GetWiki

In mathematics, an Azumaya algebra is a generalization of central simple algebras to R-algebras where R need not be a field. 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, where Alexander Grothendieck made it the basis for his geometric theory of the Brauer group in Bourbaki seminars from 1964–65. There are now several points of access to the basic definitions.

Over a ring

An Azumaya algebraBOOK, Milne, James S.,weblink Étale cohomology, 1980, Princeton University Press, 0-691-08238-3, Princeton, N.J., 5028959,weblink 21 June 2020, JOURNAL, Borceux, Francis, Vitale, Enrico, Azumaya categories, Applied Categorical Structures, 10, 449–467, 2002,weblink over a commutative ring R is an R-algebra A obeying any of the following equivalent conditions:

1. There exists an R-algebra B such that the tensor product of R-algebras B otimes_R A is Morita equivalent to R.
2. The R-algebra A^{mathrm{op}} otimes_R A is Morita equivalent to R, where A^{mathrm{op}} is the opposite algebra of A.
3. The center of A is R, and A is separable.
4. A is finitely generated, faithful, and projective as an R-module, and the tensor product A otimes_R A^{mathrm{op}} is isomorphic to text{End}_R(A) via the map sending a otimes b to the endomorphism xmapsto 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 algebras. These are algebras isomorphic to the matrix ring mathrm{M}_n(D) for some division algebra D over k whose center is just k. For example, quaternion algebras provide examples of central simple algebras.

Examples over local rings

Given a local commutative ring (R,mathfrak{m}), an R-algebra A is Azumaya if and only if A is free of positive finite rank as an R-module, and the algebra Aotimes_R(R/mathfrak{m}) is a central simple algebra over R/mathfrak{m}, hence all examples come from central simple algebras over R/mathfrak{m}.

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{Br}(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{Gal}(L/K) there is a twisted polynomial ring L[x]_sigma, also denoted A(sigma,b), generated by an element x such that and the following commutation property holds: As a vector space over L, L[x]_sigma has basis 1,x,x^2,ldots, x^{n-1} with multiplication given by x^{i + j} & text{ if } i + j < n x^{i + j - n}b & text{ if } i + j geq n end{cases}Note that give a geometrically integral varietymeaning it is an integral variety when extended to the algebraic closure of its base field X/K, there is also an associated cyclic algebra for the quotient field extension text{Frac}(X_L)/text{Frac}(X).

Brauer group of a ring

Over fields, there is a cohomological classification of Azumaya algebras using Étale cohomology. In fact, this group, called the Brauer group, can be also defined as the similarity classes{{rp|3}} of Azumaya algebras over a ring R, where rings A,A' are similar if there is an isomorphism 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_1otimes_RA_2 sim A_1'otimes_RA_2', showing is a well defined operation. This forms a group structure on the set of such equivalence classes called the Brauer group, denoted text{Br}(R). 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 R is a field.

Brauer group using Galois cohomology

There is another equivalent definition of the Brauer group using Galois cohomology. For a field extension E/F there is a cohomological Brauer group defined as and the cohomological Brauer group for F 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 F, such as the p-adic numbers mathbb{Q}_p, local class field theory gives the isomorphism of abelian groups:BOOK, Serre, Jean-Pierre.,weblink Local Fields, 1979, Springer New York, 978-1-4757-5673-9, New York, NY, 859586064, pg 193 This is because given abelian field extensions E_2/E_1/F there is a short exact sequence of Galois groups and from Local class field theory, there is the following commutative diagram:WEB, Lectures on Cohomological Class Field Theory,weblink live,weblink 22 June 2020, H^2_{text{Gal}}(text{Gal}(E_2/F),E_1^times) &to& H^2_{text{Gal}}( text{Gal}(E_1/F),E_1^times) downarrow & & downarrow left(frac{1}{[E_2:E_1]}Zright)/Z & to & left(frac{1}{[E_1:F]}Zright)/Zend{matrix}where the vertical maps are isomorphisms and the horizontal maps are injections.

n-torsion for a field

Recall that there is the Kummer sequenceBOOK, Srinivas, V.,weblink Algebraic K-Theory, 1994, Birkhäuser Boston, 978-0-8176-4739-1, Second, Boston, MA, 145–193, 8. The Merkurjev-Suslin Theorem, 853264222, giving a long exact sequence in cohomology for a field F. Since Hilbert's Theorem 90 implies H^1(F,mathbb{G}_m) = 0, there is an associated short exact sequence showing the second etale cohomology group with coefficients in the nth roots of unity mu_n is

Generators of n-torsion classes in the Brauer group over a field

The Galois symbol, or norm-residue symbol, is a map from the n-torsion Milnor K-theory group K_2^M(F)otimes Z /n to the etale cohomology group H^2_{et}(F,mu_n^{otimes 2}), 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 H^2_text{et}(F,mu_n) = text{Br}(F)_{ntext{-tors}}, whose class for {a,b } is represented by a cyclic algebra [A(sigma, b)]. For the Kummer extension E/F where E = F(sqrt[n]{a}), take a generator sigma in text{Gal}(E/F) of the cyclic group, and construct [A(sigma,b)]. There is an alternative, yet equivalent construction through Galois cohomology and etale cohomology. Consider the short exact sequence of trivial text{Gal}(overline{F}/F)-modules The long exact sequence yields a map For the unique character with chi(sigma) = 1, there is a unique lift and note the class (b) is from the Hilberts theorem 90 map chi_{n,F}(b). Then, since there exists a primitive root of unity zeta in mu_n subset F, there is also a class It turns out this is precisely the class R_{n,F}({a,b}). Because of the norm residue isomorphism theorem, R_{n,F} is an isomorphism and the n-torsion classes in text{Br}(F)_{ntext{-tors}} are generated by the cyclic algebras [A(sigma,b)].

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: where A^* is the group 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{PGL}_n for some n, and the Čech cohomology group gives a cohomological classification of such bundles. Then, this can be related to H^2_text{et}(X,mathbb{G}_m) using the exact sequence It turns out the image of H^1 is a subgroup of the torsion subgroup H^2_text{et}(X,mathbb{G}_m)_{tors}.

On a scheme

An Azumaya algebra on a scheme X with structure sheaf mathcal{O}_X, according to the original Grothendieck seminar, is a sheaf mathcal{A} of mathcal{O}_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{O}_X) into a 'twisted-form' of the sheaf M_n(mathcal{O}_X). Milne, Étale Cohomology, starts instead from the definition that it is a sheaf mathcal{A} of mathcal{O}_X-algebras whose stalk mathcal{A}_x at each point x is an Azumaya algebra over the local ring mathcal{O}_{X,x} in the sense given above.Two Azumaya algebras mathcal{A}_1 and mathcal{A}_2 are equivalent if there exist locally free sheaves mathcal{E}_1 and mathcal{E}_2 of finite positive rank at every point such that where mathrm{End}_{mathcal{O}_X}(mathcal{E}_i) is the endomorphism sheaf of mathcal{E}_i. The Brauer group B(X) of X (an analogue of the Brauer group 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{et}(X,mathbb{G}_m).

Example over Spec(Z[1/n])

The construction of a quaternion algebra over a field can be globalized to text{Spec}(Z[1/n]) by considering the noncommutative Z [1/n]-algebra then, as a sheaf of mathcal{O}_X-algebras, mathcal{A}_{a,b} has the structure of an Azumaya algebra. The reason for restricting to the open affine set text{Spec}(Z[1/n]) hookrightarrow text{Spec}(Z) is because the quaternion algebra is a division algebra over the points (p) is and only if the Hilbert symbol which is true at all but finitely many primes.

Example over Pn

Over mathbb{P}^n_k Azumaya algebras can be constructed as mathcal{End}_k(mathcal{E})otimes_k A for an Azumaya algebra A over a field k. For example, the endomorphism sheaf of mathcal{O}(a)oplus mathcal{O}(b) is the matrix sheaf end{pmatrix}so an Azumaya algebra over mathbb{P}^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, following work of Yuri Manin. The Manin obstruction to the Hasse principle is defined using the Brauer group of schemes.

See also

• Gerbe
• Class field theory
• Algebraic K-theory
• Motivic cohomology
• Norm residue isomorphism theorem

References

{{reflist}}

Brauer group and Azumaya algebras :

• Milne, John. Etale cohomology. Ch IV
• {{citation | last1=Knus | first1=Max-Albert | last2=Ojanguren | first2=Manuel | title=Théorie de la descente et algèbres d'Azumaya | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Mathematics | doi=10.1007/BFb0057799 | mr=0417149 | zbl=0284.13002 | year=1974 | volume=389}}
• Mathoverflow thread on "Explicit examples of Azumaya algebras"

Division algebras :

• {{citation | last=Knus | first=Max-Albert | title=Quadratic and Hermitian forms over rings | series=Grundlehren der Mathematischen Wissenschaften | volume=294 | location=Berlin etc. | publisher=Springer-Verlag | year=1991 | isbn=3-540-52117-8 | zbl=0756.11008 }}
• BOOK, Saltman, David J., Lectures on division algebras, Regional Conference Series in Mathematics, 94, Providence, RI, American Mathematical Society, 1999, 0-8218-0979-2, 0934.16013,