Group graded basic Morita equivalences and the Harris–Knörr correspondence

Notation 2.1.

Let 𝒪 be a complete discrete valuation ring with residue field k of characteristic p > 0 . For the sake of simplicity, we assume that k is algebraically closed.

Let H be a normal subgroup of the finite group G, and let b be a G-invariant block of 𝒪 ⁢ N , that is, { b } is a point of G on 𝒪 ⁢ N . We denote

and we regard A as a G ¯ -graded 𝒪 -algebra with identity component B.

2.2.

Take a defect pointed group P γ of G { b } . Then, by [12, Proposition 5.3], there is a defect pointed group Q δ of H { b } on the H-interior algebra 𝒪 ⁢ H such that Q δ ≤ P γ ; in this case, we may assume that Q = P ∩ H .

The Frattini argument implies that G = H ⁢ N G ⁢ ( Q δ ) ; hence

Let i ∈ γ and j ∈ δ be source idempotents such that j = i ⁢ j = j ⁢ i . Set

By [15, Proposition 3.2], both A δ and A γ are strongly G ¯ -graded 𝒪 -algebras, and we have G ¯ -graded Morita equivalences between A, A δ and A γ .

2.3.

Recall (see [13, § 5.9]) that there is a maximal ( H , b ) -Brauer pair, denoted ( Q , b δ ) , associated with Q δ . Here b δ is a block of k ⁢ C H ⁢ ( Q ) with defect group Z ⁢ ( Q ) . It is well known that b δ lifts uniquely to a block (still denoted by b δ ) of 𝒪 ⁢ C H ⁢ ( Q ) with defect group Z ⁢ ( Q ) , and that we have the equality

Moreover, b δ is also a block of 𝒪 ⁢ Q ⁢ C H ⁢ ( Q ) with defect group Q. We denote

and we will regard 𝒜 as a G ¯ -graded algebra with 1-component ℬ .

Notation 2.4.

Consider another finite group G ′ , and assume that ω : G → G ¯ and ω ′ : G ′ → G ¯ are group epimorphisms with H = Ker ⁡ ω and H ′ = Ker ⁡ ω ′ . Let b ′ be a G ′ -invariant block of 𝒪 ⁢ H ′ , and let

A ′ as a G ¯ -graded G ′ -interior algebra with 1-component B ′ . We will use ′ to denote the objects associated with b ′ .

Set G ¨ : = ( ω × ω ′ ) - 1 ( Δ ( G ¯ ) ) and Δ : = ( b ⊗ b ′ ) 𝒪 G ¨ . In particular, any Δ -module, is, by restriction of scalars, a ( B , B ′ ) -bimodule.

2.5.

Basic Morita equivalences were introduced by Puig [16]; their local properties have been further studied in [18], and these results have been generalized to block extensions in [5, 6, 7, 4]. According to [5, Definition 4.2], we say that A is G ¯ -graded basic Morita equivalent to A ′ if there is an indecomposable Δ -module M with vertex P ¨ ≤ G ¨ and endopermutation source N ¨ such that there is an embedding

of G ¯ -graded P-interior algebras, where S : = End 𝒪 ( N ¨ ) is a Dade P ¨ -algebra. Recall that, in this situation, by [5, Proposition 4.1], the projections from G × G ′ to G and G ′ induce isomorphisms P ¨ ≃ P ≃ P ′ , so we can again identify the vertex P ¨ with the defect groups P and P ′ , and this identification gives the P-interior algebra structure on S ⊗ A γ ′ ′ .

Note also that, by [5, Corollary 4.3], B and B ′ are basic Morita equivalent because the embedding f restricts to a Q-interior P-algebra embedding

Now let R ≤ P and R ′ ≤ P ′ such that R ′ corresponds to R via the identification of P ¨ with P and P ′ . By applying the Brauer construction to (2.1), we obtain the N P ⁢ ( R ) -algebra embedding

If R ϵ is a local pointed subgroup of P γ , then, by (2.2), there is a unique local pointed group R ϵ ′ ′ such that R ϵ ′ ′ ≤ P γ ′ ′ . We obtain in this way a bijection R ϵ ↔ R ϵ ′ ′ between the local pointed groups included in P γ and local pointed groups included in P γ ′ ′ . We will use this bijection in 2.6 and in Section 4, in the particular case when R ϵ = Q δ .

2.6.

Finally, recall also that the G ¯ -graded basic Morita equivalence between A and A ′ induces a G ¯ -graded basic Morita equivalence between 𝒜 = 𝒪 ⁢ N G ⁢ ( Q δ ) ⁢ b δ and 𝒜 ′ = 𝒪 ⁢ N G ′ ⁢ ( Q δ ′ ) ⁢ b δ ′ ′ , which is obtained as follows (see [5, Lemma 6.6 and 6.7] and [7, Proof of Theorem 1.1]).

Let λ = br Q ⁡ ( γ ) and λ ′ = br Q ⁡ ( γ ′ ) . Then P λ is a defect pointed group of b δ in N G ⁢ ( Q δ ) , and P λ ′ is a defect pointed group of b δ ′ ′ in N G ′ ⁢ ( Q δ ′ ) . It follows that 𝒜 λ is the source algebra of 𝒜 , and 𝒜 λ ′ ′ is the source algebra of 𝒜 ′ .

Note that S ⁢ ( Q ) is a Dade P-interior algebra. By applying the extended Brauer construction [6, Section 4] to the embedding f, we get the embedding

of G ¯ -graded P-interior algebras, which gives the G ¯ -graded basic Morita equivalence between 𝒜 and 𝒜 ′ .

3.1.

Let A = b ⁢ 𝒪 ⁢ G , B = b ⁢ 𝒪 ⁢ H , 𝒜 = 𝒪 ⁢ N G ⁢ ( Q δ ) ⁢ b δ and ℬ = 𝒪 ⁢ N H ⁢ ( Q δ ) ⁢ b δ as in Section 2, and let A ′ = b ′ ⁢ 𝒪 ⁢ N G ⁢ ( Q ) , B ′ = b ′ ⁢ 𝒪 ⁢ N H ⁢ ( Q ) , where b ′ = Tr N H ⁢ ( Q δ ) N H ⁢ ( Q ) ⁡ b δ .

The Fong–Reynolds equivalence applied in this situation says that there is a G ¯ -graded basic Morita equivalence between A ′ and 𝒜 .

3.2.

Consider the centralizer in A of the 1-component B,

We know that C is a G ¯ -graded G ¯ -algebra, which is not strongly graded in general. To obtain a crossed product, we consider the stabilizer G ¯ ⁢ [ b ] of B as a ( B , B ) -bimodule, that is (see [9, Section 2.9]),

Then G ¯ ⁢ [ b ] is a normal subgroup of G ¯ , and

is a strongly G ¯ ⁢ [ b ] -graded G ¯ -acted subalgebra of C. Regarding the graded Jacobson radical (see [9, Lemma 3.3], [14, Section 1.5.A]), we know that

hence

as G ¯ ⁢ [ b ] -graded G ¯ -acted algebras.

Recall also that J gr ⁢ ( C ) ⊆ J ⁢ ( C ) . It follows from this isomorphism that the primitive idempotents of C lie in C G ¯ ⁢ [ b ] ; hence the G-invariant primitive idempotents of C (that is, the blocks of b ⁢ 𝒪 ⁢ G ) coincide with the G-invariant primitive idempotents of C G ¯ ⁢ [ b ] .

3.3.

In a similar way, we define the group G ¯ ⁢ [ b ′ ] and the G ¯ -graded G ¯ -acted algebras

respectively the group G ¯ ⁢ [ b δ ] and the G ¯ -graded G ¯ -algebras

By [14, Corollary 5.1.4], the G ¯ -graded Morita equivalence between A ′ and 𝒜 induces the group isomorphism G ¯ ⁢ [ b ′ ] ≃ G ¯ ⁢ [ b δ ] and the following commutative diagram:

Lemma 3.4.

Let Q ≤ D ≤ P . The Brauer homomorphism br Q determines a bijection between the blocks of O ⁢ G with defect group D and the blocks of O ⁢ N G ⁢ ( Q ) with defect group D. This correspondence coincides with the Brauer correspondence determined by br D .

Proof.

By [1, Proposition 1.5], applied to the appropriate restrictions of the Brauer map, we obtain the commutative diagram

of homomorphisms of N G ⁢ ( Q ) -algebras.

Let a be a block of 𝒪 ⁢ G with defect group D. Then, by the commutativity of the above diagram, a ~ : = br Q G ( a ) is a primitive idempotent (hence block) of k ⁢ C G ⁢ ( Q ) N G ⁢ ( Q ) with defect group D. It is easy to see that a ~ lifts to a block a ′ of k ⁢ N G ⁢ ( Q ) , still with defect group D, that verifies br Q N G ⁢ ( Q ) ⁡ ( a ′ ) = a ~ . In this way, we obtain the bijection determined by br Q .

Now let a ′′ denote the Brauer correspondent block of a in N G ⁢ ( Q ) . This means that a ′′ belongs to 𝒪 ⁢ N G ⁢ ( Q ) and verifies br D N G ⁢ ( Q ) ⁡ ( a ′′ ) = br D G ⁡ ( a ) . On the other hand, we get

Finally, a ′ and a ′′ must coincide since, otherwise, we have a ′ ⋅ a ′′ = 0 , which would imply br D G ⁡ ( a ) = 0 , a contradiction. ∎

Remark 3.5.

(a) Note that the statement of Lemma 3.4 is true under more general assumptions. All we need is that D and Q are p-subgroups of G with Q ⁢ ⊴ ⁢ D and N G ⁢ ( D ) ≤ N G ⁢ ( Q ) .

(b) An immediate consequence of Lemma 3.4 is the main result of [11]: the Brauer map br Q determines a bijection between the blocks of b ⁢ 𝒪 ⁢ G and those of b ′ ⁢ 𝒪 ⁢ N G ⁢ ( Q ) , and this correspondence coincides with the Brauer correspondence between the covering blocks of b and of b ′ respectively. Recall also that any block a of 𝒪 ⁢ G covering b has a defect group D such that Q ≤ D ≤ P and D ∩ H = Q .

Remark 3.6.

The Brauer quotient of C is

Note that, since Q ⊆ H , C ⁢ ( Q ) is a G ¯ -graded G ¯ -algebra. The image of the restriction of br Q to C G ¯ ⁢ [ b ] is the strongly G ¯ ⁢ [ b ] -graded G ¯ -subalgebra ( C G ¯ ⁢ [ b ] ) ⁢ ( Q ) of b ′ ⁢ k ⁢ C G ⁢ ( Q ) N H ⁢ ( Q ) .

Theorem 3.7.

The Brauer homomorphism br Q induces an isomorphism of G ¯ -graded G ¯ -acted algebras between C ¯ ⁢ [ b ] and C ′ ¯ . Furthermore, this induces a bijection between the blocks of G covering b and the blocks of N G ⁢ ( Q ) covering b ′ , which coincides with the Harris–Knörr correspondence.

Proof.

Note that C ′ = b ′ ⁢ 𝒪 ⁢ C G ⁢ ( Q ) N H ⁢ ( Q ) ⊕ S , where S consists of N H ⁢ ( Q ) -fixed elements belonging to the radical of b ′ ⁢ 𝒪 ⁢ N G ⁢ ( Q ) . It follows that S = ⊕ x ¯ ∈ T C x ¯ ′ for some subset T of G ¯ ∖ G ¯ ⁢ [ b ′ ] . We immediately deduce the equality

hence the composition

is a well-defined epimorphism of G ¯ -graded G ¯ -algebras. Consider the diagram

For any x ¯ ∈ G ¯ ∖ G ¯ ⁢ [ b ] , we have φ ⁢ ( C x ¯ ) = 0 because, otherwise, C x ¯ would contain a homogeneous unit. If x ¯ ∈ G ¯ ⁢ [ b ] , then φ ⁢ ( C x ¯ ) is the x ¯ -component of C ¯ ′ . Furthermore, φ ⁢ ( J ⁢ ( C 1 ) ) = 0 since

This proves that the kernel of φ is J gr ⁢ ( C ) ; hence φ ¯ is an isomorphism, and then it also follows that G ¯ ⁢ [ b ] ≃ G ¯ ⁢ [ b ′ ] . The commutativity of the remaining squares of the diagram is easily checked.

The isomorphism φ ¯ induces a bijection between the blocks of G covering b and the blocks of N G ⁢ ( Q ) covering b ′ by lifting idempotents in the above diagram via the canonical maps C → C ¯ and C ′ → C ¯ ′ . The same bijection is obtained by lifting idempotents via the composition

By Lemma 3.4, this bijection coincides with the Brauer correspondence, so the theorem is proved. ∎

Corollary 3.8.

The composition of the Brauer map with the Fong–Reynolds equivalence induces an isomorphism C ¯ ≃ C ¯ of G ¯ ⁢ [ b ] -graded G ¯ -acted algebras.

4.1.

Consider the G ¯ -graded algebras A = b ⁢ 𝒪 ⁢ G and A ′ = b ′ ⁢ 𝒪 ⁢ G ′ as in Section 2. Assume that the ( A , A ′ ) -bimodule M ~ and its 𝒪 -dual M ~ * induce a G ¯ -graded Morita equivalence between A and A ′ . By [14, Corollary 5.1.4], there is an isomorphism

of G ¯ -graded G ¯ -acted algebras, which is defined as follows. By the second Morita theorem [10, Theorem 12.12], there are G ¯ -graded bimodule isomorphisms

such that

Denote by M, M * , φ and ψ the 1-components of M ~ , M ~ * , φ ~ and ψ ~ , respectively. Since M and M * induce a Morita equivalence between B and B ′ , there are finite sets I and J and the elements m j ∗ , n i ∗ ∈ M ∗ and m j , n i ∈ M , i ∈ I , j ∈ J such that

Then, for all c ∈ C A ⁢ ( B ) , we have that

4.2.

In particular, we may take an idempotent i ∈ γ and the strongly G ¯ -graded algebra A γ = i ⁢ A ⁢ i in the place of A ′ . Since A ⁢ i ⁢ A = A , there is a finite set J and elements u j , v j ∈ A such that ∑ j ∈ J u j ⁢ i ⁢ v j = 1 A . Then we get the isomorphism

with inverse given by c ′ ↦ ∑ j ∈ J u j ⁢ c ′ ⁢ v j , for all c ′ ∈ C i ⁢ A ⁢ i ⁢ ( i ⁢ B ⁢ i ) .

This idea also applies in the case of the Morita equivalence between A and S ⊗ A , where S = M r ⁢ ( 𝒪 ) is a matrix algebra.

Theorem 4.3.

With the notations of Section 2, assume that there is a G ¯ -graded basic Morita equivalence between A = b ⁢ O ⁢ G and A ′ = b ′ ⁢ O ⁢ G ′ . Denote C ¯ = C ¯ A ⁢ ( B ) , C ¯ ′ = C ¯ A ′ ⁢ ( B ′ ) , C ¯ = C ¯ A ⁢ ( B ) and C ¯ ′ = C ¯ A ′ ⁢ ( B ′ ) . Then the diagram

of G ¯ ⁢ [ b ] -graded G ¯ -algebras is commutative, where the horizontal isomorphisms are induced by the G ¯ -graded basic Morita equivalences given by the hypothesis and by 2.6, while the vertical isomorphisms come from Corollary 3.8.

Proof.

Assume that the basic Morita equivalence is induced by the G ¯ -graded ( A , A ′ ) -bimodule M ~ . Let A γ = i ⁢ A ⁢ i and A γ ′ ′ = i ′ ⁢ A ′ ⁢ i ′ . Then i ⁢ M ~ ⁢ i ′ induces a G ¯ -graded Morita equivalence between A γ and A γ ′ ′ , which is also given by an embedding f : A γ → S ⊗ A γ ′ ′ of P-interior G ¯ -graded algebras, as in 2.5.

We also use the G ¯ -graded basic Morita equivalence between 𝒜 = 𝒪 ⁢ N G ⁢ ( Q δ ) ⁢ b δ and 𝒜 ′ = 𝒪 ⁢ N G ′ ⁢ ( Q δ ′ ) ⁢ b δ ′ ′ , as recalled in 2.6. Let 𝒜 λ be the source algebra of 𝒜 , and 𝒜 λ ′ ′ the source algebra of 𝒜 ′ .

Now, consider the following diagram, in which all the maps are isomorphisms of G ¯ ⁢ [ b ] -graded G ¯ -algebras.

On the top face and the bottom face of the diagram, the maps come from the Morita equivalences A ∼ A ′ , 𝒜 ∼ 𝒜 ′ and the embedding of the respective source algebras, as in 4.2; it follows immediately that the top face and the bottom face are commutative.

On the right face and the left face, the horizontal maps come from the embeddings of the source algebras as in 4.2, while the vertical maps are induced by the Brauer map br Q , according to Theorem 3.7; hence the right face and the left face are commutative as well.

Recall that by [6, Section 3 and Remark 4.5.2], the extended Brauer quotient of A γ can be obtained by applying the usual Brauer construction to

By 4.2 and the above remarks, we deduce that the back face is commutative.

Consequently, the front face is also commutative. ∎

Corollary 4.4.

Let G ′ = N G ⁢ ( Q ) , H ′ = N H ⁢ ( Q ) , and let b ′ ∈ Z ⁢ ( O ⁢ H ′ ) be the Brauer correspondent of b. Assume that M ~ induces a G ¯ -graded basic Morita equivalences between A and A ′ . Then the bijection between the blocks of A and the block of A ′ coincides with the Harris–Knörr correspondence.

Proof.

Observe that our assumptions imply that 𝒜 = 𝒜 ′ = b δ ⁢ 𝒪 ⁢ N G ⁢ ( Q δ ) . By Theorem 3.7, Corollary 3.8 and Theorem 4.3, we have the commutative diagram

of isomorphisms of G ¯ ⁢ [ b ] -graded G ¯ -acted algebras, where the maps Φ ¯ and Φ ¯ ′ are induced by the given Morita equivalence. More precisely, Φ ¯ ′ is induced by the Fong–Reynolds equivalence between A ′ and 𝒜 , and a G ¯ -graded basic Morita auto-equivalence of 𝒜 . By [17, Lemma 4.5] (see also [4, Remark 9.6]), it follows that this auto-equivalence is the identity functor. Consequently, Φ ¯ ′ is the identity map of C ¯ ′ ; hence the isomorphism Φ ¯ is given by the Brauer map br Q . ∎