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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 R-algebras where R 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 R 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 â€“ 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 R is an R-algebra A obeying any of the following equivalent conditions: # There exists an R-algebra B 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 R-algebras B \otimes_R A 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 R. # The R-algebra A^ \otimes_R A 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 R, where A^ 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 A. # The center of A is R, and A is separable. # A is finitely generated, faithful, and projective as an R-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 ...
A \otimes_R A^ is isomorphic to \text_R(A) via the map sending a \otimes b to the endomorphism x\mapsto axb of A.


Examples over a field

Over a field k, 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 \mathrm_n(D) for some division algebra D over k whose center is just k. 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 (R,\mathfrak), an R-algebra A is Azumaya if and only if A is free of positive finite rank as an R-module, and the algebra A\otimes_R(R/\mathfrak) is a central simple algebra over R/\mathfrak, hence all examples come from central simple algebras over R/\mathfrak.


Cyclic algebras

There is a class of Azumaya algebras called cyclic algebras which generate all similarity classes of Azumaya algebras over a field K, hence all elements in the Brauer group \text(K) (defined below). Given a finite cyclic Galois field extension L/K of degree n, for every b \in K^* and any generator \sigma \in \text(L/K) there is a twisted polynomial ring L \sigma, also denoted A(\sigma,b), generated by an element x such that : x^n =b and the following commutation property holds: : l\cdot x = \sigma(x)\cdot l. As a vector space over L, L \sigma has basis 1,x,x^2,\ldots, x^ with multiplication given by : x^i \cdot x^j = \begin x^ & \text i + j < n \\ x^b & \text i + j \geq n \\ \end Note that give a geometrically integral variety X/K, there is also an associated cyclic algebra for the quotient field extension \text(X_L)/\text(X).


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 R, where rings A,A' are similar if there is an isomorphism : A\otimes_RM_n(R) \cong A'\otimes_RM_m(R) of rings for some natural numbers n,m. Then, this equivalence is in fact an equivalence relation, and if A_1 \sim A_1', A_2 \sim A_2', then A_1\otimes_RA_2 \sim A_1'\otimes_RA_2', showing : _1otimes _2= _1\otimes_R A_2 is a well defined operation. This forms a group structure on the set of such equivalence classes called the Brauer group, denoted \text(R). Another definition is given by the torsion subgroup of the etale cohomology group : \text_\text(R) := \text_^2(\text(R),\mathbb_m)_\text which is called the cohomological Brauer group. These two definitions agree when R 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 E/F there is a cohomological Brauer group defined as : \text^\text(E/F):= H^2_(\text(E/F), E^\times) and the cohomological Brauer group for F is defined as : \text^\text(F) = \underset H^2_(\text(E/F), E^\times) where the colimit is taken over all finite Galois field extensions.


Computation for a local field

Over a local non-archimedean field F, such as the ''p''-adic numbers \mathbb_p,
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 : \text^\text(F) \cong \Q/\Z. This is because given abelian field extensions E_2/E_1/F there is a short exact sequence of Galois groups : 0 \to \text(E_2/E_1) \to \text(E_2/F) \to \text(E_1/F) \to 0 and from Local class field theory, there is the following commutative diagram: : \begin H^2_(\text(E_2/F),E_1^\times) &\to& H^2_( \text(E_1/F),E_1^\times) \\ \downarrow & & \downarrow \\ \left(\frac\Z\right)/\Z & \to & \left(\frac\Z\right)/\Z \end where the vertical maps are isomorphisms and the horizontal maps are injections.


''n''-torsion for a field

Recall that there is the Kummer sequence : 1 \to \mu_n \to \mathbb_m \xrightarrow \mathbb_m \to 1 giving a long exact sequence in cohomology for a field F. 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 H^1(F,\mathbb_m) = 0, there is an associated short exact sequence : 0 \to H^2_(F,\mu_n) \to \text(F) \xrightarrow \text(F) \to 0 showing the second etale cohomology group with coefficients in the nth roots of unity \mu_n is : H^2_(F,\mu_n) = \text(F)_.


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 n-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 K_2^M(F)\otimes \Z /n to the etale cohomology group H^2_(F,\mu_n^), denoted by : R_:K_2^M(F)\otimes_\Z \Z /n\Z \to H^2_(F,\mu_n^) It comes from the composition of the cup product in etale cohomology with the Hilbert's Theorem 90 isomorphism : \chi_:F^*\otimes_\Z\Z/n \to H^1_\text(F,\mu_n) hence : R_(\) = \chi_(a)\cup \chi_(b) It turns out this map factors through H^2_\text(F,\mu_n) = \text(F)_, whose class for \ is represented by a cyclic algebra (\sigma, b)/math>. For the
Kummer extension Kummer is a German surname. Notable people with the surname include: *Bernhard Kummer (1897–1962), German Germanist * Clare Kummer (1873–1958), American composer, lyricist and playwright * Clarence Kummer (1899–1930), American jockey * Chris ...
E/F where E = F(\sqrt , take a generator \sigma \in \text(E/F) of the cyclic group, and construct (\sigma,b)/math>. There is an alternative, yet equivalent construction through
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 ...
and etale cohomology. Consider the short exact sequence of trivial \text(\overline/F)-modules : 0 \to \Z \to \Z \to \Z /n \to 0 The long exact sequence yields a map : H^1_\text(F,\Z /n) \xrightarrow H^2_\text(F,\Z ) For the unique character : \chi:\text(E/F) \to \Z /n with \chi(\sigma) = 1, there is a unique lift : \overline:\text(\overline/F) \to \Z /n and : \delta(\overline)\cup (b) = (\sigma,b)\in \text(F) note the class (b) is from the Hilberts theorem 90 map \chi_(b). Then, since there exists a primitive root of unity \zeta \in \mu_n \subset F, there is also a class : \delta(\overline)\cup(b) \cup (\zeta) \in H^2_\text(F,\mu_n^) It turns out this is precisely the class R_(\). Because of the
norm residue isomorphism theorem 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 ...
, R_ is an isomorphism and the n-torsion classes in \text(F)_ are generated by the cyclic algebras (\sigma,b)/math>.


Skolem–Noether theorem

One of the important structure results about Azumaya algebras is the Skolem–Noether theorem: given a local commutative ring R and an Azumaya algebra R \to A, the only automorphisms of A are inner. Meaning, the following map is surjective: : \begin A^* \to \text(A) \\ a \mapsto (x \mapsto a^xa) \end where A^* is the
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...
of units in A. This is important because it directly relates to the cohomological classification of similarity classes of Azumaya algebras over a scheme. In particular, it implies an Azumaya algebra has structure group \text_n for some n, and the
ÄŒech cohomology In mathematics, specifically algebraic topology, ÄŒech cohomology is a cohomology theory based on the intersection properties of open set, open cover (topology), covers of a topological space. It is named for the mathematician Eduard ÄŒech. Moti ...
group : \check^1((X)_,\text_n) gives a cohomological classification of such bundles. Then, this can be related to H^2_\text(X,\mathbb_m) using the exact sequence : 1 \to \mathbb_m \to \text_n \to \text_n \to 1 It turns out the image of H^1 is a subgroup of the torsion subgroup H^2_\text(X,\mathbb_m)_.


On a scheme

An Azumaya algebra on a scheme ''X'' with
structure sheaf In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of ...
\mathcal_X, according to the original Grothendieck seminar, is a sheaf \mathcal of \mathcal_X-algebras that is étale locally isomorphic to a matrix algebra sheaf; one should, however, add the condition that each matrix algebra sheaf is of positive rank. This definition makes an Azumaya algebra on (X,\mathcal_X) into a 'twisted-form' of the sheaf M_n(\mathcal_X). Milne, ''Étale Cohomology'', starts instead from the definition that it is a sheaf \mathcal of \mathcal_X-algebras whose stalk \mathcal_x at each point x is an Azumaya algebra over the
local ring In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of ...
\mathcal_ in the sense given above. Two Azumaya algebras \mathcal_1 and \mathcal_2 are ''equivalent'' if there exist
locally free sheaves In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
\mathcal_1 and \mathcal_2 of finite positive rank at every point such that : A_1\otimes\mathrm_(\mathcal_1) \simeq A_2\otimes\mathrm_(\mathcal_2), where \mathrm_(\mathcal_i) is the endomorphism sheaf of \mathcal_i. The Brauer group B(X) of X (an analogue 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 ...
of a field) is the set of equivalence classes of Azumaya algebras. The group operation is given by tensor product, and the inverse is given by the opposite algebra. Note that this is distinct from the cohomological Brauer group which is defined as H^2_\text(X,\mathbb_m).


Example over Spec(Z /''n''

The construction of a quaternion algebra over a field can be globalized to \text(\Z /n by considering the noncommutative \Z /n/math>-algebra : A_ = \frac then, as a sheaf of \mathcal_X-algebras, \mathcal_ has the structure of an Azumaya algebra. The reason for restricting to the open affine set \text(\Z /n \hookrightarrow \text(\Z) is because the quaternion algebra is a division algebra over the points (p) is and only if the
Hilbert symbol In mathematics, the Hilbert symbol or norm-residue symbol is a function (–, –) from ''K''× × ''K''× to the group of ''n''th roots of unity in a local field ''K'' such as the fields of real number, reals or p-adic numbers. It is related to rec ...
: (a,b)_p = 1 which is true at all but finitely many primes.


Example over P''n''

Over \mathbb^n_k Azumaya algebras can be constructed as \mathcal_k(\mathcal)\otimes_k A for an Azumaya algebra A over a field k. For example, the endomorphism sheaf of \mathcal(a)\oplus \mathcal(b) is the matrix sheaf : \mathcal_k(\mathcal(a)\oplus \mathcal(b)) = \begin \mathcal & \mathcal(b-a) \\ \mathcal(a-b) & \mathcal \end so an Azumaya algebra over \mathbb^n_k can be constructed from this sheaf tensored with an Azumaya algebra A over k, such as a quaternion algebra.


Applications

There have been significant applications of Azumaya algebras in
diophantine geometry In mathematics, Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic geometry. By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to study ...
, following work of
Yuri Manin Yuri Ivanovich Manin (; 16 February 1937 – 7 January 2023) was a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logic to theoretical physics. Life an ...
. The Manin obstruction to the
Hasse principle In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each d ...
is defined using the Brauer group of schemes.


See also

*
Gerbe In mathematics, a gerbe (; ) is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud following ideas of Alexandre Grothendieck as a tool for non-commutative cohomology in degree 2. They can be seen as an analog ...
*
Class field theory In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credit ...
*
Algebraic K-theory Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sens ...
*
Motivic cohomology Motivic cohomology is an invariant of algebraic varieties and of more general schemes. It is a type of cohomology related to motives and includes the Chow ring of algebraic cycles as a special case. Some of the deepest problems in algebraic geome ...
*
Norm residue isomorphism theorem 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 ...


References

; Brauer group and Azumaya algebras : * Milne, John
Etale cohomology
Ch IV * * Mathoverflow thread on
Explicit examples of Azumaya algebras
; Division algebras : * * {{refend Ring theory Scheme theory Algebras