G_{C}] | C_{C} ), sending any block B of ℜ lying over C to the unique block B_{C} of ℜ[G_{C}] lying over C_{C} such that

λ_{C}(1_{B}) = 1_{B}1_{C} = 1_{B_{C}} and tr_{G_{C}}^{G}(1_{B_{C}}) = 1_{B}.

Proof. This is an immediate consequence of Proposition 8.9 and of Theorem 7.10 applied to the idempotents e = 1_{C} = 1_{C_{C}} and E = 1_{C}.

We say that a block B ∈ Blk( ℜ | C) is linked by Clifford theory for C to the unique block B_{C} ∈ Blk( ℜ[G_{C}] | C_{C} ) corresponding to it in the above theorem.

Lemma 8.11. If B ∈ Blk( ℜ | C) is linked to B_{C} ∈ Blk( ℜ[G_{C}] | C_{C} ) by Clifford theory for C, then Hypothesis 6.3 holds with e = 1_{B_{C}} and H = G_{C}. In this case the idempotent E in (7.1 a) is just 1_{B}.

Proof. The idempotent 1_{B_{C}} is the product 1_{B}1_{C} of two idempotents in ℭ, and hence lies in ℭ. Since 1_{B} ∈ Z(ℜ) = ℭ^{G} is G-invariant, we have

(1_{B_{C}})^{τ} = (1_{B})^{τ}(1_{C})^{τ} = 1_{B}1_{C^{τ}}

for any τ ∈ G. In particular, G_{C} stabilizes 1_{B_{C}}. If τ ∈ G − G_{C}, then 1_{C^{τ}} is orthogonal to 1_{C}, and commutes with 1_{B} ∈ Z(ℜ). Hence

(1_{B_{C}})^{τ}1_{B_{C}} = 1_{B}1_{C^{τ}} 1_{B}1_{C} = 1_{B}1_{C^{τ}} 1_{C} = 0 ≠ 1_{B_{C}}

in this case. We conclude that G_{C} is precisely the stabilizer of 1_{B_{C}} in G, and that the distinct G-conjugates of 1_{B_{C}} are pairwise orthogonal. Thus the first statement of the lemma holds.

The idempotent corresponding to E in (7.1a) is the trace tr_{G_{C}}^{G}(1_{B_{C}}) of the idempotent 1_{B_{C}} corresponding to e, taken from its stabilizer H = G_{C} to G. By Theorem 8.10 this trace sends 1_{B_{C}} to 1_{B}. That is the remaining statement of the lemma.

Now we can state the Fong-Reynolds Theorem 7.7 in a more familiar form.

Theorem 8.12. Suppose that Hypothesis 8.5 holds, and that C belongs to a finite G-orbit C of blocks of 𝔖. If B ∈ Blk( ℜ | C) is linked to B_{C} ∈ Blk( ℜ[G_{C}] | C_{C} ) by Clifford theory for C, then multiplication by 1_{B_{C}} and induction from ℜ[G_{C}] to ℜ restrict to additive functors

(·)1_{B_{C}} : Mod(ℜ1_{B}) → Mod(ℜ[G_{C}]1_{B_{C}})

and

· ⊗ ℜ : Mod(ℜ[G_{C}]1_{B_{C}}) → Mod(ℜ1_{B}),

which form an equivalence between the abelian categories Mod(ℜ1_{B}) and Mod(ℜ[G_{C}]1_{B_{C}}).

Proof. In view of Lemma 8.11 this follows immediately from Theorem 7.7 for e = 1_{B_{C}} and E = 1_{B}.

9. Clifford theory.

In order to obtain the conclusions of the preceding section for every block B of ℜ, we shall assume from now on that G, ℜ and 𝔖 satisfy:

Hypothesis 9.1. G is a finite multiplicative group, ℜ is a fully G-graded ring having finite block theory, and 𝔖 is a unitary, G-invariant subring of ℜ.

Then we have:

Proposition 9.2. If Hypothesis 9.1 holds, then any G-subring 𝔇 of ℭ = C_{ℜ}(ℜ_{1}) has finite block theory. Hence any G-invariant subring 𝔖′ of ℜ has finite block theory. In particular, 𝔖 has finite block theory.

Proof. Since ℜ has finite block theory, Proposition 1.4 tells us that the set ZI(ℜ) of all idempotents in Z(ℜ) is finite. We know from Proposition 5.5 that ZI(ℜ) is the set of all idempotents in ℭ^{G} = Z(ℜ). If 𝔇 is a G-subring of ℭ, then its fixed subring 𝔇^{G} is a subring of ℭ^{G}. Hence 𝔇^{G} contains only a finite number of distinct idempotents, and so must have finite block theory by Proposition 1.4. Now Theorem 2.2 tells us that 𝔇 has finite block theory.

Let 𝔖′ be any G-invariant subring of ℜ. Proposition 8.3 tells us that Z(𝔖′) is a G-subring of ℭ. So Z(𝔖′) has finite block theory by the above arguments. Then 𝔖′ has finite block theory by Proposition 1.8. Thus the proposition holds.

We denote by Blk_{G}(𝔖) the set of all G-orbits under the conjugation action (8.4) of G on Blk(𝔖). We write 1_{C} for the idempotent associated with any orbit C ∈ Blk_{G}(𝔖) by (8.7).

Proposition 9.3. If Hypothesis 9.1 holds, the the ring ℜ is the direct sum

ℜ = ∑_{C∈Blk_{G}(𝔖)}^{.} ℜ1_{C} (9.4)

of G-invariant subrings ℜ1_{C}, each of which has finite block theory.

Proof. We know from Proposition 9.2 that 𝔖 has finite block theory. Since 𝔖 is also a unitary subring of ℜ (see Hypothesis 9.1), it follows from (1.3) that its identity element 1_{ℜ} is the finite sum

1_{ℜ} = ∑_{C∈Blk(𝔖)} 1_{C}

of pairwise orthogonal primitive idempotents in Z(𝔖). The finite set Blk(𝔖) is the disjoint union of its G-orbits C ∈ Blk_{G}(𝔖). So 1_{ℜ} is also the finite sum

1_{ℜ} = ∑_{C∈Blk_{G}(𝔖)} 1_{C}

of pairwise orthogonal idempotents

1_{C} = ∑_{C∈C} 1_{C}.

Each of these last idempotents lies in ℭ^{G}, which is equal to Z(ℜ) by Proposition 5.5. So this last decomposition of 1_{ℜ} implies that ℜ is the direct sum (9.4) of subrings.

The idempotent 1_{C} ∈ Z(ℜ) commutes with ℜ_{τ} for any C ∈ Blk_{G}(𝔖) and τ ∈ G. Hence

ℜ_{τ^{−1}}(ℜ1_{C})ℜ_{τ} = (ℜ_{τ^{−1}}ℜℜ_{τ})1_{C} = ℜ1_{C}.

So ℜ1_{C} is a G-invariant subring of ℜ. As such it has finite block theory by Proposition 9.2. Thus the proposition holds.

Recall from the preceding section that a block C of 𝔖 lies under a block B of ℜ if the idempotent 1_{B}1_{C} = 1_{C}1_{B} in ℜ is not zero. The final step in Clifford theory for blocks is:

Theorem 9.5. If Hypothesis 9.1 holds, then the blocks of 𝔖 lying under a fixed block B ∈ Blk(ℜ) form a single G-orbit C ∈ Blk_{G}(𝔖). This orbit C is determined by the fact that 1_{B} lies in the direct summand ℜ1_{C} in (9.4).

Proof. The primitive central idempotent 1_{B} of ℜ must lie in exactly one of the direct summands ℜ1_{C} in (9.4). Thus there is a unique G-orbit C ∈ Blk_{G}(𝔖) such that 1_{B} ∈ ℜ1_{C}. Then Proposition 8.9 tells us that B lies over every block C ∈ C and over no other block of 𝔖. So the theorem holds.

Of course Theorems 8.10 and 8.12 now hold for any block B of ℜ and any block C of 𝔖 lying under B. In particular, B is linked by Clifford theory for C to a unique block B_{C} of the restriction ℜ[G_{C}] of ℜ to the stabilizer G_{C} of C. Furthermore, the abelian categories Mod(ℜ1_{B}) and Mod(ℜ[G_{C}]1_{B_{C}}) are Morita equivalent.

10. Questions.

The above discussion raises a couple of interesting questions which we cannot answer. There is a natural duality between G-rings and G-graded rings, based on the fact that both are special cases of ℌ-rings in the sense of [CS], where the Hopf algebra ℌ is the group ring ℤG in the former case and its dual in the latter one. This allows us to formulate the dual statement to Theorem 2.2 in the form of:

Question 10.1. If G is a finite group and ℜ is any G-graded ring whose identity component ℜ_{1} has finite block theory, does ℜ have finite block theory?

Surprisingly enough, the answer to this question is yes when G has order 2 or 3. This can be shown by direct and brutal calculation of all possible central idempotents in ℜ. However the computations required for this approach become impossibly complicated as the order of G increases. So any attempt to give a positive answer to this question for all G must be based on some other idea.

Rings graded by a finite abelian group G behave very much like G^{∗}-rings, where G^{∗} is the finite dual group to G. So the fact that Question 10.1 has a positive answer for a few small abelian groups may be an accident. Before raising this question to the status of a conjecture, it would be wise to see a few non-commutative examples.

One possible consequence of a positive answer to Question 10.1 is:

Proposition 10.2. Suppose that G is a finite group and that ℜ is a fully G-graded ring with finite block theory. If the answer to Question 10.1 is always yes, then the restriction ℜ[H] of ℜ to any subgroup H of G also has finite block theory.

Proof. The identity component ℜ_{1} is a G-invariant subring of ℜ by (8.2), and so has finite block theory by Proposition 9.2. A positive answer to Question 10.1 for the H-graded ring ℜ[H] would then imply the proposition.

Of course, the conclusion of the above proposition might hold even if the answer to Question 10.1 turns out to be no in general.

It is somewhat annoying that we need to assume that the fully G-graded ring ℜ has finite block theory in order to show that each of its blocks B lies over a single G-orbit of blocks of a G-invariant subring 𝔖 (see Theorem 9.5). In view of the relations between blocks of ℜ and G-orbits of blocks of ℭ in Theorem 5.8, we might hope that this global assumption can be avoided, i.e., that we have a positive answer to:

Question 10.3. If G is a finite group and 𝔖 is a G-invariant subring of a fully G-graded ring ℜ, does every block B of ℜ lie over a G-conjugacy class C of blocks of 𝔖?

We don’t have a single example illustrating this question in a situation where ℜ does not have finite block theory. So it is much more speculative than the preceding question. We should remark that the problem is to find one block C of 𝔖 lying under a given block B of ℜ. Once such a C exists, the rest is easy.

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This research was supported by grant number DMS 96-00106 from the National Science Foundation.

University of Illinois at Urbana-Champaign 1409 W. Green St. Urbana, IL 61801, USA E-mail address: e-dade@uiuc.edu