Brauer group

Inmathematics,theBrauer groupof afieldKis anabelian groupwhose elements areMorita equivalenceclassesofcentral simple algebrasoverK,with addition given by thetensor product of algebras.It was defined by the algebraistRichard Brauer.

The Brauer group arose out of attempts to classifydivision algebrasover a field. It can also be defined in terms ofGalois cohomology.More generally, the Brauer group of aschemeis defined in terms ofAzumaya algebras,or equivalently usingprojective bundles.

Construction

Acentral simple algebra(CSA) over a fieldKis afinite-dimensionalassociativeK-algebraAsuch thatAis asimple ringand thecenterofAis equal toK.Note that CSAs are in generalnotdivision algebras, though CSAs can be used to classify division algebras.

For example, thecomplex numbersCform a CSA over themselves, but not overR(the center isCitself, hence too large to be CSA overR). The finite-dimensional division algebras with centerR(that means the dimension overRis finite) are thereal numbersand thequaternionsbya theorem of Frobenius,while anymatrix ringover the reals or quaternions –M(n,R)orM(n,H)– is a CSA over the reals, but not a division algebra (ifn> 1).

We obtain anequivalence relationon CSAs overKby theArtin–Wedderburn theorem(Wedderburn's part, in fact), to express any CSA as aM(n,D)for some division algebraD.If we look just atD,that is, if we impose an equivalence relation identifyingM(m,D)withM(n,D)for all positiveintegersmandn,we get theBrauer equivalencerelation on CSAs overK.The elements of the Brauer group are the Brauer equivalence classes of CSAs overK.

Given central simple algebrasAandB,one can look at theirtensor productA⊗Bas aK-algebra.It turns out that this is always central simple. A slick way to see this is to use a characterization: a central simple algebraAoverKis aK-algebrathat becomes a matrix ring when we extend the field of scalars to analgebraic closureofK.This result also shows that the dimension of a central simple algebraAas aK-vector space is always asquare.ThedegreeofAis defined to be thesquare rootof its dimension.

As a result, theisomorphism classesof CSAs overKform amonoidunder tensor product, compatible with Brauer equivalence, and the Brauer classes are allinvertible:the inverse of an algebraAis given by its opposite algebraA^op(theopposite ringwith the same action byKsince the image ofK→Ais in the center ofA). Explicitly, for a CSAAwe haveA⊗A^op= M(n²,K),wherenis the degree ofAoverK.

The Brauer group of any field is atorsion group.In more detail, define theperiodof a central simple algebraAoverKto be itsorderas an element of the Brauer group. Define theindexofAto be the degree of the division algebra that is Brauer equivalent toA.Then the period ofAdivides the index ofA(and hence is finite).

Examples

• In the following cases, every finite-dimensional central division algebra over a fieldKisKitself, so that the Brauer group Br(K) istrivial:
  • Kis analgebraically closed field.
  • Kis afinite field(Wedderburn's theorem).Equivalently, every finite division ring iscommutative.
  • Kis thefunction fieldof analgebraic curveover an algebraically closed field (Tsen's theorem).More generally, the Brauer group vanishes for anyC₁field.
  • Kis an algebraic extension ofQcontaining allroots of unity.
• The Brauer group Br Rof the real numbers is thecyclic groupofordertwo. There are just two non-isomorphicreal division algebras with centerR:Ritself and the quaternion algebraH.SinceH ⊗ H≅ M(4,R),the class ofHhas order two in the Brauer group.
• LetKbe anon-Archimedeanlocal field,meaning thatKis complete under adiscrete valuationwith finite residue field. Then Br Kis isomorphic toQ/Z.

Severi–Brauer varieties

Another important interpretation of the Brauer group of a fieldKis that it classifies theprojective varietiesoverKthat become isomorphic toprojective spaceover an algebraic closure ofK.Such a variety is called aSeveri–Brauer variety,and there is aone-to-one correspondencebetween the isomorphism classes of Severi–Brauer varieties of dimensionn− 1overKand the central simple algebras of degreenoverK.

For example, the Severi–Brauer varieties of dimension 1 are exactly thesmoothconicsin the projective plane overK.For a fieldKofcharacteristicnot 2, every conic overKis isomorphic to one of the formax²+by²=z²for some nonzero elementsaandbofK.The corresponding central simple algebra is thequaternion algebra

$(a,b)=K\langle i,j\rangle /(i^{2}=a,j^{2}=b,ij=-ji).$

The conic is isomorphic to the projective lineP¹overKif and only ifthe corresponding quaternion algebra is isomorphic to the matrix algebra M(2,K).

Cyclic algebras

For a positive integern,letKbe a field in whichnis invertible such thatKcontains aprimitiventh root of unityζ.For nonzero elementsaandbofK,the associatedcyclic algebrais the central simple algebra of degreenoverKdefined by

$(a,b)_{\zeta }=K\langle u,v\rangle /(u^{n}=a,v^{n}=b,uv=\zeta vu).$

Cyclic algebras are the best-understood central simple algebras. (Whennis not invertible inKorKdoes not have a primitiventh root of unity, a similar construction gives the cyclic algebra(χ,a)associated to a cyclicZ/n-extensionχofKand a nonzero elementaofK.)

TheMerkurjev–Suslin theoreminalgebraic K-theoryhas a strong consequence about the Brauer group. Namely, for a positive integern,letKbe a field in whichnis invertible such thatKcontains a primitiventh root of unity. Then thesubgroupof the Brauer group ofKkilled bynis generated by cyclic algebras of degreen.Equivalently, any division algebra of period dividingnis Brauer equivalent to a tensor product of cyclic algebras of degreen.Even for aprime numberp,there are examples showing that a division algebra of periodpneed not be actually isomorphic to a tensor product of cyclic algebras of degreep.

It is a majoropen problem(raised byAlbert) whether every division algebra of prime degree over a field is cyclic. This is true if the degree is 2 or 3, but the problem is wide open for primes at least 5. The known results are only for special classes of fields. For example, ifKis aglobal fieldorlocal field,then a division algebra of any degree overKis cyclic, by Albert–Brauer–Hasse–Noether.A "higher-dimensional" result in the same direction wasprovedby Saltman: ifKis a field oftranscendence degree1 over the local fieldQ_p,then every division algebra of prime degreel≠poverKis cyclic.

The period-index problem

For any central simple algebraAover a fieldK,the period ofAdivides the index ofA,and the two numbers have the same prime factors.Theperiod-index problemis to bound the index in terms of the period, for fieldsKof interest. For example, ifAis a central simple algebra over a local field or global field, then Albert–Brauer–Hasse–Noether showed that the index ofAis equal to the period ofA.

For a central simple algebraAover a fieldKof transcendence degreenover an algebraically closed field, it isconjecturedthat ind(A) divides per(A)ⁿ⁻¹.This is true forn≤ 2,the casen= 2being an important advance byde Jong,sharpened in positive characteristic by de Jong–Starr and Lieblich.

Class field theory

The Brauer group plays an important role in the modern formulation ofclass field theory.IfK_vis a non-Archimedean local field,local class field theorygives a canonical isomorphisminv_v:Br K_v→Q/Z,theHasse invariant.

The case of a global fieldK(such as anumber field) is addressed byglobal class field theory.IfDis a central simple algebra overKandvis aplaceofK,thenD ⊗ K_vis a central simple algebra overK_v,the completion ofKatv.This defines ahomomorphismfrom the Brauer group ofKinto the Brauer group ofK_v.A given central simple algebraDsplits for all but finitely manyv,so that the image ofDunder almost all such homomorphisms is 0. The Brauer group Br Kfits into anexact sequenceconstructed by Hasse:

$0\rightarrow \operatorname {Br} K\rightarrow \bigoplus _{v\in S}\operatorname {Br} K_{v}\rightarrow \mathbf {Q} /\mathbf {Z} \rightarrow 0,$

whereSis the set of all places ofKand the right arrow is the sum of the local invariants; the Brauer group of the real numbers is identified with (1/2)Z/Z.Theinjectivityof the left arrow is the content of theAlbert–Brauer–Hasse–Noether theorem.

The fact that the sum of all local invariants of a central simple algebra overKis zero is a typicalreciprocity law.For example, applying this to a quaternion algebra(a,b)overQgives thequadratic reciprocity law.

Galois cohomology

For an arbitrary fieldK,the Brauer group can be expressed in terms ofGalois cohomologyas follows:

$\operatorname {Br} K\cong H^{2}(K,G_{\text{m}}),$

whereG_mdenotes themultiplicative group,viewed as analgebraic groupoverK.More concretely, the cohomology group indicated meansH²(Gal(K_s/K),K_s^*),whereK_sdenotes aseparable closureofK.

The isomorphism of the Brauer group with a Galois cohomology group can be described as follows. Theautomorphism groupof the algebra ofn-by-nmatricesis theprojective linear groupPGL(n). Since all central simple algebras overKbecome isomorphic to the matrix algebra over a separable closure ofK,the set of isomorphism classes of central simple algebras of degreenoverKcan be identified with the Galois cohomology setH¹(K,PGL(n)).The class of a central simple algebra inH²(K,G_m)is the image of its class inH¹under the boundary homomorphism

$H^{1}(K,\operatorname {PGL} (n))\rightarrow H^{2}(K,G_{\text{m}})$

associated to theshort exact sequence1 →G_m→ GL(n) → PGL(n) → 1.

The Brauer group of a scheme

The Brauer group was generalized from fields tocommutative ringsbyAuslanderandGoldman.Grothendieckwent further by defining the Brauer group of anyscheme.

There are two ways of defining the Brauer group of a schemeX,using eitherAzumaya algebrasoverXorprojective bundlesoverX.The second definition involves projective bundles that are locally trivial in theétale topology,not necessarily in theZariski topology.In particular, a projective bundle is defined to be zero in the Brauer group if and only if it is the projectivization of some vector bundle.

Thecohomological Brauer groupof aquasi-compactschemeXis defined to be thetorsion subgroupof theétale cohomologygroupH²(X,G_m).(The whole groupH²(X,G_m)need not be torsion, although it is torsion forregular schemesX.) The Brauer group is always a subgroup of the cohomological Brauer group.Gabbershowed that the Brauer group is equal to the cohomological Brauer group for any scheme with an ample line bundle (for example, anyquasi-projectivescheme over a commutative ring).

The whole groupH²(X,G_m)can be viewed as classifying thegerbesoverXwith structure groupG_m.

For smooth projective varieties over a field, the Brauer group is abirationalinvariant. It has been fruitful. For example, whenXis alsorationally connectedover the complex numbers, the Brauer group ofXis isomorphic to the torsion subgroup of thesingular cohomologygroupH³(X,Z),which is therefore a birational invariant.ArtinandMumfordused this description of the Brauer group to give the first example of aunirational varietyXoverCthat is not stably rational (that is, no product ofXwith a projective space is rational).

Relation to the Tate conjecture

Artin conjectured that everyproper schemeover the integers has finite Brauer group.This is far from known even in the special case of a smooth projective varietyXover a finite field. Indeed, the finiteness of the Brauer group for surfaces in that case is equivalent to theTate conjecturefordivisorsonX,one of the main problems in the theory ofalgebraic cycles.

For a regularintegralscheme of dimension 2 which isflatand proper over thering of integersof a number field, and which has asection,the finiteness of the Brauer group is equivalent to the finiteness of theTate–Shafarevich groupШ for theJacobian varietyof the general fiber (a curve over a number field).The finiteness of Ш is a central problem in the arithmetic ofelliptic curvesand more generallyabelian varieties.

The Brauer–Manin obstruction

LetXbe a smooth projective variety over a number fieldK.TheHasse principlewould predict that ifXhas arational pointover all completionsK_vofK,thenXhas aK-rational point. The Hasse principle holds for some special classes of varieties, but not in general.Maninused the Brauer group ofXto define theBrauer–Manin obstruction,which can be applied in many cases to show thatXhas noK-points even whenXhas points over all completions ofK.

References

• Colliot-Thélène, Jean-Louis(1995),"Birational invariants, purity and the Gersten conjecture"(PDF),K-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras (Santa Barbara, 1992),Proceedings of Symposia in Pure Mathematics, vol. 58, Part 1,American Mathematical Society,pp. 1–64,ISBN978-0821803394,MR1327280
• de Jong, Aise Johan(2004), "The period-index problem for the Brauer group of an algebraic surface",Duke Mathematical Journal,123:71–94,doi:10.1215/S0012-7094-04-12313-9,MR2060023
• de Jong, A. J."A result of Gabber"(PDF).
• Farb, Benson;Dennis, R. Keith (1993).Noncommutative Algebra.Graduate Texts in Mathematics. Vol. 144.Springer-Verlag.ISBN978-0387940571.MR1233388.
• Gille, Philippe; Szamuely, Tamás (2006).Central Simple Algebras and Galois Cohomology.Cambridge Studies in Advanced Mathematics. Vol. 101. Cambridge:Cambridge University Press.ISBN0-521-86103-9.MR2266528.
• Grothendieck, Alexander(1968),"Le groupe de Brauer, I–III"(PDF),inGiraud, Jean;Grothendieck, Alexander;Kleiman, Steven L.;et al. (eds.),Dix Exposés sur la Cohomologie des Schémas,Advanced Studies in Pure Mathematics, vol. 3, Amsterdam: North-Holland, pp. 46–66, 67–87, 88–188,MR0244271
• V.A. Iskovskikh (2001) [1994],k"Brauer group of a fieldk",Encyclopedia of Mathematics,EMS Press
• Milne, James S.(1980),Étale Cohomology,Princeton Mathematical Series, vol. 33,Princeton University Press,ISBN978-0-691-08238-7,MR0559531
• Pierce, Richard (1982).Associative Algebras.Graduate Texts in Mathematics.Vol. 88. New York–Berlin:Springer-Verlag.ISBN0-387-90693-2.MR0674652.
• Saltman, David J. (1999).Lectures on Division Algebras.Regional Conference Series in Mathematics. Vol. 94. Providence, RI:American Mathematical Society.ISBN0-8218-0979-2.MR1692654.
• Saltman, David J. (2007), "Cyclic algebras overp-adic curves ",Journal of Algebra,314(2): 817–843,arXiv:math/0604409,doi:10.1016/j.jalgebra.2007.03.003,MR2344586,S2CID119160155
• Serre, Jean-Pierre(1979).Local Fields.Graduate Texts in Mathematics.Vol. 67. Translated byGreenberg, Marvin Jay.Springer-Verlag.ISBN0-387-90424-7.MR0554237.
• Tate, John(1994), "Conjectures on algebraic cycles inl-adic cohomology ",Motives,Proceedings of Symposia in Pure Mathematics, vol. 55, Part 1, American Mathematical Society, pp. 71–83,ISBN0-8218-1636-5,MR1265523