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A generalization of the topological Brauer group
A.V. Ershov
Let X be a nite CW -complex, and k a positive integer greater than 1. By A k m ; B k r we denote
locally trivial bundles over X with bers the matrix algebras M k m(C ); M k r (C ), respectively. On the
set of such bundles consider the following equivalence relation:
A k m  B k r , there are vector bundles  k t ;  k v with bers C k t
; C k v
; respectively,
such that A k
m
End( k t )  = B k
r
End( k v ):
Then the k-primary component Br k
(X) of the topological Brauer group Br(X) of X is the group
of equivalence classes of this relation with respect to the operation induced by the tensor product of
bundles.
In my talk some generalization of this group will be introduced. More precisely, let k; l be coprime
positive integers greater than 1. By M k; l
denote the union of all subalgebras in the matrix algebra
M kl (C ) that are isomorphic to the matrix algebra M k (C ). Consider M k; l as a topological space with
respect to the topology induced by the canonical embedding M k; l
 M kl
(C ): Let Aut(M k; l
) be the
group of homeomorphisms of M k; l
such that their restriction to any subalgebra A  M k; l
; A  = M k
(C ),
is a C -algebras isomorphism with a subalgebra B  M k; l ; B  = M k (C ): We equip Aut(M k; l ) with the
compact-open topology related to the natural action Aut(M k; l
)  M k; l
! M k; l
:
We construct our generalized Brauer group GBr by means of locally trivial bundles with bers
M k; l for dierent k; l as above. The natural embedding M k m(C )  = M k m(C
)
C
C E l n ,! M k m ; l n can be
extended to the functor F k m ; l n which to any M k m(C )-bundle A k m over X associates the M k m ; l n-bundle
F k m ; l n(A k m) over X.
On the set of M k m ; l n-bundles (for arbitrary m; n  1) over X we consider the following equivalence
relation
A k m ; l n  B k r ; l s , there are vector bundles  k t ;  k v
with bers C k t
; C k v
; respectively, and integers u; w  1 such that
A k m ; l
n
F k t ; l u(End( k t ))  = B k r ; l
s
F k v ; l w(End( k v )): (1)
Then the k-primary component GBr k (X) of GBr(X) is dened to be the group of equivalence classes
of this relation with respect to the product induced by the tensor product of M k; l
-bundles.
Note that the k-primary component GBr k
(X) of the generalized Brauer group, similarly to Br k
(X),
can be treated as the group of obstructions to the lifting of locally trivial bundles A k m ; l n with bers
M k m ; l n to bundles of the form F k m ; l n(End( k m)) for some vector C k m
-bundle  k m : Compared to the
classical case, we see that a whole new step is added to the procedure of lifting in (1): rst of all, we
have to lift locally trivial bundles A k m ; l n to bundles of the form F k m ; l n(A k m) for some M k m(C )-bundle
A k m (the second step essentially coincides with the classical case).
Suppose a nite CW-complex X is obtained by applying the forgetful functor to some algebraic
variety e
X over C . Then the group Br(X) is not only a homotopy invariant of X but also a birational
invariant of e
X. I believe that the group GBr(X) should be a birational invariant of e
X, too.
1