In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an Azumaya algebra is a generalization of
central simple algebra
In ring theory and related areas of mathematics a central simple algebra (CSA) over a field ''K'' is a finite-dimensional associative ''K''-algebra ''A'' that is simple, and for which the center is exactly ''K''. (Note that ''not'' every simple ...
s to
-algebras where
need not be a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
. Such a notion was introduced in a 1951 paper of
Goro Azumaya, for the case where
is a
commutative local ring. The notion was developed further in
ring theory, and in
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, where
Alexander Grothendieck
Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research ext ...
made it the basis for his geometric theory of the
Brauer group
In mathematics, the Brauer group of a field ''K'' is an abelian group whose elements are Morita equivalence classes of central simple algebras over ''K'', with addition given by the tensor product of algebras. It was defined by the algebraist ...
in
Bourbaki seminars from 1964–65. There are now several points of access to the basic definitions.
Over a ring
An Azumaya algebra
over a commutative ring
is an
-algebra
obeying any of the following equivalent conditions:
# There exists an
-algebra
such that the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...
of
-algebras
is
Morita equivalent
In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. More precisely, two rings ''R'', ''S'' are Morita equivalent (denoted by R\approx S) if their categories of modules ar ...
to
.
# The
-algebra
is
Morita equivalent
In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. More precisely, two rings ''R'', ''S'' are Morita equivalent (denoted by R\approx S) if their categories of modules ar ...
to
, where
is the
opposite
In lexical semantics, opposites are words lying in an inherently incompatible binary relationship. For example, something that is ''even'' entails that it is not ''odd''. It is referred to as a 'binary' relationship because there are two members i ...
algebra of
.
# The
center of
is
, and
is
separable.
#
is
finitely generated, faithful, and
projective as an
-module, and the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...
is isomorphic to
via the map sending
to the endomorphism
of
.
Examples over a field
Over a field
, Azumaya algebras are completely classified by the
Artin–Wedderburn theorem since they are the same as
central simple algebra
In ring theory and related areas of mathematics a central simple algebra (CSA) over a field ''K'' is a finite-dimensional associative ''K''-algebra ''A'' that is simple, and for which the center is exactly ''K''. (Note that ''not'' every simple ...
s. These are algebras isomorphic to the matrix ring
for some division algebra
over
whose center is just
. For example,
quaternion algebra
In mathematics, a quaternion algebra over a field (mathematics), field ''F'' is a central simple algebra ''A'' over ''F''See Milies & Sehgal, An introduction to group rings, exercise 17, chapter 2. that has dimension (vector space), dimension 4 ove ...
s provide examples of central simple algebras.
Examples over local rings
Given a local commutative ring
, an
-algebra
is Azumaya if and only if
is free of positive finite rank as an
-module, and the algebra
is a central simple algebra over
, hence all examples come from central simple algebras over
.
Cyclic algebras
There is a class of Azumaya algebras called cyclic algebras which generate all similarity classes of Azumaya algebras over a field
, hence all elements in the Brauer group
(defined below). Given a finite cyclic Galois field extension
of degree
, for every
and any generator
there is a twisted polynomial ring
, also denoted
, generated by an element
such that
:
and the following commutation property holds:
:
As a vector space over
,
has basis
with multiplication given by
:
Note that give a geometrically integral variety
, there is also an associated cyclic algebra for the quotient field extension
.
Brauer group of a ring
Over fields, there is a cohomological classification of Azumaya algebras using
Étale cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectu ...
. In fact, this group, called the
Brauer group
In mathematics, the Brauer group of a field ''K'' is an abelian group whose elements are Morita equivalence classes of central simple algebras over ''K'', with addition given by the tensor product of algebras. It was defined by the algebraist ...
, can be also defined as the similarity classes
of Azumaya algebras over a ring
, where rings
are similar if there is an isomorphism
:
of rings for some natural numbers
. Then, this equivalence is in fact an equivalence relation, and if
,
, then
, showing
:
is a well defined operation. This forms a group structure on the set of such equivalence classes called the Brauer group, denoted
. Another definition is given by the torsion subgroup of the etale cohomology group
:
which is called the cohomological Brauer group. These two definitions agree when
is a field.
Brauer group using Galois cohomology
There is another equivalent definition of the Brauer group using
Galois cohomology In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated with a field extension ''L''/''K'' acts in a na ...
. For a field extension
there is a cohomological Brauer group defined as
:
and the cohomological Brauer group for
is defined as
:
where the colimit is taken over all finite Galois field extensions.
Computation for a local field
Over a local non-archimedean field
, such as the
''p''-adic numbers
,
local class field theory In mathematics, local class field theory, introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite re ...
gives the isomorphism of abelian groups:
pg 193
:
This is because given abelian field extensions
there is a short exact sequence of Galois groups
:
and from Local class field theory, there is the following commutative diagram:
:
where the vertical maps are isomorphisms and the horizontal maps are injections.
''n''-torsion for a field
Recall that there is the Kummer sequence
:
giving a long exact sequence in cohomology for a field
. Since
Hilbert's Theorem 90
In abstract algebra, Hilbert's Theorem 90 (or Satz 90) is an important result on cyclic extensions of field (mathematics), fields (or to one of its generalizations) that leads to Kummer theory. In its most basic form, it states that if ''L''/''K'' ...
implies
, there is an associated short exact sequence
:
showing the second etale cohomology group with coefficients in the
th roots of unity
is
:
Generators of ''n''-torsion classes in the Brauer group over a field
The
Galois symbol
In mathematics, the norm residue isomorphism theorem is a long-sought result relating Milnor ''K''-theory and Galois cohomology. The result has a relatively elementary formulation and at the same time represents the key juncture in the proofs of ...
, or norm-residue symbol, is a map from the
-torsion
Milnor K-theory In mathematics, Milnor K-theory is an algebraic invariant (denoted K_*(F) for a field F) defined by as an attempt to study higher algebraic K-theory in the special case of fields. It was hoped this would help illuminate the structure for algebraic ...
group
to the etale cohomology group
, denoted by
:
It comes from the composition of the cup product in etale cohomology with the Hilbert's Theorem 90 isomorphism
:
hence
:
It turns out this map factors through
, whose class for
is represented by a cyclic algebra