aa r X i v : . [ m a t h . R A ] S e p DIFFERENTIAL BRAUER MONOIDS
ANDY R. MAGID
Dedicated to the memory of Professor Raymond T. Hoobler
Abstract.
The differential Brauer monoid of a differential commutative ring R s defined. Its elements are the isomorphism classes of differential Azumaya R algebras with operation from tensor product subject to the relation thattwo such algebras are equivalent if matrix algebras over them are differentiallyisomorphic. The Bauer monoid, which is a group, is the same thing without thedifferential requirement. The differential Brauer monoid is then determinedfrom the Brauer monoids of R and R D and the submonoid whose underlyingAzumaya algebras are matrix rings. Introduction
Let R be a commutative ring with a derivation D R , or just D if it is clear fromcontext. A differential R algebra is an R algebra A with a derivation D A whichextends the derivation D R on R . If A is a differential R algebra, an A module M isa differential A module if there is an additive endomorphism D M of M satisfying D M ( am ) = D A ( a ) m + aD M ( m ) for all a ∈ A and m ∈ M . If A and C aredifferential R algebras, an R algebra homomorphism f : A → C is differential provided D C f = f D A . Thus we may speak of isomorphism classes of differentialalgebras. If A and B are differential algebras, so is A ⊗ R B , where the derivationof the tensor product is D A ⊗ ⊗ D B . The set of isomorphism classes is closedunder the operation induced from tensor product, forming a commutative monoidwith identity the class of R .This paper is concerned with differential R algebras which are Azumaya as R algebras, or differential Azumaya algebras . The monoid of isomorphism classes ofdifferential Azumaya algebras iunder the operation induced from tensor product isdenoted M diff ( R ). Inspired by definitions of the Brauer group, we call two differen-tial Azumaya algebras A and B equivalent provided there are integers m and n suchthat the algebras M m ( A ) and M n ( B ), with entrywise derivations, are isomorphicdifferential Azumaya R algebras. This relation is an equivalence relation compati-ble with the operation in the monoid M diff ( R ); we denote the resulting monoid ofequivalence classes BM diff ( R ) and call it the differential Brauer monoid of R .The similar construction carried out for the ordinary Azumaya algebras overany commutative ring R (no derivations) also produces a monoid of equivalenceclasses of isomorphism classes BM ( R ). In fact this monoid is a group. Because theequivalence relation is in general close to but finer than that which produces theBrauer group we also denote BM ( R ) as Br ∗ ( R ). Date : September 7, 2020.1991
Mathematics Subject Classification.
The Brauer group of a commutative ring is made up of equivalence classes of Azu-maya algebras over the ring, with the operation induced from the tensor productof algebras, where two algebras are equivalent if after tensoring each with endo-morphism rings of projective modules the resulting algebras are isomorphic. Thisdefinition, introduced by Auslander and Goldman in 1960 [2] (and anticipated forlocal rings by Azumaya nine years earlier) generalized the construction of the Brauergroup of a field. In 1967 Bass [3] showed how the Brauer group of a commutativering could be defined using the K-theory of the category of Azumaya algebras overthe ring. Building on this latter idea, Juan and the author defined a Brauer groupfor differential Azumaya algebras over a differential field F [8]: we took K of thecategory of differential central simple algebras over F and passed to the quotient bythe subgroup generated by the matrix algebras over F with entry-wise derivation,terming this quotient the differential Brauer group. The same definition applied toa differential ring R defines what could be called the differential Brauer group of R . Although this differential Brauer group maps (surjectively) to the usual Brauergroup by forgetting the derivations, its origin in K-theory presents difficulties. Un-like in the Brauer group of a commutative ring, not every element of this differentialBrauer group is the image of an isomorphism class of a differential Azumaya alge-bra; this is due to the fact that the inverse of the class of a differential Azumayaalgebra need not be the class of a differential Azumaya algebra. By staying withmonoids, instead of passing to K groups, this difficulty is avoided. The idea of usingmonoids of isomorphism classes instead of groups is found, in a different context,in [6, Section 3 p. 297]By this approach, we seek to isolate in algebraic structure the additional con-tribution of “differential” to the classification of differential Azumaya algebras, byseparating out the contributions from the classification of the underlying Azumayaalgebras captured in Br ∗ ( R ) as well as the contribution from the ring of invariantscaptured in Br ∗ ( R D ).This process begins with the homomorphism BM diff ( R ) → Br ∗ ( R ). By Knus’sTheorem [10, Th`eor´eme 3, p. 639] if A is a differential Azumaya which is a sub-algebra of an Azumaya algebra B , then there is a derivation of B which extendsthat of A . In particular, every Azumaya algebra A over the differential ring R hasa derivation which extends that of R so the above homomorphism is surjective.Although BM diff ( R ) is not a group in general, we can look at its subgroupof invertible elements. As we will see, this subgroup is an isomorphic image of Br ∗ ( R D ).The foundational results on differential Azumaya algebras on which these factsdepend are presented in Section 1 below, while the results on the monoids arecovered in Section 2. Section 3 looks at the case when R is a differential field andother examples.We retain throughout the terminology and notations of this introduction. Inparticular, R is always a commutative differential ring.This work is a collaborative project of the author and the late Raymond T.Hoobler, undertaken with the idea of publishing it on the 50th anniversary of ourfirst collaboration [7]. Except for the fact that Ray is no longer here to correct theproofs, in both senses, he is in every sense a coauthor. IFFERENTIAL BRAUER MONOIDS 3 Differential Azumaya Algebras
It is a basic property of an Azumaya algebra A that derivations which are trivialon the center are inner. For a ∈ A , we let I a denote inner derivation by a . Thismeans that if we have one derivation of an Azumaya algebra we in effect have themall: Proposition 1.
Let A be a differential Azumaya R algebra with derivation D A . If D is a derivation of A which agrees with D R on R then D = D A + I a for some a ∈ A .Proof. Since D and D A on R are D R , D − D A is a derivation of A which is trivialon R so D − D A = I a for some a ∈ A . Thus D = D A + I a . (cid:3) Two important special cases of Proposition 1 occur when A is the endomorphismring of a projective module and when A is the endomorphism ring of a free module(matrix ring).We recall that a differential projective R modules is a differential R modulewhich is finitely generated and projective as an R module [8, Definition 1, p. 4338].If P is such a differential projective module then End R ( P ) is a differential Azumayaalgebra with induced derivation D ( S ) = D P S − SD P . Corollary 1.
Let P be a finitely generated projective R module and let D be aderivation of End R ( P ) . Then there is a differential structure on P such that thederivation induced from D P is D .Proof. By [8, Theorem 2, p. 4341], there is a differential structure D on P . Let D be the induced derivation on End R ( P ). By Proposition 1 D = D + I T forsome T ∈ End R ( P ). Consider the additive endomorphism D + T . It is easy tocheck, or we can cite [8, p. 4338], that D + T is also a differential structure on P ,and the derivation it induces on End R ( P ) sends S to ( D + T ) S − S ( D + T ) = D S − SD + T S − ST = D ( S ) + I T ( S ) = D ( S ). (cid:3) Let P be a projective R module. If an R algebra A is a subalgebra of End R ( P ),we can view P as a faithful A module via the endomorphisms given by left multi-plication, and conversely. Suppose A is also Azumaya over R . Then we have thefollowing consequence of Corollary 1: Corollary 2.
Let A be a differential Azumaya R algebra, and let P be an A mod-ule which is finitely generated and projective as an R module. Then there is adifferential structure on P making P a differential A module.Proof. As noted, P gives an algebra embedding A → End R ( P ), which we regardas an inclusion. Both End R ( P ) and its subalgebra A are Azumaya over R .By [10, Th`eor´eme 3, p. 639], the derivation of A extends to a derivation ofEnd R ( P ), making the inclusion one of differential algebras. By Corollary 1, thereis a differential R module structure on P which induces the given derivation onEnd R ( P ). Then for p ∈ P and a ∈ A (or indeed for any endomorphism of P ) D ( ap ) = D ( a ) p + aD ( p ). (cid:3) If M n ( R ) is a matrix ring over R , then applying D R entry-wise gives a derivation,which we denote ( · ) ′ . And once we have one derivation, by Proposition 1, we havethen all. ANDY R. MAGID
Corollary 3.
Let D be a derivation of M n ( R ) , Then there is a matrix X ∈ M n ( R ) such that D ( Y ) = Y ′ + XY − Y X for Y ∈ M n ( R ) . Conversely, any matrix Z defines a derivation ∂ Z of M n ( R ) by the formula ∂ Z ( Y ) = Y ′ + ZY − Y Z .Proof.
By Proposition 1, D = ( · ) ′ + I X for some X ∈ M n ( R ), which is the firstassertion. The second follows since the sun of the two derivations ( · ) ′ and I Z is aderivation which restricts to D R on R . (cid:3) Notation 1.
For A ∈ M n ( R ) we denote M n ( R ) with the derivation ∂ A by ( M n ( R ) , A )The matrix algebra M n ( R ) is isomorphic to the endomorphism ring of the freemodule R n (column vectors). By Corollary 1, the differential structure ( M n ( R ) , A )is induced by a differential structure on R n . It is easy to see, using the proofof the Corollary as a guide, that in this structure the derivative of x is x ′ + Ax ,where x ′ means coordinate wise derivation by D R . We abbreviate this structure( R n , A ). Note that this is the dual of the similar structure in [9, p. 4338], so wecan use the calculations from [9] after changing rows to columns. In particular, if Y ∈ M n ( R ) is invertible and Y ′ = − AY then Y defines a differential isomorphism( R n , A ) → ( R n , Y defines a differential isomorphism ( M n ( R ) , A ) → M n ( R ) , M n ( R ) , Z ) differentially isomorphic to ( M n ( R ) , T .Then T satisfies a differential equation of the form T ′ = ZT + cI where c ∈ R . (In[8, Proposition 2, p.1923] we show how to make T , there called H , determinate,using the fact that the base ring R is a field of characteristic zero. In the generalcase we need to keep the possibility of any c .)An important special case is where the matrix Z in Corollary 3 is zero: thederivation ∂ is just another name for ( · ) ′ . Because Z = 0, we will refer to this asthe trivial derivation on M n ( R ).Every Azumaya algebra appears as a tensor factor in a matrix ring. However fora differential Azumaya algebra to appear as a tensor factor of a matrix ring withtrivial derivation is a very restrictive condition, as we see in Theorem 1.The proof of this Theorem will use the result, [9, Theorem 1, p. 4340] that adifferential projective module P is a differential direct summand of R n , where thelatter has the component-wise differential structure, if and only if P is of the form R ⊗ R D P where P is a finitely generated projective R D module. If is further shownin [9, Lemma 1, p. 4340] that if P = R ⊗ R D P for finitely generated projective P then P = P D . Theorem 1.
Let A be a differential Azumaya algebra and suppose there is a dif-ferential Azumaya algebra B such that A ⊗ R B is isomorphic, as a differential R algebra, to M n ( R ) with the trivial derivation. Then A D is an Azumaya algebraover R D (in fact A D ⊗ B D is isomorphic to M n ( R D ) ) and R ⊗ R D A D → A by r ⊗ a ra is an isomorphism. Conversely, if A is an Azumaya algebra over R D then the Azumaya R algebra R ⊗ R D A is a differential tensor factor of an R matrixalgebra with trivial derivation.Proof. We assume both A and B are subalgebras of M n ( R ). Consider the map p : M n ( R ) → R given by projection on the entry of the first row and column of amatrix. Because the derivation on M n ( R ) is the trivial one, p is a homomorphismof differential modules which sends the identity to one. Let p be the restriction of IFFERENTIAL BRAUER MONOIDS 5 p to B . The kernel B of p is then a differential module, and since p is onto wehave B = R ⊕ B as differential modules. Thus as differential modules M n ( R ) is( A ⊗ R R ) ⊕ ( A ⊗ R B ). In particular, A is a differential direct summand of M n ( R ),which as a differential module is R free with coordinate-wise derivation. By [9,Theorem 1, p. 4340], this means that A D is a finitely generated projective R D module and R ⊗ R D A D → A is an isomorphism. The same applies to B D . Notethat both A D and B D are subalgebras of M n ( R ) D = M n ( R D ). To see that A D isAzumaya, we consider the multiplication map m : A D ⊗ B D → M n ( R D ). Then1 ⊗ m , under the multiplication isomorphisms R ⊗ A D → A and R ⊗ B B → B ,becomes the multiplication map m : A ⊗ B → M n ( R ), which is a differential algebraisomorphism. Since 1 ⊗ m : R ⊗ R D ( A D ⊗ B D ) → M n ( R ) is a differential algebraisomorphism, it induces an isomorphism of the constants of its domain, namely A D ⊗ B D , and the constants of its range, M n ( R ) D = M n ( R D ). Thus A D is anAzumaya R D algebra. The final statement is a consequence of the fact, appliedto R D , that every Azumaya algebra is a tensor factor of a matrix algebra, andthe fact that R ⊗ M n ( R D ) is isomorphic to M n ( R ) with the trivial derivation as adifferential R algebra. (cid:3) If A is an Azumaya R algebra, and P is an A module which is finitely generatedand free of rank n as an R module, by Corollary 2 P has a differential modulestructure as an A module. If we regard R n (column n tuples) as a left M n ( R )module via matrix multiplication, then the differential structure of P induces aderivation of M n ( R ), and the differential inclusion A → M n ( R ) implies that A isa differential tensor factor of M n ( R ), the other factor being the commutator of A ,which is also a differential subalgebra. By Theorem 1, if the derivation of M n ( R )is trivial, A is induced up from R D . Hence: Corollary 4.
Let A be an Azumaya R algebra. Then A has a faithful differentialmodule P , finitely generated and projective over R , with an R basis of constants (andhence free), if and only if A is isomorphic, as a differential algebra, to R ⊗ R D A for some Azumaya R D algebra A .Proof. If A has a faithful differential module P , finitely generated and projectiveover R , with an R basis of constants. then A is a differential subalgebra of M n ( R )with the trivial derivation, and hence of the desired form. Convesely, if A is iso-morphic to R ⊗ R D A for some Azumaya R D algebra A and if P is a faithful A module, finitely generated and free as an R D module, then P = R ⊗ P is afaithful A module, finitely generated and projective over R , with an R basis ofconstants. (cid:3) Brauer Monoids
Throughout this section we will be concerned with monoids of isomorphismclasses and various quotients thereof. To fix terminology, by a monoid M we meana commutative semigroup with identity 1, with the operation written multiplica-tively. If N ⊆ M is a submonoid, by M/N we mean the set of equivalence classeson M under the equivalence relation m ∼ m if there are n , n ∈ N such that m n = m n . Products of equivalent elements are equivalent, which defines a com-mutative operation on M/N by multiplying representatives of equivalence classes,and the equivalence class of the identity is an identity. Thus
M/N is a monoid.Note that we can have
M/N = 1 but N = M , for example Q ∗ / Z × . ANDY R. MAGID
The set of invertible elements U ( M ) of M is its maximal subgroup. The setelements a ∈ M whose equivalence classes in M/N are invertible is { a ∈ M |∃ b ∈ M, n ∈ N such that abn ∈ N } .We will be considering the following monoids: Definition 1.
M A ( R ) denotes the set of isomorphism classes of Azumaya R alge-bras with the operation induced by tensor product and the identity the isomorphismclass of R . M A diff ( R ) denotes denotes the set of isomorphism classes of differential Azu-maya R algebras with the operation induced by tensor product and the identity theisomorphism class of R . M E ( R ) denotes the submonoid of M A ( R ) whose elements are isomorphismclasses of endomorphism rings of faithful finitely generated projective R modules. M M ( R ) denotes the submonoid of M A ( R ) whose elements are isomorphismclasses of matrix algebras. M M diff ( R ) denotes the submonoid of M A diff ( R ) whose elements are isomor-phism classes of Azumaya algebras of the form ( M n ( R ) , Z ) . M M diff ( R ) denotes the submonoid of M A diff whose elements are isomorphismclasses of Azumaya algebras of the form ( M n ( R ) , . BM ( R ) and Br ∗ ( R ) both denote M A ( R ) /M M ( R ) BM diff ( R ) denotes M A diff ( R ) /M M diff ( R ) M BM diff ( R ) denotes M M diff ( R ) /M M diff ( R ) BM ( R ) is called the Brauer monoid of R . BM diff ( R ) is called the differential Brauer monoid of R . M BM diff ( R ) is called the matrix diffferential Brauer monoid of R . All of the monoids in Definition 1 are functors.Note that an alternative definition of BM diff ( R ) is the monoid, under the oper-ation induced by tensor product, of the equivalence classes of differential Azumayaalgebras under the relation A ≡ B whenever there are integers m, n with M m ( A )isomorphic to M n ( B ) as differential R algebras. This follows since M m ( A ) is iso-morphic to A ⊗ ( M m ( R ) ,
0) as differential algebras. The same reasoning showsthat BM ( R ) is the monoid, under the operation induced by tensor product, of theequivalence classes of Azumaya algebras under the relation A ≡ B whenever thereare integers m, n with M m ( A ) isomorphic to M n ( B ) as R algebras. When R is afield, this is one of the defintions of the Brauer group. In any event, we always havea surjection Br ∗ ( R ) → BrR ); this is a group homomorphism with an identifiablekernel.
Proposition 2.
The Brauer monoid Br ∗ ( R ) is a group. The group homomorphism Br ∗ ( R ) → Br ( R ) is surjective with kernel M E ( R ) /M M ( R ) . In particular, thelatter is a group. The proof of the proposition will use the fact, already mentioned, that an Azu-maya R algebra A is a tensor factor of a matrix algebra. This follows from Bass’sTheorem, [4, Proposition 4.6, p. 476], that if P is a faithful finitely generated pro-jective R module then there is a faithful finitely generated projective R module Q such that P ⊗ Q is free: A is a subalgebra and tensor factor of End R ( A ) by leftmultiplications and since A is faithful and finitely generated projective as an R module there is a Q as above, and then End R ( A ) ⊗ End R ( Q ) is a matrix algebra, IFFERENTIAL BRAUER MONOIDS 7
Proof.
As remarked above, the invertible elements of Br ∗ ( R ) are the classes ofthe Azumaya algebras A for which there exists an Azumaya algebra B and matrixalgebras M and M such that A ⊗ B ⊗ M is isomorhpic to M . Since any Azumayaalgebra is a tensor factor of a matrix algebra, this condition holds. Thus Br ∗ ( R ) is agroup. Since matrix algebras are trivial in the Brauer group, the monoid surjection M A ( R ) → Br ( R ) factors through Br ∗ ( R ). Suppose A in M A ( R ) represents a classin Br ∗ ( R ) which goes to the identity in Br ( R ). That is, suppose the class of A istrivial in the Brauer group of R . Then A is the endomorphism ring of a projective,and hence lies in M E ( R ). It follows that M E ( R ) /M M ( R ) is the kernel. (cid:3) Proposition 2 shows that the Brauer monoid of a commutative ring differs fromthe Brauer group of the ring by an identifiable subgroup. If this subgroup is trivial,then Br ∗ ( R ) and Br ( R ) coincide. To describe this situation, we introduce someterminology: Definition 2. An R algebra A is said to be a stably matrix algebra if there are n and m such that A ⊗ M n ( R ) is isomorphic to M m ( R ) . A differential R algebra A is said to be a differentially stably matrix algebra if there are n and m such that A ⊗ ( M n ( R ) , is differentially isomorphic to ( M m ( R ) , . A faithfully projective R module P is said to be stably tensor free is there are n and m such that P ⊗ R n is isomorphic to R m . A differential Azumaya algebra A represents the identity element of BM diff ( R )if and only if A is a differentially stably matrix algebra. If so, and A ⊗ ( M n ( R ) , M m ( R ) , A is differentiallyisomorphic to R ⊗ A D . This implies that A D is a projective R D module of thesame rank as that of A over R . If the ranks differ, then the class of A in BM diff ( R )is not the identity element. Corollary 5.
The map Br ∗ ( R ) → Br ( R ) is an isomorphism if and only if endo-morphism rings of faithfully projective R modules are stably matrix algebras. If allfaithfully projective R modules are free, or even stably tensor free, then all theirendomorphism rings are stably matrix algebras. It is not in general the case that the differential Brauer monoid is a group. Ournext result can be understood as saying that the differential Brauer monoid differsfrom the Brauer monoid by the matrix differential Brauer monoid, and hence thelatter captures the differential classification of Azumaya algebras:
Theorem 2.
The monoid surjection BM diff ( R ) → BM ( R ) induces an isomor-phism BM diff ( R ) /M BM diff ( R ) → BM ( R ) Proof.
For the purposes of the proof, we will write ∼ = for R algebra isomorphismand ∼ = diff for differential R algebra isomorphism. Suppose A and B are differentialAzumaya R algebras whose images are equal in BM ( R ). Let C be an Azumaya R algebra whose image in BM ( R ) is the inverse of the class of B . Then there arethere are matrix R algebras M c and M d such that (1) B ⊗ C ⊗ M c ∼ = M d (since C is the inverse of B ) and, since A and B are equal in the Brauer monoid, thereare also matrix algebras M a and M b such that (2) A ⊗ C ⊗ M a ∼ = M b . Now A and B already have differential structures. Put a differential structure on C and M c , and use the isomorphism (1) to give M d a differential structure so that (1) is adifferential isomorphism. Put a differential structure on M a , and use the differential ANDY R. MAGID structure already placed on C and the isomorphism isomorphism (2) to give M b adifferential structure so that (2) is a differential isomorphism. We tensor (2) with B ⊗ M c to obtain A ⊗ C ⊗ M a ⊗ B ⊗ M c ∼ = diff M b ⊗ B ⊗ M c . Then we use (1) to replace B ⊗ C ⊗ M c on the left with M d to obtain A ⊗ M d ⊗ M a ∼ = diff B ⊗ M a ⊗ M c which shows that A and B are equivalent in BM diff ( R ), Thus the surjection BM diff ( R ) → BM ( R )is also an injection. (cid:3) As a consequence of Theorem 2, we know that, modulo the matrix differentialBrauer monoid, the differential Brauer monoid and the Brauer monoid coincide,so that in particular (as noted in the following corollary) if the matrix differentialBrauer monoid is trivial the differential Brauer monoid and the Brauer monoid co-incide. In this sense, the matrix differential Brauer monoid captures the additionalstructure of the differential case.
Corollary 6. If M BM diff ( R ) = 1 then BM diff ( R ) = BM ( R ) . It is also true that if the Brauer monoid of R is trivial then the differentialBrauer monoid of R coincides with the matrix differential Brauer monoid of R .This is not quite a corollary of the statement of Theorem 2, but follows from thesame techniques used in its proof: Corollary 7. If BM ( R ) = 1 then M BM diff ( R ) = BM diff ( R ) Proof. If BM ( R ) = 1 then for any Azumaya R algebra A there are p and q suchthat A ⊗ M p ( R ) ∼ = M q ( R ). Then if A is a differential Azumaya algebra, so is A ⊗ ( M p ( R ) ,
0) and hence for suitable
Z A ⊗ ( M p ( R ) , ∼ = diff ( M q ( R ) , Z ). Since( M q ( R ) , Z ) ∈ M BM diff , so is A . (cid:3) The hypothesis of Corollary 7 is satisfied, for example, using Proposition 2, when Br ( R ) = 1 and all faithful projective R modules are free.Theorem 2 and Corollary 7 are about the contribution of BM ( R ) to BM diff ( R ).Our next result is about the contribution of BM ( R D ). Theorem 3.
The map BM ( R D ) → BM diff ( R ) by A R ⊗ A is an isomorphismfrom BM ( R ) to the group of units of BM diff .Proof. By Proposition 2 BM ( R D ) is a group, and so the monoid homomorphismcarries it to the group of units U of BM diff ( R ). Suppose A is a differential Azumaya R algebra whose class in BM diff ( R ) is a unit. This means there is a differentialAzumaya algebra B and p , q such that A ⊗ B ⊗ ( M p ( R ) ,
0) is differentially isomorphicto ( M q ( R ) , A ia a tensor factor of a matrix algebra with trivial derivation.By Theorem 1 this means that A D is an Azumaya R D algebra and that A isdifferentially isomorphic to R ⊗ A D . Thus A lies in the image of BM ( R D ). To seethat the onto group homomorphism BM ( R D ) → U is an injection, let A be anAzumaya algebra representiing an element of the kernel. Then there are p and q suchthat ( R ⊗ A ) ⊗ ( M p ( R ) ,
0) is differentially isomorphic to ( M q ( R ) , R D isomorphism of A ⊗ M p ( R D ) to M q ( R D ) which means the class of A is the identity in BM ( R D ). (cid:3) IFFERENTIAL BRAUER MONOIDS 9
The kernel of the group homomorphism Φ : BM ( R D ) → BM ( R ) induced fromthe inclusion consists of classes represented by Azumaya R D algebras A suchthat R ⊗ A is a stably matrix algebra, say ( R ⊗ A ) ⊗ M n ( R ) ∼ = M m ( R ). Since A ⊗ M n ( R D ) represents the same class as A , and R ⊗ ( A ⊗ M n ( R D )) = ( R ⊗ A ) ⊗ M n ( R ) is isomorphic to M m ( R ), we can replace A with A ⊗ M n ( R D ) andassume that Φ( A ) is a matrix algebra. Now Φ factors as BM ( R D ) → BM diff ( R ) → BM ( R ). The first map is injective by Theorem 3, and by the remarks just noted thekernel of Φ is BM ( R D ) ∩ M BM diff ( R ), which is the group of units of M BM diff ( R ).An interesting special case occurs when BM ( R D ) = 1. Then Theorem 3 impliesthat BM diff ( R ) has no units, and in particular no torsion elements, in contrastwith the situation of the Brauer group of R , which is torsion [5, Corollary 11.2.5,p. 417]. Thus any non-trivial element of BM diff ( R ) generates a cyclic submonoidof infinite order. In particular, if BM diff ( R ) is not trivial, it is infinite.When BM diff ( R ) has no units except 1, the monoid surjection BM diff ( R ) → BM ( R ) can’t split. This implies that the existence of differential structures onAzumaya algebras can’t be made canonical.We also have the following consequences of Theorem 2, Theorem 3, Corollary 6,and Corollary 7: Corollary 8. If BM ( R D ) = BM ( R ) = 1 then BM diff ( R ) = M BM diff ( R ) andhas no units. If M BM ( R ) diff = 1 then BM ( R D ) → BM ( R ) is an isomorphism. Fields and Examples
We consider the case that R D = C is an algebraically closed field of characteristiczero. Since C is a field, Br ∗ ( C ) = Br ( C ) and since C is algebraically closed Br ( C ) = 1. Example 1.
We consider the case R = C . Then by Corollary 8 BM diff ( C ) = M BM diff ( C ). Consider the element repre-sented by ( M n ( C ) , Z ). As a linear transformation on M n ( R ) we have I Z = L Z − R Z where L Z (respectively R Z ) means left (respectively right) multiplication by Z . Any C polynomial satisfied by Z is satisfied by both L Z and R Z , and since L Z and R Z commute we conclude that every eigenvalue of I Z is a difference of eigenvalues of Z . For any p , the linear transformation I Z ⊗ ⊗ I = I Z ⊗ M n ( C ) ⊗ M p ( C )will have the same eigenvalues as I Z . It follows that the set of eigenvalues of I Z is an invariant of the class of ( M n ( C ) , Z ) in BM diff ( C ), and these eigenvalues areamong the set of differences of eigenvalues of Z . In particular, if Z ∈ M n ( C ) and W ∈ M m ( C ) are such that the differences of their eigenvalues are disjoint, then( M n ( C ) , Z ) and M m ( C ) , W ) represent different elements of BM diff ( C ). We con-clude that the cardinality of BM diff ( C ) is that of C . Note also that if ( M n ( C ) , Z )represents the trivial element then all the eigenvalue differences of Z are zero, so Z is a scalar matrix. Example 2.
We consider the case that R = C ( x ) with derivation d/dx . Then by [1], Br ( C ( x )) = 1, and C ( x ) being a field Br ∗ ( C ( x ) = Br ( C ( x )) = 1as well. By Corollary 8, BM diff ( C ( x )) = M BM diff ( C ( x )). Consider the map BM diff ( C ) → BM diff ( C ( x )). Let Z = [ z ij ] ∈ M ( C ) be the matrix with z = 1 andthe rest of its entries zero. Calculation shows that I Z = 0 (although I Z = 0, so theeigenvalue differences of I Z are all zero). If there were p, q such that ( M ( C ) , Z ) ⊗ ( M p ( C ) ,
0) were differentially isomorphic to ( M q ( C ) ,
0) then I Z ⊗ I on M q ( C ), but the since the square of the former is not zero andthe square of the latter is, this is impossible. Thus the class of ( M ( C ) , Z ) is nottrivial in BM diff ( C ). However for the class of C ( x ) ⊗ ( M ( C ) , Z ) = ( M ( C ( x )) , Z )in BM diff ( C ( x )), there is a Y = [ y ij ] ∈ M ( C ( x )) such that Y ′ = ZY , namely where y = x , y = − y = 1 and y = 0. So in BM diff ( C ( x )), C ( x ) ⊗ ( M ( C ) , Z ) =1. Now let λ ∈ C and let Z ( λ ) ∈ M ( C ) be the matrix with diagonal elements 1 + λ and 1, with λ = 0. For A = (cid:20) a bc d (cid:21) , [ Z ( λ ) , A ] = (cid:20) λb − λc (cid:21) . The standard unitmatrices e (2) ij , i = 1 , j = 1 , Z ( λ ) , · ], with eigenvalues 0 , λ, − λ .Let e ( p ) ij , 1 ≤ i, j ≤ p be the standard unit matrices for M p ( C ). Then e (2) ij ⊗ e ( p ) kl are C basis of M ( C ) ⊗ M p ( C ) of eigenvectors of [ Z ( λ ) ⊗ , · ], the eigenvalues begin0 , λ, − λ .Now consider A = ( M ( C ( x )) ⊗ ( M p ( C ( x )) , Z ( λ ) ⊗ D . Then D ( e (2)12 ⊗ e ( p )11 ) = λ ( e (2)12 ⊗ e ( p )11 ). Thus A has a non-zeroelement a with D ( a ) = λa . We will call a an e-element and λ an e-value . Similarly,there are e-elements with e-values 0 and − λ . Now suppose b = P β ij,kl e (2) ij ⊗ e ( p ) kl is an e-vector in A with e-value µ = 0, so D ( b ) = µb . We calculate D ( b ): this is X β ′ ij,kl e (2) ⊗ e ( p ) kl + [ Z ( λ ) ⊗ , X β ij,lk e (2) ij ⊗ e ( p ) kl ] = X β ij,lk [ Z ( λ ) , e (2) ij ] ⊗ e ( p ) kl . Setting D ( b ) equal to µb and comparing coefficients gives β ′ ,kl = µβ ,kl , β ′ ,kl = µβ ,kl , β ′ ,kl + λβ ,kl = µβ ,kl , β ′ ,kl − λβ ,kl = µβ ,kl . We rewrite the last two equations as β ′ ,kl = − ( λ − µ ) β ,kl , β ′ ,kl = ( λ + µ ) β ,kl .In the field C ( x ) if y ′ = cy for c ∈ C − { } then y = 0. Thus β ,kl = β ,kl = 0,and either λ = µ , β ,kl ∈ C and β ,kl = 0 or λ = − µ , β ,kl ∈ C and β ,kl = 0 or λ = − µβ ,kl ∈ C and β ,kl = 0.Thus the only non-zero e-values in A are λ, − λ , and the corresponding e-vectorsare in ( M ( C ) , Z ( λ )) ⊗ ( M p ( C ) , A = A ( λ ) to be explicit about thevalue λ , and let A ( λ ) denote ( M ( C ) , Z ( λ )). Finally, let λ and λ be non-zeroelements of C and suppose that | λ | 6 = | λ | . If A ( λ i ), i = 1 , BM diff ( C ) → BM diff ( C ( x )) then there is a p such that A ( λ ) is differen-tially isomorphic to A ( λ ). But then the above calculation on e-values shows that | λ | = | λ | , contrary to assumption. Thus the A ( λ ) represent distinct elements of BM diff ( C ( x )) (and a fortiori different elements of BM diff ( C )). Example 3.
We consider the case that F is an algebraically closed differential fieldwith C = F D . Let R be the Picard–Vessiot ring of a differential Galois extension E of F withdifferential Galois group G , which is a connected linear algebraic group over C .Then also E D = R D = C . We suppose the extension is given by a matrix dif-ferential equation X ′ = AX for A ∈ M n ( F ), n >
1. Then E contains, theentries of, and is generated by the entries of, an invertible matrix Y ∈ M n ( R )with Y ′ = AY . So A = Y ′ Y − , which means that ( M n ( F ) , A ) goes to the trivialclass in BM diff ( F ) → BM diff ( R ). (If ( M n ( F ) , A ) were already trivial then thePicard–Vessiot extension would be trivial.) Since F is algebraically closed, R , as adifferential ring, is isomorphic to F [ G ]. IFFERENTIAL BRAUER MONOIDS 11 If G is unipotent, then R is a polynomial ring so all projective R modules arefree, so BM ( R ) = Br ( R ), and since R has trivial Brauer group, BM ( R ) = 1. Thus BM diff ( R ) = M BM diff ( R ).If G is a torus of rank m , R is a Laurent polynomial ring, so all projective R modules are free so BM ( R ) = Br ( R ). Since R D = C , BM ( R D ) = 1, and BM diff ( R ) has no units. In this case Br ( R ) is a product of m ( m − / Q / Z [11, Corollary 7, p. 166] so M BM diff ( BM diff ( R ). Example 4.
Let F be a differential field with F D = C and let R be the Picard–Vessiot ring of a Picard–Vessiot closure E of F . Then for every n and every A ∈ M n ( F ) a Picard–Vessiot extension for X ′ = AX appears in E , and thus there is an invertible Y ∈ M n ( R ) with A = Y ′ Y − ,which means that ( M n ( F ) , A ) goes to the trivial class in BM diff ( F ) → BM diff ( R ).(Neither monoid has any units since BM ( F D ) = BM ( R D ) = BM ( C ) = 1.) Inparticular, all of M BM diff ( F ) becomes trivial in BM diff ( R ).A Picard–Vessiot closure may have proper Paicard–Vessiot extensions. By takingthe Picard–Vessiot closure of the closure, and iterating the process a countablenumber of times, one obtains a differential extension E of F such that every elementlies in a finite tower of Picard–Vessiot extensions and such that for every A ∈ M n ( E )there is an invertible Y ∈ M n ( E ) with A = Y ′ Y − , which means that ( M n ( E ) , A )goes to the trivial class in BM diff ( E ), so M BM diff ( R ) = 1. Since E is algebraicallyclosed (algebraic extensions are Picard–Vessiot), BM ( E ) = 1. Thus BM diff ( E ) = 1. References [1] Auslander, M., and Brumer, A.
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Brauer groups of linear algebraic groups with characters , Proc. Amer. MathSoc. (1978), 164-168 Department of Mathematics, University of Oklahoma, Norman OK 73019,
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