Azumaya algebras on local rings

MINISTRY OF EDUCATION AND TRAINING HANOI PEDAGOGICAL UNIVERSITY DEPARTMENT OF MATHEMATICS NGUYEN PHUONG ANH AZUMAYA ALGEBRAS ON LOCAL RINGS BACHELOR THESIS Hanoi – 2019 MINISTRY OF EDUCATION AND TRAINING HANOI PEDAGOGICAL UNIVERSITY DEPARTMENT OF MATHEMATICS NGUYEN PHUONG ANH AZUMAYA ALGEBRAS ON LOCAL RINGS BACHELOR THESIS Major: Geometry SUPERVISOR: PHAM THANH TAM Hanoi – 2019 Assurance I hereby declare that this thesis is my own work and to the best of my knowledge, it contains no material previously published or written by another person, nor material which to a substantial extent has been accepted for the award of any other degree or diploma at any educational institution, except where due acknowledgement is made in the thesis I also assure that all the help for this thesis has been acknowledged and that the results presented in the thesis has been identified clearly Ha Noi, May 16, 2019 Student Nguyen Phuong Anh Bachelor thesis NGUYEN PHUONG ANH Acknowledgement Firstly, I would like to express my deep respect and sincere gratitude to my supervisor MCs Pham Thanh Tam for the continuous support of my study as well as related research, for his patience, motivation and immense knowledge Without his precious guidance in all the time of research, it would not be possible to complete this thesis Besides my advisor, I would like to take this opportunity to thank to all teachers of the Department of Mathematics, Hanoi Pedagogical University No.2, the teachers in the geometry group as well as the teachers involved The lecturers have imparted valuable knowledge and facilitated for me to complete the course and the thesis Due to time, capacity and conditions are limited, so the thesis cannot avoid errors Therefore, I look forward to receiving valuable comments and recommendations from teachers and friends Ha Noi, May 16, 2019 Student Nguyen Phuong Anh Contents Assurance Acknowledgement Notation Preface Preliminaries 1.1 Modules and related notions 1.2 Endomorphism and matrix algebras 1.3 Tensor products of algebras 1.4 Semisimplicity 1.5 Central simple algebras 1.5.1 Simple rings 1.5.2 Center of algebras 4 10 12 13 14 15 Azumaya algebras over local rings 2.1 Azumaya algebras 2.2 Azumaya algebras over local rings 2.3 Some examples of Azumaya algebras 19 19 23 26 Conclusion 28 Bibliography 29 Bachelor thesis NGUYEN PHUONG ANH Notation CSA ⊕ Mi Central simple algebras The direct sum of Mi Π Mi The direct product of Mi i∈I i∈I Z(A) Center of A + A The additive group A-module B AB EndA (M ) The ring of endomorphisms of M over A Mn (A) The ring of n × n-matrices with coefficients in A o R The opposite ring of R J (R) The Jacobson radical of R Bachelor thesis NGUYEN PHUONG ANH Preface Mathematics is one of the important fields and serves so many of the sciences that it is a prerequisite for studying every scientific discipline The matrix algebras over a field have many especially applications in studying the structure of the linear automorphism of finite dimensional vector spaces that is generated to center simple algebras Azumaya algebras over field is a generalization of center simple algebras In particular, it is a generalization of matrix algebras on a field Such notion was introduced in [GA51] for the case of a field Azumaya algebras over local ring is an extension of Azumaya algebras over field to local rings The studying of Azumaya algebras on a local ring is really significant in extending the research of central simple algebras and matrix algebras An Azumaya algebra over a local ring R is a R-algebra A that is free and of finite rank r ≥ as an R-module, such that the tensor product with to Ao , the opposite algebra, is isomorphic to the matrix algebra Mr (R) Two Azumaya algebras A1 and A2 over R are similar if there exist locally free R-modules E1 and E2 of finite positive rank such that A1 ⊗End(E1 ) A2 ⊗End(E2 ), where End(Ei ) is the endomorphism ring of Ej The set of similar classes of Azumaya algebras with operation given by tensor product has form a group This group is called Brauer group of field R Based on the basic knowledge about algebraic geometry and desiring comprehensive improvement of Mathematics, researcher has selected a topic “Azumaya Algebras on local rings” for Bachelor Thesis Let us describe the content of this thesis In Chapter 1, our focus is the theory of central simple algebras over a ring First of all, we begin by reviewing, without proofs, some facts about modules and related notions Next, we will introduce algebras over arbitrary commutative rings as well as some examples of algebras Namely endomorphism algebras, matrix algebras, and tensor products of algebras will be presented in this chapter Furthermore, a brief explanation of the theory of semi-simple rings and modules, and its connection with central simple algebras is given In particular, we research the technique of changing the base ring of an algebra and more properties of central simple algebras Then splitting fields are studied, consisting of strictly maximal subfields of a central simple algebra The Skolem-Noether theorem will be concluded in this chapter In the second chapter, we will introduce about Azumaya Algebras and develop some of their properties Standard references for Azumaya algebras are [DI71] and Bachelor thesis NGUYEN PHUONG ANH [OS75] Azumaya algebras are a generalization of central simple algebras in which the center is allowed to be an arbitrary communication ring instead of a field Moreover, some interesting examples are given in this chapter to illustrate for Azumaya algebras In addition, central simple algebras over a field will be generalized to Azumaya algebras over local rings There are several equivalent ways to define Azumaya algebras Finally, we would like to present some interesting proofs and some examples related to Azumaya algebras as well Last but not least, the conclusion and bibliography about related problems to Azumaya algebras on local rings are presented to complete for Bachelor Thesis Chapter Preliminaries Azumaya algebras are introduced as generalized or global versions of central simple algebras So the first part of this research will be about central simple algebras In this chapter, we will recall and introduce some notions, namely modules and related notions, endomorphism algebras, matrix algebras, and tensor products of algebras will be presented in this chapter A brief explanation of the theory of semi-simple rings and modules, and its connection with central simple algebras is given Besides, we research the technique of changing the base ring of an algebra and more properties of central simple algebras 1.1 Modules and related notions In what follow R be a commutative ring with = Definition 1.1.1 Let (M, +) be an abelian group We say that M is an R-module (or a module over R) if there is a map (the action of ring elements): :RìM M (r, m) rm satisfying four axioms (i) (rs)m = r(sm) (ii) (r + s)m = rm + sm (iii) r(m + m ) = rm + rm (iv) 1R m = m for all r, s ∈ R and m, m ∈ M Example 1.1.2 Every k-vector space is k-module Bachelor thesis NGUYEN PHUONG ANH Every abelian additive        ax =       group M is Z-module with the action of ring elements: x + + x if a > 0, a times (−x) + + (−x) if a < (−a) times for every a ∈ Z, x ∈ M Every ideal of R is R-module Definition 1.1.3 Let M be an R-module A subgroup N of M is said to be a submodule of M if for all n ∈ N, r ∈ R then r.n ∈ N Remark 1.1.4 • A nonempty subset N of a module M is a submodule of M if and only if for all a, b ∈ N , r, s ∈ R then + sb ∈ N • A submodule R-module then it is also an R-module Example 1.1.5 Every module M which has two submodules and M is called trivial submodule For simplicity, we denote := {0} Every ideal I of R is submodule R-module of R-module R Definition 1.1.6 Let {Mi }i∈I be a family of submodules of an R-module M then Mi = { xα + xβ + + xγ | xα ∈ Mα , , xγ ∈ Mγ , α, β, , γ ∈ I} i∈I is a submodule of M It is said to be the sum of Mi n n xi xi ∈ Mi , i = 1, , n} and ∩ Mi is a submodule of M Mi = { In particular, i=1 i∈I i=1 Definition 1.1.7 Let {Mi }i∈I be a family of R-module Let Π Mi := {(xi )i∈I : xi ∈ i∈I Mi , ∀i ∈ I} Then Π Mi is an R-module under: i∈I • Addition: (xi )I + (yi )I := (xi + yi )I • Action: r(xi )i∈I := (rxi )i∈I We call Π Mi be the direct product of Mi i∈I • ⊕Mi := {(xi ) ∈ Π Mi xi = 0} is a submodule of Π Mi , for almost i We say i∈I i∈I that ⊕ Mi is the direct sum of Mi i∈I Remark 1.1.8 If I is finite then ⊕ Mi = Π Mi Sometimes, we say ⊕ Mi is the i∈I i∈I i∈I external direct sum of Mi Definition 1.1.9 An exact sequence of modules Mi is a sequence of the form fi−1 fi −→ Mi −−−→ Mi−1 −−−−→ Mi−2 −→ (∗) Bachelor thesis NGUYEN PHUONG ANH We see that ϕ : R → M at(2, R) a→ then ∀a ∈ R, ∀M ∈ A we get a.M = a 0 a a 0 a m11 m12 m21 m22 = am11 am12 am21 am22 Remark 1.5.10 If given an R-algebra A then a ring homomorphism ϕ : R → Z(A) is called the structure map of A over R Indeed, we obtain r ∈ R, ∀a ∈ A : ϕ(r).a = r.ϕ(1).a = r.a = a.r From all above, we come to the following general conclusion: Let R be a commutative ring An algebra over R, or R- algebra, is a pair (A, ϕ) consisting of a ring A and a ring homomorphism ϕ : R → Z(A) called the structure map of A over R, where Z(A) = {x ∈ A : xa = ax} for all a ∈ A is called the center of A, which is a subring of A One usually refers to A as the R-algebra, and keeps the structure map in mind If A is an R-algebra that is a division ring, we say that A is a division algebra 1.5.2 Center of algebras Definition 1.5.11 Let R be a commutative ring and an R-algebra A The center of A is given by Z(A) = {x ∈ A : xa = ax} for all a ∈ A Example 1.5.12 Consider M at2 (K) for some field K We have known that A ∈ M at2 (K) : AX = XA, ∀X ∈ M at2 (K) if and only if a A= for a ∈ K So, we infer that a Z(A) = a 0 a a∈K Definition 1.5.13 Let K be a field and A be a finite dimensional associative Kalgebra Then A is called a central simple algebra (CSA) over K if A is a simple ring and Z(A) = K Note that the inclusion of K in the center of A is automatic as A is a K-algebra Example 1.5.14 Every field is a central simple algebra over itself 15 Bachelor thesis NGUYEN PHUONG ANH Convention: Let k be a field, and observe that the additive group A+ of a kalgebra A is a k-vector space The k-dimension that A+ hereby gets is called the rank of the algebra A, denoted [A : k] If [A : k] is finite, we say that A has finite rank over k or that A is of finite rank over k Example 1.5.15 Let K = R; A = M at(2, R) First, A is R-algebra since R → M at(2, R) a→ a 0 a , ∀a ∈ R From the previous example, A is simple Moreover, center of A is written by Z(A) = {M ∈ M at(2, R)| X.M = M.X, ∀X ∈ M at(2, R)} Moreover, X.M = M.X, ∀X ∈ M at(2, R) because X= x11 x12 x21 x22 = x11 E11 + x12 E12 + x21 E21 + x22 E22 i.e, Eij M = M.Eij , ∀i, j = 1, (1) m11 m12 Write M = , then m21 m22 E11 M = M E11 ⇔ E11 M = m11 m12 0 = m11 m21 = M E11 ⇔ m12 = m21 = E22 M = M E22 ⇔ E22 M = 0 m21 m22 = m12 m22 = M E22 ⇔ m12 = m21 = E12 M = M E21 ⇔ E12 M = m21 m22 0 = m11 m21 = M E21 ⇔ It infers that the matrix M has form M = Z(A) = M= m 0 m m 0 m m21 = m11 = m22 = m for all m ∈ R So, m∈R −→ R M →m Therefore, M at(2, R) is central simple algebra over R and dimR M at(2, R) = Example 1.5.16 Let n be some natural number, then the matrix ring Mn (K) is a central simple algebra over K It obviously has dimension n2 over K so we only need to check that it is central and simple To see this, let eij denote the matrix with a at position (i, j) and zeroes at all other positions, i.e 16 Bachelor thesis NGUYEN PHUONG ANH  ···     eij =     0          Then for a matrix eii M = M eii for all i implies that M is diagonal and eij M = M eij for all i, j implies that all entries on the diagonal must be the same Thus, a central matrix must be a scalar matrix and obviously all scalar matrices are central In a similar way one can show that any nonzero ideal must be Mn (K) because suppose I is some nonzero ideal and M ∈ I\{0} Suppose mij = then eii = (mij )−1 eii M eij ∈ I and similarly for all l : ell = eli eii eil ∈ I, hence n ell ∈ I Idn = l=1 Theorem 1.5.17 (Wedderburn (it is a special case of the more general Artin-Wedderburn Theorem)) Let k be a field and A be a CSA over k Then there is a unique division algebra D (i.e a division ring which is a algebra over k) and a positive integer n such that A∼ = Mn (D) Remark 1.5.18 The division algebra D in the above theorem is automatically a central k-algebra because k = Z(A) = Z(Mn (D)) = Z(D) Recall : Let A be a ring, K ⊂ A be an ideal We say that K is minimal if it only contains two ideals and K Example 1.5.19 For A = Z, we see that for every ideal I ⊂ Z then I = n Thus, K = m is minimal It is inferred that K = since K = m ⊃ m, n , ∀n Lemma 1.5.20 Let K ⊂ A be a minimal and K = Then there exists e ∈ A such that e2 = e (e is an idempotent) and eAe is division ring Proof Because K = {a1 b1 + a2 b2 + + am bm } = then there exists u ∈ K : Ku = In addition, K ⊂ A is a minimal, Ku ⊂ K ⊂ K ⇒ Ku = K L⊂K Put L = {a ∈ K : au = 0} ⇒ ∀r ∈ A, ∀a, b ∈ L : (ra − b).u = rau − bu L⊂K ∀r ∈ A; ∀a, b ∈ L : − b ∈ L Because K is minimal then either L = or L = K Neverthless, eu = u = 0; e ∈ K → e ∈ / L so L = (for L = {a ∈ K : au = 0}) Furthermore, we have L=0 2 (e − e)u = e u − eu = e(eu) − eu = eu − eu = ⇒ e2 − e ∈ L −−→ e2 = e ⇒ 17 Bachelor thesis NGUYEN PHUONG ANH • We will prove that DA = eAe is division ring The identity of DA = e.1.e Indeed, for all α ∈ DA , α = e.a.e ⇒ (e.1.e).α = (e.1.e).(e.a.e) = e.1.e2 a.e = e.e.a.e = e.a.e = α for every β ∈ DA , β = 0, we see that A.e = 0, A.e ⊂ K Thus A.e = K (because K is minimal) Moreover, A.β = 0, since β = e.b.e for b ∈ A ⇒ A.β = (A.e.b).e ⊂ Ae = K Because K is minimal, A.β = K = A.e ⇒ ∃s ∈ A : s.β = e Then (ese).β = (e.s.e).(e.b.e) = e.s.e2 b.e = e.s.e.b.e = e.(s.β) = e.e = e.1.e ⇒ γ.β = 1DA Similarly, β.γ = (e.b.e).(e.s.e) = e.b.e2 s.e = e.b.e.s.e = e.(b.γ) = e.e = e.1.e ⇒ β.γ = 1DA Therefore, DA is division ring Theorem 1.5.21 (The Skolem-Noether theorem) It is known that any k-linear automorphism of the matrix ring Mn (k) is an inner automorphism That means there is an invertible matric C such that the k-linear automorphism is given by M → CM C −1 We will give the generalization of this statement to arbitrary central simple algebras, formulated by the Skolem-Noether Theorem 18 Chapter Azumaya algebras over local rings In this chapter, we begin by reviewing without proofs Some facts about projective modules are shown as follows 2.1 Azumaya algebras Theorem 2.1.1 Let R be a ring and let M be an R-module Then the following statements are equivalent (i) Every short exact sequence −→ S −→ T −→ M −→ of R-modules splits (ii) M is a direct summand of a free R-module Definition 2.1.2 An R-module M is projective if it satisfies the equivalent conditions of theorem 2.1.1 Theorem 2.1.3 If R is a local ring, then every projective module over R is free (A ring R is local if R/J (R) is a division ring, where J (R) is the Jacobson radical of R.) Notation 2.1.4 If A is a commutative ring and P is a prime ideal in A, then denotes the localization of A at the multiplicative set A − P , where A − P = {a ∈ A| a ∈ / P } If M is an A-module, then Mp denotes the Ap -module Ap ⊗A M If A is a commutative ring, P is a prime ideal of A, and M is a projective A-module, then Mp is a projective module over the local ring Ap , and hence free If M is finitely generated as an A-module, then Mp is finitely generated as an Ap -module, so it is free of finite rank The rank is well-defined since if a free module of rank r is isomorphic to a free module of rank s, then r = s (Warning: This is true since A is commutative There exist noncommutative rings for which a free module of rank r is isomorphic to a free module of rank s, with r = s.) 19 Bachelor thesis NGUYEN PHUONG ANH Exercise 2.1.5 Let A be a commutative ring Prove that if r and s are positive integers and Ar and As are isomorphic as A-modules, then r = s Hint: Note that EndA (Ar ) is isomorphic to Mr (A) Note that the rank of Mp as an Ap -module is the same as the dimension of MP /P MP as a vector space over AP /P AP If P is a maximal ideal, then AP /P AP ∼ = A/P In general,AP /P AP is isomorphic to the field of quotients of A/P Definition 2.1.6 Let M be a finitely generated projective module over a commutative ring A The rank of M is the function rankM : {prime ideals of A} → N defined by rankM (P ) = rank of Mp as an Ap -module If rankM (P ) = r for all prime ideals P in A, then M is said to be a projective module of constant rank r Theorem 2.1.7 Let M be a finitely generated projective module over a commutative ring A Then there exist orthogonal idempotents e1 , , ek in A such that e1 + e2 + + ek = and er M , regarded as an er M -module, has constant rank r An equivalent but fancier reformulation of above theorem is that rankM is a continuous function when the set of prime ideals of A is given the Zariski topology (closed sets are the sets V(I) = {prime ideals P of A such that P ⊇ I}, where I is a subset of A) and N is given the discrete topology We now introduce Azumaya algebras and develop some of their properties Standard references for Azumaya algebras are [DI71] and [OS75] Our convention that rings have unit elements has not been essential so far, but it is now, because Azumaya algebras are projective modules over their centers, and projective modules are only useful over rings with a unit Azumaya algebras are a generalization of central simple algebras in which the center is allowed to be an arbitrary commutative ring instead of a field Definition 2.1.8 If R is a ring, the opposite of R, denoted R◦ is a ring which is equal to R as an additive abelian group, and whose multiplication, ◦, is defined by r ◦ s = sr, where juxtaposition denotes the ordinary multiplication in R The A-algebra R⊗A R◦ is called the enveloping algebra of R The A-module map µ : R⊗A R◦ → R defined on elementary tensors by µ(r ⊗ s) = rs is the evaluation map Exercise 2.1.9 Show that defining (r⊗s)∗t = rts makes R into a left R⊗A R◦ -module Show that the evaluation map µ is a homomorphism of left R⊗A R◦ -modules Show that Ker(µ) is generated as a left ideal in R⊗A R◦ by the set {r ⊗ − ⊗ r| r ∈ R} Theorem 2.1.10 Let R be an algebra over the commutative ring A Make R into a left R⊗A R◦ -modules by (r ⊗ s) ∗ t = rts, and let µ : R⊗A R◦ → R be the evaluation map Then the following are equivalent (a) R is a projective left R⊗A R◦ -module 20 Bachelor thesis NGUYEN PHUONG ANH (b) The exact sequence → Ker(µ) → R⊗A R◦ → R → splits as a sequence of R⊗A R◦ -modules (c) There exists e ∈ R⊗A R◦ such that µ(e) = and Ker(µ)e = Proof The equivalence of (a) and (b) is immediate from the definition of projective (b) ⇒ (c) Let ϕ : R → R⊗A R◦ be a homomorphism of left R⊗A R◦ -modules such that is the identity map, and let e = ϕ(1) Then µ(e) = µϕ(1) = Moreover, for any r ∈ R, (r ⊗ − ⊗ r) ∗ = r − r = Applying ϕ shows that (r ⊗ − ⊗ r)e = for any r ∈ R Hence Ker(µ)e = 0, since Ker(µ) is generated as a left ideal in R⊗A R◦ by the set {r ⊗ − ⊗ r| r ∈ R} (c) ⇒ (b) If such an e exists, then ϕ(r) = (r ⊗ 1)e defines an R⊗A R◦ -module homomorphism ϕ : R → R⊗A R◦ such that µϕ is the identity Definition 2.1.11 An A-algebra R over a commutative ring A is a separable Aalgebra if any of the three equivalent conditions of Theorem 2.1.10 are satisfied If A is equal to the center of R, then R is an Azumaya A-algebra (or central separable A-algebra) The notion of separability makes sense when A is not the center of R, as, for example, when one considers separability of field extensions The following result, which will not be used in the sequel, shows that separable extensions can be divided into a central separable part and a separable extension of commutative rings Theorem 2.1.12 Let R be a ring, and suppose that A ⊆ Z ⊆ R, where Z is the center of R and A is a subring of Z Then R is separable over A if and only if R is central separable over Z and Z is separable over A Example 2.1.13 If A is a field, then R is Azumaya over A if and only if R is a finite-dimensional central simple A-algebra If A is a commutative ring, Mn (A) is Azumaya over A More generally, if P is a faithful finitely generated projective A-module, then EndA (P ) is Azumaya over A If R and S are Azumaya algebras over A, so is R⊗A S Exercise 2.1.14 Show that if A is a field, then R is an Azumaya A-algebra if and only if R is a finite-dimensional central simple A-algebra The following surprising result of A Braun shows that the classical definition of Azumaya algebra can be weakened in a small but significant way In conjunction with central polynomials, it is just what is needed for the quite simple proof of some theorems 21 Bachelor thesis NGUYEN PHUONG ANH Theorem 2.1.15 Let R be a ring with center A, let µ : R⊗A R◦ → R be the evaluation map, and let R be a left R⊗A R◦ -module with action induced by (r ⊗ s) ∗ t = rts Then the following are equivalent (a) R is an Azumaya algebra over A (b) There is e ∈ R⊗A R◦ such that µ(e) = and Ker(µ)e = (c) There is e ∈ R⊗A R◦ such that µ(e) = and (Ker(µ)e) ∗ R = (d) There exists a1 , a2 , , ak , b1 , , bk ∈ R such that bi = and rbi ∈ A for all r ∈ R Proof The equivalence of (a) and (b) is the definition of Azumaya algebra, and (b) trivially implies (c) (c) ⇔ (d) If e = ⊗ bi , then µ(e) = bi and e∗r = rbi Noting that A = {r ∈ R| Ker(µ) ∗ r = 0}, we have µ(e) = and = (Ker(µ)e) ∗ R = Ker(µ) ∗ (e ∗ R) ⇔ µ(e) = and e ∗ R ⊆ A ⇔ bi = and rbi ∈ A for all r ∈ R (c) ⇒ (b) Set U = R⊗A R◦ , J = Ker(µ), and suppose that e ∈ U satisfies µ(e) = 1, (Je) ∗ R = Make U into a left U -module actions ∗1 and ∗2 defined on elementary tensors by u∗1 (a ⊗ b) = (u ∗ a) ⊗ b, u∗2 (a ⊗ b) = a ⊗ (u ∗ b), where u ∈ U, a, b ∈ R We first show that U eU = U , where U eU is the two-sided ideal of U generated by e Let N = {r ∈ R| r ⊗ ∈ U eU }, a two-sided ideal of R Note that e∗2 (a ⊗ b) = a ⊗ (e ∗ b) = a(e ∗ b) ⊗ and U ∗2 (U eU ) ⊆ U eU Thus, every element of e∗2 (U eU ) is of the form r ⊗ 1, where r ∈ N We claim that N = R Assume not, and let M be a maximal ideal of R which contains N Consider the composition – : R⊗A R◦ → (R/M )⊗A (R/M )◦ → (R/M )⊗A (R/M )◦ = U , where A is the center of R/M (The right arrow is an isomorphism because A is the image of A in R/M , but we not need this fact.) Observe that ∗2 is compatible with –, in the sense that u∗2 v = u∗2 v for all u, v ∈ U , where ∗2 is defined analogously to ∗2 Since R/M is a simple ring, U = (R/M )⊗A (R/M )⊗A (R/M )◦ is a simple ring Now = ⊗ = e∗2 (1 ⊗ 1) = e∗2 (1 ⊗ 1) In particular, e = 0, so U = U eU = U eU Then e∗2 U = e∗2 U = e∗2 (U eU ) ⊆ N ⊗ ⊆ M ⊗ = 0, a contradiction Thus N = R, and U eU = U We claim that for any u ∈ U, (U eU )∗2 u ⊆ (u∗1 U )U Let v, w ∈ U, let ew = cj ⊗ dj and let u = gk ⊗ hk Then 22 Bachelor thesis NGUYEN PHUONG ANH vew∗2 u = vew∗2 ( gk ⊗ hk ) = gk ⊗ (vew ∗ hk ) = gk ⊗ (v ∗ 1)(ew ∗ hk ) = gk (ew ∗ hk ) ⊗ (v ∗ 1) = j,k gk (cj hk dj ) ⊗ (v ∗ 1) = [ j,k gk cj hk ⊗ (v ∗ 1)](dj ⊗ 1) = [u∗1 (cj ⊗ (v ∗ 1))](dj ⊗ 1) ⊆ (u∗1 U )U Since (Je) ∗ R = by (c), (Je)∗1 U = Hence Je ⊆ U ∗2 Je = (U eU )∗2 Je ⊆ ((Je)∗1 U )U = 0, so Je = 0, which establishes (b) Exercise 2.1.16 Show that if R is an Azumaya A-algebra and N is a two-sided ideal in R, then the center of R/N is image of A under the canonical map R → R/N We conclude this section by recording some facts about Azumaya algebras which will be useful later Theorem 2.1.17 Let R be an Azumaya algebra over A Then (a) The map R⊗A R◦ → HomA (R, R) induced by the *-action of R⊗A R◦ on R, is an isomorphism of A-algebras (b) Expansion-contraction gives a − correspondence between ideals of A and two-sided ideals of R (I.e, For any ideal I of A, IR ∩ A = I, and for any two-sided ideal J of R, J = (J ∩ A)R The − correspondence preserves sums, products and intersections of ideals, and carries prime ideals to prime ideals (c) R is finitely generated projective A-module (d) For some k there exist orthogonal idempotents e1 , , ek of A such that e1 + e2 + + ek = and each Rej is Azumaya over constant rank j over Aej Proof of (d) Since R is a finitely generated projective A-module, such a decomposition of R into summands Re of constant rank over Ae exists This constant rank is the same as the rank of RP over AP where P is any prime ideal of A such that e ∈ P This in turn is the dimension of RP /P RP is central simple over A/P , this dimension is the square of an integer, and the theorem follows 2.2 Azumaya algebras over local rings We now generalize CSAs over a field to Azumaya algebras over local rings There are several equivalent ways to define Azumaya algebras We start with the following rather technical definition: Definition 2.2.1 Let R be a commutative local ring (this will be the case throughout this section) and let A be an associative R-algebra such that R → A identifies R with a subring of Z(A) (i.e, the structure morphism is injective) Then A is called an Azumaya algebra over R is A is free of finite rank l as an R-module and if the following map is an isomorphism: 23 Bachelor thesis NGUYEN PHUONG ANH φA : A⊗R Aop → EndR (A) : a ⊗ a → (x → axa ) where Aop is the opposite algebra to A (i.e the same additive structure and the multiplicative structure given by a • b := b · a) Remark 2.2.2 • As we require A to be free over R, the inclusion R ⊂ Z(A) is automatic • φA always is an R-algebra morphism, so only the bijectivity in the definition is a nontrivial condition In the case where R = k is a field we have the following: Proposition 2.2.3 If A is a CSA over a field k, then A is Azumaya over k Proof Let dimk (A) = l then φA is a morphism between k-algebras which both have dimension l2 over k Hence, it suffices to check injectivity Note that A⊗k Aop is a CSA over k hence ker(φA ) = or A⊗k Aop As the second option is obviously false we have proven injectivity of φA The other direction is also true and follows from the following proposition Proposition 2.2.4 Let A be an Azumaya algebra over R, then Z(A) = R and there is a bijection between the (two-sided) ideals of A and the ideals of R: {Ideals of A} ←→ {Ideals of R} I →I ∩R JA ← J Proof Let ψ ∈ EndR (A) and c ∈ Z(A) then for all a ∈ A we have cψ(a) = ψ(ca) = ψ(ac) = ψ(a)c because ψ is given by multiplication be elements in A as A is Azumaya Similarly, ψ(I) ⊂ I for each ideal I of A Now let = a1 , , al be a basis for A as an R-module and define χi ∈ EndR (A) by χi (aj ) = δij Write c= i ri with all ri ∈ R, then l c = · c = χ1 (a1 )c = χ1 (a1 c) = χ1 (1 · c) = χ1 ( ri ) = r1 ∈ R i=1 Now we check the bijection between the sets of ideals As the maps are well defined, it suffices to prove I = (I ∩ R)A and J = J ∪ R Both equalities are trivial to check Corollary 2.2.5 An Azumaya algebra over a field is a CSA 24 Bachelor thesis NGUYEN PHUONG ANH Proposition 2.2.6 Let (R, m), (R , m ) be commutative local rings and let A be a free R-module of rank l Assume there is a morphism R → R then: i) If A is Azumaya over R then A⊗R R is Azumaya over R ii) If A ⊗ R/m is Azumaya (hence CSA) over R/m then A is Azumaya over R Proof We have the following commutative diagram: φA ⊗ R : (A ⊗R Aop ) ⊗R R EndR (A) ⊗R R ∼ = ∼ = φA⊗R : (A ⊗R R ) ⊗R (A ⊗R R )op EndR (A ⊗R R ) Figure 2.1: This first statement follows immediately from this diagram For the second statement note that surjectivity of φA ⊗R R/m implies surjectivity of φA by Nakayama’s Lemma For the injectivity we need a technical Lemma Corollary 2.2.7 Let A be a free module of rank l over (R, m) and let k = R/m, then the following are equivalent: (i) A is Azumaya over R (ii) A ⊗ k is a CSA over k (iii) A ⊗ k ∼ = Mn (k) In particular, l = n2 for some n ∈ N Corollary 2.2.8 • The tensor product of two Azumaya algebras is an Azumaya algebra • Mn (R) is Azumaya over R We now state the main result of this section Proposition 2.2.9 (Skolem-Noether) Let A be Azumaya over R, then every ψ ∈ AutR (A) is inner I.e, for any such ψ there is a unit u ∈ A∗ such that ψ(a) = uau−1 Proof Given ψ ∈ AutR (A), there are two different ways to turn A into an A⊗R Aop module: (a1 ⊗ a2 )a = a1 aa2 (a1 ⊗ a2 )a = ψ(a1 )aa2 Denote the resulting A⊗R Aop -modules by A, respectively A Both A := A ⊗R R/m op op and A are simple A⊗R A -modules This is based on the fact that A⊗R A -submodules of A or A correspond to two-sided ideals of the central simple algebra A (the argument 25 Bachelor thesis NGUYEN PHUONG ANH for A uses the fact that ψ is not just an endomorphism but an automorphism) op By Proposition: A⊗R A is a CSA over R = R/m and thus it is of the form Mn (D) for some division algebra D over R All simple modules overMn (D) are of the form Dn , op so there must be an isomorphism of A⊗R A -modules: χ:A→A We now claim that this lifts to a surjective A⊗R Aop -module morphism χ : A → A Firstly, suppose the claim holds, then setting u = ψ(1) gives: ψ(a)u = (a ⊗ 1)u = χ((a ⊗ 1)1) = χ(a) = χ((1 ⊗ a)1) = (1 ⊗ a)χ(1) = ua Surjectivity of χ gives the existence of an a0 ∈ A such that χ(a0 ) = 1, hence = χ(a0 ) = χ((1 ⊗ a0 )1) = ua0 implying that u is invertible in A Now we prove the claim: Note that we have the following diagram of A⊗R Aop module morphisms: A A χ A A Figure 2.2: so the existence of χ follows if we can prove that A is a projective A⊗R A -module As A is free as an R-module there is an R-module morphism g : A → R such that g(r) = r As A is Azumaya we have A⊗R Aop ∼ = EndR (A) and A is a direct summand of EndR (A) via a→(a →g(a )a) f →(f (1)) A −−−−−−−−→ EndR (A) −−−−−−−−→ A Finally, surjectivity of χ follows from Nakayama’s Lemma 2.3 Some examples of Azumaya algebras Example 2.3.1 Any finite dimensional central simple algebra A over a field F is an Azumaya algebra A central simple algebra ´ı free, so it is projective and we know A⊗F Aop ∼ = Mn (F ) ∼ = EndF (A) Example 2.3.2 For any commutative ring R, Mn (R) is an Azumaya algebra over R Since Rn is free and therefore projective over R, EndR (Rn ) is an Azumaya algebra over 26 Bachelor thesis NGUYEN PHUONG ANH R, and we know that Mn (R) ∼ = EndR (Rn ) as R-algebras Example 2.3.3 Let R be a commutative ring in which is invertible Define the quaternion algebra Q over R to be the free R-module with basis 1, i, j, k and with multiplication satisfying i2 = j = k = −1 and ij = −ji = k As for quaternion algebras over fields, it follows that Q is a central R-algebra Then Q is separable over R The proof considers the following element of Q: e = (1 ⊗ − i ⊗ i − j ⊗ j − k ⊗ k) It is routine to show that e is a separability idempotent for Q, so Q is separable over R It follows that Q is an Azumaya algebra over R 27 Bachelor thesis NGUYEN PHUONG ANH Conclusion In this thesis, we present systematically the following results: In the chapter 1, we present algebras over arbitrary commutative rings such as endomorphism algebras, matrix algebras, tensor products of algebras and their properties In the chapter 2, we present about Azumaya algebras over local rings and their properties in Proposition 2.2.6 and its corollaries In particular, in Proposition 2.2.9, we show about the structure of the linear automorphism of Azumaya algebras over local rings Finally, we give some interesting examples to illustrate for knowledge which we present in this bachelor thesis 28 Bibliography [Ve04] V Drensky, Edward Formanek, Polynomial Identity Rings, Advanced Courses in Mathematics: CRM Barcelona, 2004 [Ja11] A Javanpeykar, The Brauer group of a field, Bachelor’s thesis, 2011 [GA51] G Azumaya, On maximally central algebras, Nagoya Mathematical Journal, 1951 [Co03] P M Cohn, Further Algebra and Applications, Springer, 2003 [DI71] F DeMeyer and E Ingraham, Separable Algebras over Commutative Rings, Springer-Verlag, 1971 [OS75] M Orzech and C Small, The Brauer Group of Commutative Rings, Marcel Dekker, 1975 29 ... [GA51] for the case of a field Azumaya algebras over local ring is an extension of Azumaya algebras over field to local rings The studying of Azumaya algebras on a local ring is really significant... simple algebras 1.5.1 Simple rings 1.5.2 Center of algebras 4 10 12 13 14 15 Azumaya algebras over local rings 2.1 Azumaya algebras 2.2 Azumaya. .. proofs and some examples related to Azumaya algebras as well Last but not least, the conclusion and bibliography about related problems to Azumaya algebras on local rings are presented to complete