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In [[mathematics]], the '''Brauer group''' of a [[field (mathematics)|field]] ''K'' is an [[abelian group]] whose elements are [[Morita equivalence]] classes of [[central simple algebra]]s of finite rank over ''K'' and addition is induced by the [[tensor product]] of algebras. It arose out of attempts to classify [[division algebra]]s over a field and is named after the algebraist [[Richard Brauer]]. The group may also be defined in terms of [[Galois cohomology]]. More generally, the Brauer group of a [[scheme (mathematics)|scheme]] is defined in terms of [[Azumaya algebra]]s. | |||
== Construction == | |||
A [[central simple algebra]] (CSA) over a field ''K'' is a finite-dimensional associative [[algebra over a field|''K''-algebra]] ''A'', which is a [[simple ring]], and for which the [[center of an algebra|center]] is exactly ''K''. Note that CSAs are in general ''not'' division algebras, though CSAs can be used to classify division algebras. | |||
For example, the complex numbers '''C''' form a CSA over themselves, but not over '''R''' (the center is '''C''' itself, hence too large to be CSA over '''R'''). The finite-dimensional division algebras with center '''R''' (that means the dimension over '''R''' is finite) are the real numbers and the quaternions by a [[Frobenius theorem (real division algebras)|theorem of Frobenius]], while any matrix ring over the reals or quaternions – M(''n'','''R''') or M(''n'','''H''') – is a CSA over the reals, but not a division algebra (if <math>n > 1</math>). | |||
We obtain an [[equivalence relation]] on CSAs over ''K'' by the [[Artin–Wedderburn theorem]] ([[Joseph Wedderburn|Wedderburn]]'s part, in fact), to express any CSA as a [[matrix ring|M(''n'',''D'')]] for some division algebra ''D''. If we look just at ''D'', that is, if we impose an equivalence relation identifying M(''m'',''D'') with M(''n'',''D'') for all integers ''m'' and ''n'' at least 1, we get the '''Brauer equivalence''' and the '''Brauer classes'''. | |||
Given central simple algebras ''A'' and ''B'', one can look at the their tensor product ''A'' ⊗ ''B'' as a ''K''-algebra (see [[tensor product of R-algebras]]). It turns out that this is always central simple. A slick way to see this is to use a characterisation: a central simple algebra over ''K'' is a ''K''-algebra that becomes a [[matrix ring]] when we extend the field of scalars to an [[algebraic closure]] of ''K''. | |||
2. | Given this closure property for CSAs, they form a [[monoid]] under tensor product, compatible with Brauer equivalence, and the Brauer classes are all [[invertible]]: the inverse class to that of an algebra ''A'' is the one containing the [[opposite algebra]] ''A''<sup>op</sup> (the [[opposite ring]] with the same action by ''K'' since the image of ''K'' → ''A'' is in the [[center (algebra)|center]] of ''A''). In other words, for a CSA ''A'' we have ''A'' ⊗ ''A''<sup>op</sup> = M(''n''<sup>2</sup>,''K''), where ''n'' is the degree of ''A'' over ''K''. (This provides a substantial reason for caring about the notion of an opposite algebra: it provides the inverse in the Brauer group.) | ||
== Examples == | |||
* In the following cases, every finite-dimensional central division algebra over a field ''K'' is ''K'' itself, so that the Brauer group Br(''K'') is [[trivial group|trivial]]: | |||
** ''K'' is an [[algebraically closed field]]:<ref name=L164>Lorenz (2008) p.164</ref> more generally, this is true for any [[pseudo algebraically closed field]]<ref>{{cite book | last1=Fried | first1=Michael D. | last2=Jarden | first2=Moshe | title=Field arithmetic | edition=3rd revised | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge | volume=11 | publisher=[[Springer-Verlag]] | year=2008 | isbn=978-3-540-77269-9 | zbl=1145.12001 | page=209 }}</ref> or [[quasi-algebraically closed field]].<ref name=S161>Serre (1979) p.161</ref> | |||
** ''K'' is a [[finite field]] ([[Wedderburn's little theorem|Wedderburn's theorem]]);<ref name=L164/><ref name=S162>Serre (1979) p.162</ref> | |||
** ''K'' is the [[function field of an algebraic variety|function field]] of an [[algebraic curve]] over an algebraically closed field ([[Tsen's theorem]]);<ref name=S162/> | |||
** An algebraic extension of '''Q''' containing all [[roots of unity]].<ref name=S162/> | |||
* The Brauer group Br('''R''') of the field '''R''' of [[real number]]s is the [[cyclic group]] of order two. There are just two non-isomorphic real division algebras with center '''R''': the algebra '''R''' itself and the [[quaternion]] algebra '''H'''.<ref name=S163>Serre (1979) p.163</ref> Since '''H''' ⊗ '''H''' ≅ M(4,'''R'''), the class of '''H''' has order two in the Brauer group. More generally, any [[real closed field]] has Brauer group of order two.<ref name=L164/> | |||
* ''K'' is complete under a discrete valuation with finite residue field. Br(''K'') is isomorphic to '''Q'''/'''Z'''.<ref name=S163/> | |||
== Brauer group and class field theory == | |||
The notion of Brauer group plays an important role in the modern formulation of the [[class field theory]]. If ''K''<sub>''v''</sub> is a non-archimedean [[local field]], the [[Hasse invariant of an algebra|Hasse invariants]] gives a canonical isomorphism inv<sub>''v''</sub>: Br(''K''<sub>''v''</sub>) → '''Q'''/'''Z''' constructed in [[local class field theory]].<ref name=L232>Lorenz (2008) p.232</ref><ref name=S67137>Serre (1967) p.137</ref><ref name=Sha155>Shatz (1972) p.155</ref> An element of the Brauer group of order ''n'' can be represented by a cyclic division algebra of dimension ''n''<sup>2</sup>.<ref name=L226>Lorenz (2008) p.226</ref> | |||
The case of a [[global field]] ''K'' is addressed by the [[global class field theory]]. If ''D'' is a central simple algebra over ''K'' and ''v'' is a valuation then ''D'' ⊗ ''K''<sub>''v''</sub> is a central simple algebra over ''K''<sub>''v''</sub>, the local completion of ''K'' at ''v''. This defines a homomorphism from the Brauer group of ''K'' into the Brauer group of ''K''<sub>''v''</sub>. A given central simple algebra ''D'' splits for all but finitely many ''v'', so that the image of ''D'' under almost all such homomorphisms is 0. The Brauer group Br(''K'') fits into an [[exact sequence]]<ref name=S163/><ref name=GS159>Gille & Szamuely (2006) p.159</ref> | |||
:<math> 0\rightarrow \textrm{Br}(K)\rightarrow \bigoplus_{v\in S} \textrm{Br}(K_v)\rightarrow \mathbf{Q}/\mathbf{Z} \rightarrow 0,</math> | |||
where ''S'' is the set of all valuations of ''K'' and the right arrow is the direct sum of the local invariants: the Brauer group of the real numbers is identified with (1/2)'''Z'''/'''Z'''. The injectivity of the left arrow is the content of the [[Albert–Brauer–Hasse–Noether theorem]]. Exactness in the middle term is a deep fact from the global class field theory. The group '''Q'''/'''Z''' on the right may be interpreted as the "Brauer group" of the [[class formation]] of idele classes associated to ''K''. | |||
==Properties== | |||
* [[Base change]] from a field ''K'' to an [[extension field]] ''L'' gives a ''restriction'' map from Br(''K'') to Br(''L''). The kernel is the group Br(''L''/''K'') of classes of ''K''-algebras that split over ''L''. | |||
* The Brauer group of any field is a torsion group.<ref>Lorenz (2008) p.194</ref> | |||
== General theory == | |||
For an arbitrary field ''K'', the Brauer group may be expressed in terms of [[Galois cohomology]] as follows:<ref name=S1579>Serre (1979) pp.157-159</ref> | |||
:<math>\textrm{Br}(K) \cong H^2(\textrm{Gal} (K^s/K), {K^s}^*).</math> | |||
Here, ''K''<sup>''s''</sup> is the [[separable closure]] of ''K'', which coincides with the algebraic closure when ''K'' is a [[perfect field]]. | |||
The isomorphism of the Brauer group with a Galois cohomology group can be described as follows. If ''D'' is a division algebra over ''K'' of dimension ''n''<sup>2</sup> containing a Galois extension ''L'' of degree ''n'' over ''K'', then the subgroup of elements of ''D''* that normalize ''L'' is an extension of the Galois group Gal(''L''/''K'') by the nonzero elements ''L''* of ''L'', so corresponds to an element of H<sup>2</sup>(Gal(''L''/''K''), ''L''*). | |||
A generalisation of the Brauer group to the case of [[commutative ring]]s was introduced by [[Maurice Auslander]] and [[Oscar Goldman (mathematician)|Oscar Goldman]],<ref>{{cite journal | last1=Auslander | first1=Maurice | author1-link=Maurice Auslander | last2=Goldman | first2=Oscar | author2-link=Oscar Goldman (mathematician) | title=The Brauer group of a commutative ring | journal=Trans. Am. Math. Soc. | volume=97 | pages=367–409 | year=1961 | issn=0002-9947 | zbl=0100.26304 }}</ref> and more generally for [[scheme (algebraic geometry)|schemes]] by [[Alexander Grothendieck]]. In their approach, central simple algebras over a field are replaced with [[Azumaya algebra]]s.<ref name=Salt21>Saltman (1999) p.21</ref> | |||
== See also == | |||
* [[Algebraic K-theory]] | |||
== References == | |||
{{reflist}} | |||
* {{eom|id=B/b017610|author=V.A. Iskovskikh|title=Brauer group of a field ''k''}} | |||
* {{cite book | first=Falko | last=Lorenz | title=Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics | year=2008 | publisher=Springer | isbn=978-0-387-72487-4 | zbl=1130.12001 }} | |||
* {{cite book | last1=Gille | first1=Philippe | last2=Szamuely | first2=Tamás | title=Central simple algebras and Galois cohomology | series=Cambridge Studies in Advanced Mathematics | volume=101 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2006 | isbn=0-521-86103-9 | zbl=1137.12001 }} | |||
* {{cite book | first=Richard | last=Pierce | title=Associative algebras | series=[[Graduate Texts in Mathematics]] | volume=88 | publisher=[[Springer-Verlag]] | location=New York-Berlin | year=1982 | isbn=0-387-90693-2 | zbl=0497.16001 }} | |||
* {{cite book | last=Reiner | first=I. | authorlink=Irving Reiner | title=Maximal Orders | series=London Mathematical Society Monographs. New Series | volume=28 | publisher=[[Oxford University Press]] | year=2003 | isbn=0-19-852673-3 | zbl=1024.16008 | pages=237–241 }} | |||
* {{cite book | last=Saltman | first=David J. | title=Lectures on division algebras | series=Regional Conference Series in Mathematics | volume=94 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=1999 | isbn=0-8218-0979-2 | zbl=0934.16013 }} | |||
* {{cite book | last=Serre | first=Jean-Pierre | authorlink=Jean-Pierre Serre | chapter=VI. Local class field theory | pages=128–161 | editor1-last=Cassels | editor1-first=J.W.S. | editor1-link=J. W. S. Cassels | editor2-last=Fröhlich | editor2-first=A. | editor2-link=Albrecht Fröhlich | title=Algebraic number theory. Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the International Mathematical Union | location=London | publisher=Academic Press | year=1967 | zbl=0153.07403 }} | |||
* {{cite book | last=Serre | first=Jean-Pierre | authorlink=Jean-Pierre Serre | title=[[Local Fields (book)|Local Fields]] | others=Translated from the French by Marvin Jay Greenberg | series=[[Graduate Texts in Mathematics]] | volume=67 | publisher=[[Springer-Verlag]] | year=1979 | isbn=0-387-90424-7 | zbl=0423.12016 }} | |||
* {{cite book | last=Shatz | first=Stephen S. | title=Profinite groups, arithmetic, and geometry | series=Annals of Mathematics Studies | volume=67 | location=Princeton, NJ | publisher=[[Princeton University Press]] | year=1972 | isbn=0-691-08017-8 | zbl=0236.12002 | mr=0347778 }} | |||
==Further reading== | |||
* {{cite book | last1=DeMeyer | first1=F. | last2=Ingraham | first2=E. | title=Separable algebras over commutative rings | series=Lecture Notes in Mathematics | volume=181 | location=Berlin-Heidelberg-New York | location=[[Springer-Verlag]] | year=1971 | isbn=978-3-540-05371-2 | zbl=0215.36602 }} | |||
==External links== | |||
*[http://planetmath.org/encyclopedia/BrauerGroup.html PlanetMath page] | |||
*[http://mathworld.wolfram.com/BrauerGroup.html MathWorld page] | |||
[[Category:Ring theory]] | |||
[[Category:Algebraic number theory]] | |||
Revision as of 16:31, 14 May 2013
In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras of finite rank over K and addition is induced by the tensor product of algebras. It arose out of attempts to classify division algebras over a field and is named after the algebraist Richard Brauer. The group may also be defined in terms of Galois cohomology. More generally, the Brauer group of a scheme is defined in terms of Azumaya algebras.
Construction
A central simple algebra (CSA) over a field K is a finite-dimensional associative K-algebra A, which is a simple ring, and for which the center is exactly K. Note that CSAs are in general not division algebras, though CSAs can be used to classify division algebras.
For example, the complex numbers C form a CSA over themselves, but not over R (the center is C itself, hence too large to be CSA over R). The finite-dimensional division algebras with center R (that means the dimension over R is finite) are the real numbers and the quaternions by a theorem of Frobenius, while any matrix ring over the reals or quaternions – M(n,R) or M(n,H) – is a CSA over the reals, but not a division algebra (if ).
We obtain an equivalence relation on CSAs over K by the Artin–Wedderburn theorem (Wedderburn's part, in fact), to express any CSA as a M(n,D) for some division algebra D. If we look just at D, that is, if we impose an equivalence relation identifying M(m,D) with M(n,D) for all integers m and n at least 1, we get the Brauer equivalence and the Brauer classes.
Given central simple algebras A and B, one can look at the their tensor product A ⊗ B as a K-algebra (see tensor product of R-algebras). It turns out that this is always central simple. A slick way to see this is to use a characterisation: a central simple algebra over K is a K-algebra that becomes a matrix ring when we extend the field of scalars to an algebraic closure of K.
Given this closure property for CSAs, they form a monoid under tensor product, compatible with Brauer equivalence, and the Brauer classes are all invertible: the inverse class to that of an algebra A is the one containing the opposite algebra Aop (the opposite ring with the same action by K since the image of K → A is in the center of A). In other words, for a CSA A we have A ⊗ Aop = M(n2,K), where n is the degree of A over K. (This provides a substantial reason for caring about the notion of an opposite algebra: it provides the inverse in the Brauer group.)
Examples
- In the following cases, every finite-dimensional central division algebra over a field K is K itself, so that the Brauer group Br(K) is trivial:
- K is an algebraically closed field:[1] more generally, this is true for any pseudo algebraically closed field[2] or quasi-algebraically closed field.[3]
- K is a finite field (Wedderburn's theorem);[1][4]
- K is the function field of an algebraic curve over an algebraically closed field (Tsen's theorem);[4]
- An algebraic extension of Q containing all roots of unity.[4]
- The Brauer group Br(R) of the field R of real numbers is the cyclic group of order two. There are just two non-isomorphic real division algebras with center R: the algebra R itself and the quaternion algebra H.[5] Since H ⊗ H ≅ M(4,R), the class of H has order two in the Brauer group. More generally, any real closed field has Brauer group of order two.[1]
- K is complete under a discrete valuation with finite residue field. Br(K) is isomorphic to Q/Z.[5]
Brauer group and class field theory
The notion of Brauer group plays an important role in the modern formulation of the class field theory. If Kv is a non-archimedean local field, the Hasse invariants gives a canonical isomorphism invv: Br(Kv) → Q/Z constructed in local class field theory.[6][7][8] An element of the Brauer group of order n can be represented by a cyclic division algebra of dimension n2.[9]
The case of a global field K is addressed by the global class field theory. If D is a central simple algebra over K and v is a valuation then D ⊗ Kv is a central simple algebra over Kv, the local completion of K at v. This defines a homomorphism from the Brauer group of K into the Brauer group of Kv. A given central simple algebra D splits for all but finitely many v, so that the image of D under almost all such homomorphisms is 0. The Brauer group Br(K) fits into an exact sequence[5][10]
where S is the set of all valuations of K and the right arrow is the direct sum of the local invariants: the Brauer group of the real numbers is identified with (1/2)Z/Z. The injectivity of the left arrow is the content of the Albert–Brauer–Hasse–Noether theorem. Exactness in the middle term is a deep fact from the global class field theory. The group Q/Z on the right may be interpreted as the "Brauer group" of the class formation of idele classes associated to K.
Properties
- Base change from a field K to an extension field L gives a restriction map from Br(K) to Br(L). The kernel is the group Br(L/K) of classes of K-algebras that split over L.
- The Brauer group of any field is a torsion group.[11]
General theory
For an arbitrary field K, the Brauer group may be expressed in terms of Galois cohomology as follows:[12]
Here, Ks is the separable closure of K, which coincides with the algebraic closure when K is a perfect field.
The isomorphism of the Brauer group with a Galois cohomology group can be described as follows. If D is a division algebra over K of dimension n2 containing a Galois extension L of degree n over K, then the subgroup of elements of D* that normalize L is an extension of the Galois group Gal(L/K) by the nonzero elements L* of L, so corresponds to an element of H2(Gal(L/K), L*).
A generalisation of the Brauer group to the case of commutative rings was introduced by Maurice Auslander and Oscar Goldman,[13] and more generally for schemes by Alexander Grothendieck. In their approach, central simple algebras over a field are replaced with Azumaya algebras.[14]
See also
References
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- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
Further reading
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
External links
- ↑ 1.0 1.1 1.2 Lorenz (2008) p.164
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ Serre (1979) p.161
- ↑ 4.0 4.1 4.2 Serre (1979) p.162
- ↑ 5.0 5.1 5.2 Serre (1979) p.163
- ↑ Lorenz (2008) p.232
- ↑ Serre (1967) p.137
- ↑ Shatz (1972) p.155
- ↑ Lorenz (2008) p.226
- ↑ Gille & Szamuely (2006) p.159
- ↑ Lorenz (2008) p.194
- ↑ Serre (1979) pp.157-159
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