On Graded Simple Algebras

Let R be a commutative ring and A be an algebra over R which is finitely generated as an R-module. If for any maximal ideal m of R, the algebra A ⊗ R R/m is a central simple R/m-algebra, then A is called an Azumaya algebra. This is equivalent to saying that A is a faithfully projective R-module, and the natural R-algebra homomorphism A ⊗ R A op → End R (A) is an isomorphism (see [9, Thm. III.5.1.1]). In [6] it was proven that for an Azumaya algebra A free over its centre R of rank n, the Quillen K-groups of A are isomorphic to the K-groups of its centre up to n-torsion, i.e., (I) Boulagouaz [2,Prop. 5.1] and Hwang and Wadsworth [8,Cor. 1.2] observed that a graded central simple algebra, graded by a torsion free abelian group, is an Azumaya algebra; thus its K-theory can be estimated by the above result. This note studies graded central simple algebras graded by an arbitrary abelian group. We observe that a graded central simple algebra, graded by an abelian group, is a (graded) Azumaya algebra (Theorem 2.4), which extends the result of [2,8] to graded rings in which the grade group is not totally ordered. Thus its K-theory can also be estimated by (I). We then study the graded K-theory of graded Azumaya algebras. We introduce an abstract functor called a graded D-functor defined on the category of graded Azumaya algebras over a commutative graded ring R (Definition 3.3), and show that the range of this functor is the category of bounded torsion abelian groups (Theorem 3.4). We then prove that the kernel and cokernel of the K-groups are graded D-functors, which allows us to show that, for a graded Azumaya algebra A free over R, we have a relation similar to (I) in the graded setting (see Theorem 3.5). This note is organised as follows. We begin Section 2 by recalling some definitions, many of which can be found in [8,10], though not always in the generality that we require. We then study graded central simple algebras graded by an arbitrary abelian group and observe that they are Azumaya algebras (Theorem 2.4). In order to do so, we need to rewrite the standard results from the literature in the setting of arbitrary graded rings. We observe that the tensor product of two graded central simple R-algebras is graded central simple (Propositions 2.2 and 2.3). This result has been proven by Wall for Z/2Z -graded central simple algebras (see [11,Thm. 2]), and by Hwang and Wadsworth for R-algebras with a totally ordered, and hence torsion-free, grade group (see [8,Prop. 1.1]). In Section 3 we study the graded K-theory of graded Azumaya algebras by introducing an abstract functor called a graded D-functor, which is defined on the category of graded Azumaya algebras over a commutative graded ring R (Definition 3.3). Similar concepts have been studied in [4,5,6], where functors have been defined on the category of central simple algebras and the category of Azumaya algebras. We begin this section by recalling some basic definitions in the graded setting. A unital ring R = γ∈Γ R γ is called a graded ring if Γ is a group, each R γ is a subgroup of (R, +) and R γ • R δ ⊆ R γ+δ for all γ, δ ∈ Γ. We remark that although Γ is initially an arbitrary group which is not necessarily abelian, we will write Γ as an additive group. The elements of R γ are called homogeneous of degree γ and we write deg the set of homogeneous elements of R. Here R * is the set of invertible elements of R. Note that the support of R is not necessarily a group, and that 1 R is homogeneous of degree zero. An ideal Let S = γ∈Γ ′ S γ be another graded ring and suppose there is a group ∆ containing Γ and Γ ′ as subgroups. The graded ring R can be written as R = γ∈∆ R γ with R γ = 0 if γ ∈ ∆ \ Γ R , and similarly for S. Then a graded ring homomorphism f : R → S is a ring homomorphism such that f (R γ ) ⊆ S γ for all γ ∈ ∆. If f is bijective, then f is a graded isomorphism. A graded ring R is said to be graded simple if the only graded two-sided ideals of R are {0} and R. A graded ring D = γ∈Γ D γ is called a graded division ring if every non-zero homogeneous element has a multiplicative inverse, where it follows easily that Γ D is a group. We say that a group (Γ, +) acts freely (as a left action) on a set Γ ′ if for all γ, γ ′ ∈ Γ, δ ∈ Γ ′ , we have γ + δ = γ ′ + δ implies γ = γ ′ , where γ + δ denotes the image of δ under the action of γ. A graded left R-module M is defined to be an R-module with a direct sum decomposition M = γ∈Γ ′ M γ , where M γ are abelian groups and Γ acts freely on the set Γ ′ , such that R γ • M λ ⊆ M γ+λ for all γ ∈ Γ R , λ ∈ Γ ′ . From now on, unless otherwise stated, a graded module will mean a graded left module. A graded R-module M is said to be graded simple if the only graded submodules of M are {0} and M, where graded submodules are defined in the same way as graded ideals. A graded free R-module M is defined to be a graded R-module which is free as an R-module with a homogeneous base. Let N = γ∈Γ ′′ N γ be another graded R-module, such that there is a group ∆ containing Γ ′ and Γ ′′ as subgroups, where Γ acts freely on ∆. A graded R-module homomorphism f : M → N is an R-module homomorphism such that f (M δ ) ⊆ N δ for all δ ∈ ∆. Let Hom R-gr-Mod (M, N) denote the group of graded R-module homomorphisms, which is an additive subgroup of Hom R (M, N). A graded R-module homomorphism may also shift the grading on N. For each δ ∈ ∆, we have a subgroup of Hom R (M, N) of δ-shifted homomorphisms Hom R (M, If M is finitely generated, then HOM R (M, N) = Hom R (M, N) (see [10,Cor. 2.4.4]). Note that R(δ) ∼ = gr R as graded R-modules if and only if δ ∈ Γ * R . In the following Proposition we are considering graded modules over graded division rings. We note that the grade groups here are defined as above; that is, we do not initially assume them to be abelian or torsion-free. Proposition 2.1. Let Γ be a group which acts freely on a set Γ ′ . Let R = γ∈Γ R γ be a graded division ring and M = γ∈Γ ′ M γ be a graded module over R. Then M is a graded free R-module. More generally, any linearly independent subset of M consisting of homogeneous elements can be extended to form a homogeneous basis of M. Furthermore, any two homogenous bases have the same cardinality and if N is a graded submodule of M, then Proof. The proof follows the standard proof in the non-graded setting (see for example [ A graded field R = γ∈Γ R γ is defined to be a commutative graded division ring. Note that the support of a graded field is an abelian group. Let Γ ′ be another group such that there is a group ∆ containing Γ and Γ ′ as subgroups. A graded R-algebra A = γ∈Γ ′ A γ is a graded ring which is an R-algebra such that the associated ring homomorphism ϕ : R → Z(A) is a graded homomorphism. A graded algebra A over R is said to be a graded central simple algebra over R if A is a graded simple ring, Z(A) = R, and [A : R] < ∞. Note that since the centre of A is a graded field, by Proposition 2.1, A is graded free over its centre, so the dimension of A over R is uniquely defined. Let A = γ∈Γ ′ A γ and B = γ∈Γ ′′ B γ be graded R-algebras, such that there is a group ∆ containing Γ ′ and Γ ′′ as subgroups with Γ ′ ⊆ Z ∆ (Γ ′′ ), where Z ∆ (Γ ′′ ) is the set of elements of ∆ which commute with Γ ′′ . Then A ⊗ R B has a natural grading as a graded R-algebra given by A ⊗ R B = γ∈∆ (A ⊗ R B) γ where: Note that the condition Γ ′ ⊆ Z ∆ (Γ ′′ ) is needed to ensure that the multiplication on A ⊗ R B is well defined. Moreover, for the following Proposition, we require that the group ∆ is an abelian group. For a graded ring A = γ∈Γ ′ A γ , let A op denote the opposite graded ring, where the grade group of A op is the opposite group Γ ′ op . So, for a graded R-algebra A, in order to define A ⊗ R A op , we note that the grade group of A must be abelian. Thus we will now assume that for a graded R-algebra A = γ∈Γ ′ A γ , the group Γ ′ is in fact an abelian group. By combining Propositions 2.2 and 2.3, we show that the tensor product of two graded central simple R-algebras is graded central simple, where the grade groups Γ ′ and Γ ′′ , as below, are abelian but not necessarily torsion-free. This has been proven by Wall for graded central simple algebras with Z/2Z as the support (see [11,Thm. 2]), and by Hwang and Wadsworth for R-algebras with a torsion-free grade group (see [8,Prop. 1.1]). Proposition 2.2. Let Γ, Γ ′ and Γ ′′ be abelian groups such that there is an abelian group ∆ containing Γ, Γ ′ and Γ ′′ as subgroups. Let R = γ∈Γ R γ be a graded field and let A = γ∈Γ ′ A γ and B = γ∈Γ ′′ B γ be graded R-algebras. If A is graded central simple over R and B is graded simple, then A ⊗ R B is graded simple. Proof. Let I be a graded two-sided ideal of A ⊗ B, with I = 0. We will show that A ⊗ B = I. First suppose a ⊗ b is a homogeneous element of I, where a ∈ A h and b ∈ B h . Then A is the graded two-sided ideal generated by a, so there exist a i , a Similarly, B is the graded two-sided ideal generated by b. Repeating the above argument shows that 1 ⊗ 1 is an element of I, proving I = A ⊗ B in this case. Now suppose there is an element and k as small as possible. Note that since x is homogeneous, deg(a j ) + deg(b j ) = deg(x) for all j. By the above argument we can suppose that k > 1. As above, since Thus, without loss of generality, we can assume that a k = 1. Then a k and a k-1 are linearly independent, since if which is homogeneous and thus gives a smaller value of k. Thus a k-1 / ∈ R = Z(A), and so there is a homogeneous element a ∈ A with aa k-1 -a k-1 a = 0. Consider the commutator where the last summand is not zero. If the whole sum is not zero, then we have constructed a homogeneous element in I with a smaller k. Otherwise suppose the whole sum is zero, and write c = aa k-1 -a k-1 a. Then we can write c ⊗b k-1 = k-2 j=1 x j ⊗b j where x j = -(aa j -a j a). Since 0 = c ∈ A h and A is the graded two-sided ideal generated by c, using the same argument as above, we have . . , b k-1 are linearly independent homogeneous elements of B, they can be extended to form a homogeneous basis of B, say {b i }, by Proposition 2.1. Then {1 ⊗ b i } forms a homogeneous basis of A ⊗ R B as a graded A-module, so in particular they are A-linearly independent, which is a contradiction to equation (III). This reduces the proof to the first case. Proposition 2.3. Let Γ, Γ ′ and Γ ′′ be abelian groups such that there is an abelian group ∆ containing Γ, Γ ′ and Γ ′′ as subgroups. Let R be a graded field and let A = γ∈Γ ′ A γ and B = γ∈Γ ′′ B γ be graded R-algebras. If A ′ ⊆ A and B ′ ⊆ B are graded subalgebras, then In particular, if A and B are central over R, then A ⊗ R B is central. Proof. First note that by Proposition 2.1, A ′ , B ′ , Z A (A ′ ) and Z B (B ′ ) are free over R, and thus one can consider The inclusion ⊇ follows immediately. For the reverse inclusion, let x ∈ Z A⊗B (A ′ ⊗B ′ ). Let b 1 , . . . , b n be a homogeneous basis for B over R which exists thanks to Proposition 2.1. Then x can be written uniquely as [7,Thm. IV.5.11]). For every a ∈ A ′ , (a ⊗ 1)x = x(a ⊗ 1), so By the uniqueness of this representation we have x i a = ax i , so that x i ∈ Z A (A ′ ) for each i. Thus we have shown that x ∈ Z A (A ′ ) ⊗ R B. Similarly, let c 1 , . . . , c k be a homogeneous basis of Z A (A ′ ). Then we can write x uniquely as A similar argument to above shows that y i ∈ Z B (B ′ ), completing the proof. Theorem 2.4. Let Γ and Γ ′ be abelian groups such that there is an abelian group ∆ containing Γ and Γ ′ as subgroups. Let A = γ∈Γ ′ A γ be a graded central simple algebra over the graded field R = γ∈Γ R γ . Then A is an Azumaya algebra over R. Proof. Since A is graded free of finite rank, it follows that A is faithfully projective over R. There is a natural graded R-algebra homomorphism ψ : A ⊗ R A op → End R (A) defined by ψ(a ⊗ b)(x) = axb where a, x ∈ A, b ∈ A op . By Proposition 2.2, the domain is graded simple, so ψ is injective. Hence the map is surjective by dimension count, using equation (II). This shows that A is an Azumaya algebra over R, as required. For a graded field R, this theorem shows that a graded central simple R-algebra, graded by an abelian group Γ ′ , is an Azumaya algebra over R. One can not extend the theorem to non-abelian grading. Consider a finite dimensional division algebra D and a group G and consider the group ring DG. This is clearly a graded simple algebra (in fact a graded division ring) and if G is abelian the above theorem implies that DG is an Azumaya algebra. However in general, for an arbitrary group G, DG is not always an Azumaya algebra. In fact DeMeyer and Janusz [3] have shown the following: the group ring RG is an Azumaya algebra if and only if R is Azumaya, [G : Z(G)] < ∞ and [G, G], the commutator subgroup of G, has finite order m and m is invertible in R. Corollary 2.5. Let Γ and Γ ′ be abelian groups such that there is an abelian group ∆ containing Γ and Γ ′ as subgroups. Let A = γ∈Γ ′ A γ be a graded central simple algebra over its graded centre R = γ∈Γ R γ of degree n. Then for any i ≥ 0, Proof. By Theorem 2.4, a graded central simple algebra A over R is an Azumaya algebra. From Proposition 2.1, since R is a graded field, A is a free R-module. The corollary now follows immediately from [6, Thm. 6] (or see (I)), since A is an Azumaya algebra free over its centre. Corollary above shows that the K-theory of a graded division algebra is very close to the K-theory of its centre, where this follows immediately from the corresponding result in the non-graded setting (see [6,Thm. 6]). Note that for the K-theory of a graded central simple algebra A, we are considering K i (A) = K i (P(A)), where P(A) denotes the category of finitely generated projective A-modules. But in the graded setting, there is also the category of graded finitely generated projective modules over a given graded ring, which is what we consider here. Below we define an abstract functor called a graded D-functor (Definition 3.3), and show that its range is the category of bounded torsion abelian groups. We use this to show that a similar result to the above Corollary also holds when we consider graded projective modules over a graded ring. Let Γ be an abelian group and let R = γ∈Γ R γ be a commutative Γ-graded ring. We will consider the category R-gr-Mod which is defined as follows: the objects are Γ-graded left R-modules, and for two objects M, N in R-gr-Mod, the morphisms are defined as Through out this section, unless otherwise stated, we will assume that Γ is an abelian group, R is a fixed commutative Γ-graded ring and all graded rings, graded modules and graded algebras are also Γ-graded. Let A be a graded ring and let (d) = (δ 1 , . . . , δ n ), where each δ i ∈ Γ. Then we have a graded ring M n (A)(d), where M n (A)(d) means the n × n-matrices over A with the degree of the ij-entry shifted by δ iδ j . Thus, the ε-component of M n (A)(d) consists of matrices with the ij-entry in A ε+δ i -δ j . Consider where A(δ i ) γ is the γ-component of the δ i -shifted graded A-module A(δ i ). Note that for each i, 1 ≤ i ≤ n, the basis element e i of A n (d) is homogeneous of degree -δ i . Suppose M is a graded left A-module which is graded free with a finite homogeneous base {b 1 , . . . , b n }, where deg(b i ) = δ i . If we ignore the grading, it is well-known that End A (M) ∼ = M n (A). When we take the grading into account, we have that End A (M) ∼ = gr M n (A)(d) for (d) = (δ 1 , . . . , δ n ) (see [10,Prop. 2.10.5]). Note that this isomorphism does not depend on the order that the elements in the basis are listed. For some permutation π ∈ S n we have that {b π(1) , . . . , b π(n) } is also a homogeneous base of M. So for For a Γ-graded ring A, and as graded A-modules if and only if there exists , then there is a graded A-module homomorphism Since r is invertible, there is a matrix t ∈ GL m×n (A) with rt = I n and tr = I m . So there is an A-module homomorphism R t : A m (a) → A n (d), which is an inverse of R r . This proves that R r is bijective, and therefore it is a graded A-module isomorphism. Conversely, if φ : A n (d) ∼ = gr A m (a), then we can construct a matrix as follows. Let e i denote the basis element of A n (d) with 1 in the i-th entry and 0 elsewhere. Then let φ(e i ) = (r i1 , r i2 , . . . , r im ), and let r = (r ij ) n×m . It can be easily verified that r In the same way, using φ -1 : A m (a) → A n (d) construct a matrix t. Let e ′ i denote the i-th element of the standard basis for A m (a). Since for each i, and in a similar way for φ•φ -1 , we can show that rt = I n and tr Then it is immediate from the above Proposition that A n (d) ∼ = gr A n as graded A-modules if and only if (d) ∈ Γ * Mn(A) . A graded A-module P = γ∈Γ P γ is said to be graded projective (resp. graded faithfully projective) if P is projective (resp. faithfully projective) as an A-module. Note that a graded A-module P is projective as an A-module if and only if Hom A-gr-Mod (P, -) is an exact functor in A-gr-Mod. We use Pgr(A) to denote the category of graded finitely generated projective modules over A. The following Proposition proves a partial result of Morita equivalence (only in one direction), which we will use later in this section (after Definition 3.3). a -→ (a, 0, . . . , 0) ⊗ (1, 0, . . . 0) are graded A-module homomorphisms with σ • θ = id and θ • σ = id. Further shows that ψ and ϕ are mutually inverse equivalences of categories. A graded R-algebra A is called a graded Azumaya algebra if A is graded faithfully projective and A ⊗ R A op ∼ = gr End R (A). We let Az gr (R) denote the category of graded Azumaya algebras over R and Ab the category of abelian groups. Note that a graded R-algebra which is an Azumaya algebra (in the non-graded sense) is also a graded Azumaya algebra, since it is faithfully projective as an R-module, and the natural homomorphism A ⊗ R A op → End R (A) is clearly graded. So a graded central simple algebra over a graded field (as in Theorem 2.4) is in fact a graded Azumaya algebra. Definition 3.3. An abstract functor F : Az gr (R) → Ab is defined to be a graded Dfunctor if it satisfies the three properties below: (1) F(R) is the trivial group. (2) For any graded R-Azumaya algebra A and for any Note that these properties are well-defined since both R and M k (A)(d) are graded Azumaya algebras over R. We set K gr i (R) = K i (Pgr(R)), where Pgr(R) is the category of graded finitely generated projective R-modules and K i are the Quillen K-groups. Let A be a graded ring with graded centre R. Then the graded R-linear homomorphism R → A induces an exact functor Pgr(R) → Pgr(A), which, in turn, induces a group homomorphism K gr i (R) → K gr i (A). Then we have an exact sequence (IV) 1 → ZK gr i (A) → K gr i (R) → K gr i (A) → CK gr i (A) → 1 where ZK gr i (A) and CK gr i (A) are the kernel and cokernel of the map K gr i (R) → K gr i (A) respectively. Then CK gr i can be regarded as the following functor CK gr i : Az gr (R) -→ Ab A -→ CK gr i (A), and similarly for ZK gr i . We will now show that CK gr i is a graded D-functor. Property (1) is clear, since R is commutative so Proof. The argument before Theorem 3.4 shows that CK gr i (and in the same manner ZK gr i ) is a graded D-functor, and thus by the theorem CK gr i (A) and ZK gr i (A) are n 2 -torsion abelian groups. Tensoring the exact sequence (IV) by Z[1/n], since CK gr i (A) ⊗ Z[1/n] and ZK gr i (A) ⊗ Z[1/n] vanish, the result follows. Corollary 3.6 ([6], Thm. 6). Let A be an Azumaya algebra free over its centre R of rank n. Then for any i ≥ 0, Proof. By taking Γ to be the trivial group, this follows immediately from Theorem 3.5. Remark 3.7. Note that a graded division algebra A is strongly graded. By Dade's Theorem [10,Thm. 3.1.1], there is an additive functor from the category of A 0 -modules to the category of graded A-modules which induces an equivalence of categories. This implies that (VII) K i (A 0 ) ∼ = K gr i (A). Let D be a tame and Henselian valued division algebra with centre F of index n. Consider the associated graded division algebra gr(D) with centre gr(F ). We know gr (see [8]). If D is unramified over F , i.e., Γ D = Γ F , then the assumption of Theorem 3.5 on the homogenous basis is satisfied, so We end the note with an example of a graded Azumaya algebra such that its graded K-theory is not the same as the graded K-theory of its centre. Example 3.8. Consider the quaternion algebra H = R⊕Ri⊕Rj ⊕Rk. Then H is an Azumaya algebra over R and it is a Z 2 ×Z 2 -graded division ring. So H is in fact a graded Azumaya algebra, which is strongly Z 2 ×Z 2 -graded. By Dade's Theorem, K gr 0 (H) ∼ = K 0 (H 0 ) ∼ = K 0 (R) ∼ = Z. The centre Z(H) = R is a field and is trivially graded by Z 2 ×Z 2 . So K gr 0 Z(H) = K gr 0 (R) ∼ = Z⊕Z⊕Z⊕Z. We remark that the graded Azumaya algebra H does not satisfy the conditions of Theorem 3.5.