The Endomorphism Ring of a Galois Azumaya Extension

Transcription

1 International Journal of Algebra, Vol. 7, 2013, no. 11, HIKARI Ltd, The Endomorphism Ring of a Galois Azumaya Extension Xiaolong Jiang Department of Mathematics Sun Yat-Sen University, Guangzhou, P.R. China mcsjxl@mail.sysu.edu.cn George Szeto Department of Mathematics Bradley University, Peoria, Illinois USA szeto@bradley.edu Copyright c 2013 Xiaolong Jiang and George Szeto. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Let B be a ring, C the center of B, G a finite automorphism group of B, and B G = {b B g(b) =b} for each g G denoted by D. If B is a Galois extension of D with Galois group G such that D is an Azumaya C G -algebra, then two expressions of the ring Hom( D B, D B) of the left D-module endomorphisms of B are given, one in terms of the skew group ring of G over V B (D), where V B (D) is the commutator subring of D in B and G is the automorphism group of V B (D) induced by G, and the other one in terms of the opposite ring of B. Mathematics Subject Classification: 13B05 Keywords: Endomorphism rings, Azumaya algebras, Galois extensions, Galois Azumaya extensions, Skew group rings

2 528 Xiaolong Jiang and George Szeto 1 Introduction Let B be a ring with 1, G a finite automorphism group of B, C the center of B, and B G = {b B g(b) =b} for each g G. As defined in [1], B is called a Galois Azumaya extension of B G with Galois group G if B is a Galois extension of B G which is an Azumaya C G -algebra. A lot of properties of a Galois Azumaya extension are given in [1, 4, 5, 7]. It is shown that B is a Galois Azumaya extension if and only if the ring Hom C G(B,B) of the C G -module endomorphisms of B is an Azumaya C G -algebra (([1, Theorem 1]) such that Hom C G(B,B) = B G Hom C G(Δ, Δ) as Azumaya C G -algebras, where Δ is the commutator subring of B G in B (([1, Theorem 1])). Denote B G by D. The purpose of the present paper is to study Hom( D B, D B) of the left D-module endomorphisms of B. Denote the subring of Hom( D B, D B)of the right multiplications by the elements of B by μ B, and the subring of the endomorphisms of B as a D-bimodule by Hom( D B D, D B D ). We shall show that Hom( D B, D B) = μ D Hom( D B D, D B D ) as Azumaya C G -algebras. Thus two expressions of Hom( D B, D B) are derived, one in terms of skew group ring of G over Δ where G is induced by G, and the other one in terms of the opposite ring of B. 2 Preliminary Throughout the paper, let B be a ring with 1, C the center of B, A a subring of B with the same 1. As given in [1, 2], B is called a separable extension of A if the multiplication map: B A B B splits as a B-bimodule homomorphism. In particular, if A C, a separable extension B of A is called a separable A- algebra, and if A = C, a separable extension B of A is called an Azumaya C-algebra. If B A B is isomorphic to a direct summand of a finite direct sum of B as a B-bimodule, then B is called a Hirata separable extension of A. Itis known that an Azumaya algebra is a Hirata separable extension. Let G be a finite automorphism group of B and B G = {b B g(b) =b} for each g G. If there exist elements {a i,b i in B, i =1, 2,,m for some integer m} such that m i=1 a ig(b i )=δ 1,g for each g G, then B is called a Galois extension of B G with Galois group G, and {a i,b i } is called a G-Galois system for B. A Galois extension B of B G is called a Galois Azumaya extension if B G is an Azumaya C G -Azumaya algebra as studied in [1, 4, 5, 7]. The commutator subring of A in B is denoted by V B (A), and the opposite ring of B by B o. The endomorphism ring of a left B-module M is denoted by Hom( B M, B M), and the endomorphism ring of M as a B-bimodule by Hom( B M B, B M B ).

3 The endomorphism ring of a Galois Azumaya extension Endomorphism Rings In this section, we keep the definitions and notations in Section 2, and let B be a Galois Azumaya extension of B G with Galois group G. Denote B G by D, and the commutator subring of B G in B by Δ. We shall show two expressions for Hom( D B, D B), one in terms of the skew group ring of G over Δ where G is the automorphism group of Δ induced by G, and the other one in terms of the opposite ring of B. We begin with an expression of Hom( A M, A M), the endomorphism ring of a finitely generated and projective left module M over a separable algebra A. Lemma 3.1 Let A be a separable algebra over a commutative ring R and M a finitely generated and projective left A-module. Then Hom( A M, A M) is an Azumaya-C-algebra where C is the center of A. Proof. Since M is a finitely generated and projective left module over a separable algebra A, the opposite ring of Hom( A M, A M) is an Azumaya algebra over the center of A ([8, Theorem 5-(2)]). Thus Hom( A M, A M) is an Azumaya C-algebra where C is the center of A. Lemma 3.2 Let M be a unitary bimodule over an algebra A, and μ A the subring of Hom( A M, A M) of the right multiplications by the elements of A on M. Then A o = μa where A o is the opposite ring of A. Proof. Let r μ r for each r A o. Then it is straightforward to verify that A o = μa. Theorem 3.3 Let A be a separable R-algebra, M a unitary A-bimodule and a finitely generated and projective left A-module, and Ω = Hom( A M, A M). Then Ω = μ A Hom( A M A, A M A ) as Azumaya C-algebras where C is the center of A. Proof. By Lemma 3.1, Ω is an Azumaya C-algebra, and by Lemma 3.2, A o = μ A. Since A is a separable R-algebra by hypothesis, A is an Azumaya C- algebra ([2, Theorem 3.8]). Hence A o is an Azumaya C-algebra; and so μ A is an Azumaya C-subalgebra of Ω. Thus Ω = μ A V Ω (μ A ) as Azumaya C- algebras ([2, Theorem 4.3]). Next we claim that V Ω (μ A ) = Hom( A M A, A M A ). In fact, for any f V Ω (μ A ), d A, we have f(μ d )=(μ d )f, sofμ d (m) = μ d f(m) for all m M. Hence f(md) =(f(m))d. Thusf Hom( A M A, A M A ). Conversely, for any f Hom( A M A, A M A ),d A, we have fμ d (m) =f(md) = (f(m))d = μ d (f(m)) = μ d f(m) for all d A; and so f V Ω (μ A ). Therefore V Ω (μ A ) = Hom( A M A, A M A ). But then Ω = μ A Hom( A M A, A M A ) as Azumaya C-algebras.

4 530 Xiaolong Jiang and George Szeto Since a Galois Azumaya extension B of B G with Galois group G is a left finitely generated and projective B G -module, the following corollary is immediate. Corollary 3.4 Let B be a Galois Azumaya extension of B G with Galois group G, and C the center of B. Denote B G by D. Then Hom( D B, D B) = μ D Hom( D B D, D B D ) as Azumaya C G -algebras. Keeping the notations in the above corollary, we show an expression of Hom( D B, D B) in terms of the skew group ring (V B (D) G )ofg over V B (D) where D = B G and G is the automorphism group of V B (D) induced by G. Theorem 3.5 Let B be a Galois Azumaya extension of B G which is denoted by D. Then Hom( D B, D B) = μ D (V B (D) G ) as Azumaya C G -algebras where G is the automorphism group of V B (D) induced by G. Proof. By Corollary 3.4, Hom( D B, D B) = μ D Hom( D B D, D B D ), so it suffices to show that Hom( D B D, D B D ) = (V B (D) G ). Since B is a Galois Azumaya extension of D with Galois group G, α : B G = Hom(B D,B D )byσ i r i g i Σ i r i g i (b) for all b B. But Hom( D B D, D B D ) Hom(B D,B D ), so for each d D, Σ i r i g i V B (D) G, Σ i r i g i (db) =d(σ i r i g i (b)) for all b B. Hence α(v B (D) G ) Hom( D B D, D B D ) Hom(B D,B D ). On the other hand, since B is a Galois Azumaya extension again, B = D C G (V B (D) ([1, Theorem 3.2]). Hence Hom(B D,B D ) = B G = D C G (V B (D) G ). Similar to Corollary 3.4, let λ B be the subring of Hom(B D,B D ) which are the left multiplications by the elements in B. Then D = λ D Hom(B D,B D ) as an Azumaya C G -subalgebra. But then Hom(B D,B D ) = D C G (V B (D) G ) = λ D C G Hom( D B D, D B D ) as Azumaya C G -algebras. Therefore Hom( D B D, D B D ) = (V B (D) G ). Consequently, Hom( D B, D B) = μ D Hom( D B D, D B D ) = μ D (V B (D) G )as Azumaya C G -algebras. Next we show another expression of Hom( D B, D B) in terms of the opposite ring of B. Theorem 3.6 Let B be a Galois Azumaya extension of B G denoted by D, B o the opposite ring of B and Ω the endomorphism ring Hom( D B, D B). Then Ω = (B o G ) o, where G is the automorphism group of B o induced by G. Proof. We define an automorphism group G of B o induced by G such that for each g G,g (b) =g(b), b B o,g G. That is, for x, y B o,g (xoy) = g(yx) =g(y)g(x) =g(x)og(y) =g (x)og (y) and g (x + y) =g(x + y) = g (x)+g (y). Next we claim that B o is a Galois extension of D o with Galois group G. In fact, let {a i,b i B, i =1, 2,,m} for some integer m such

5 The endomorphism ring of a Galois Azumaya extension 531 that m i=1 a ig(b i )=δ 1,g for each g G. Then m i=1 g 1 (a i )(b i )=δ 1,g. Hence m i=1 (b i)og 1 (a i )=δ 1,g.Thus m i=1 (b i)o(g ) 1 (a i )=δ 1,g ; and so {b i,a i B o } is a Galois system for the Galois extension B o of D o with Galois group G. Since Ω = Hom( D B, D B), Ω o = Hom((B o ) D o, (B o ) D o) = B o G. Therefore Ω = (B o G ) o. 4 Hirata Separable Extensions As given in Section 3, let B be a Galois Azumaya extension of D with Galois group G, C the center of B, Δ=V B (D), and Ω = Hom( D B, D B). Then Ω is an Azumaya C G -algebra by Lemma 3.1, and by Lemma 3.2, μ Δ μ B Ω and μ D Ω such that μ Δ = Δ o, μ B = B o and μ D = D o by Lemma 3.2. In this section, we shall show that Ω is a Hirata separable extension of μ B,μ Δ and μ D respectively. We shall employ the following property due to S. Ikehata. Lemma 4.1 ([3, Theorem 1]) Let A be an Azumaya algebra and E a subalgebra of A. IfA is a left projective E-module, then A is a Hirata separable extension of E. Lemma 4.2 By keeping the notations in this section, μ B, μ Δ and μ D are separable subalgebras of Ω. Proof. By Corollary 3.5, Ω is an Azumaya C G -algebra containing μ B which is isomorphic with B o. Since B is a Galois extension of D such that D is an Azumaya C G -algebra, B is a separable C G -algebra. Hence B o is a separable C G -algebra. Thus μ B and μ D are separable subalgebras of Ω. Similarly, noting that Δ is a separable C G -algebra, we have that μ Δ is a separable subalgebra of Ω. Theorem 4.3 By keeping the notations in this section, if B is a Galois Azumaya extension of D with Galois group G, then Ω is a Hirata separable extension of μ B, μ Δ and μ D respectively. Proof. By Lemma 4.2, μ B,μ Δ and μ D are separable subalgebras of Ω. Since Ω is an Azumaya C G -algebra by Corollary 3.5, Ω is a projective left module over μ B,μ Δ and μ D respectively ([2, Proposition 2.3]). Thus, by Lemma 4.1, Ωis a Hirata separable extension of μ B,μ Δ and μ D respectively.

6 532 Xiaolong Jiang and George Szeto Acknowdegement: This work was done in Summer, 2012 when the second author visited the Department of Mathematics, Sun Yat-Sen University, China and participated in the International Conference on Algebras and Rings at Sun Yat-Sen University. The second author would like to thank Sun Yat-Sen University for her hospitality. References [1] R. Alfaro and G. Szeto, Skew Group Rings Which Are Azumaya, Comm. in Algebra, 22(6) 1995, [2] F.R. DeMeyer and E. Ingraham, Separable Algebras over Commutative Rings, 181, Springer Verlag, Berlin, Heidelberg, New York, [3] S. Ikehata, Note on Azumaya Algebras and H-Separable Extensions, Math. J. Okayama Univ., , [4] X.L. Jiang and G. Szeto, On Composition Series of a General Azumaya Galois Extension, South Asian J. Math., , [5] X.L. Jiang and G. Szeto, On Galois Skew Group Rings of a Separable Algebra, Pure Math. Sci., , [6] T. Kanzaki, On Commutator Ring and Galois Theory of Separable Algebras, Osaka J. Math., , [7] A. Paques, V. Rodrigues and A. Sant ana, Galois Correspondences for Partial Galois Azumaya Extensions, J. Algebra and Its Applications, , [8] K. Sugano, Note on Separability of Endomorphism Rings, Hokkaido Math. J., XXI 1971, Received: September 15, 2013