forms of orthogonal groups in Chapter ??.

§1. Central Simple Algebras

Unless otherwise mentioned, all the algebras we consider in this work are ¬nite-

dimensional with 1. For any algebra A over a ¬eld F and any ¬eld extension K/F ,

we write AK for the K-algebra obtained from A by extending scalars to K:

AK = A —F K.

We also de¬ne the opposite algebra Aop by

Aop = { aop | a ∈ A },

with the operations de¬ned as follows:

aop + bop = (a + b)op , aop bop = (ba)op , ± · aop = (± · a)op

for a, b ∈ A and ± ∈ F .

A central simple algebra over a ¬eld F is a (¬nite dimensional) algebra A = {0}

with center F (= F · 1) which has no two-sided ideals except {0} and A. An algebra

A = {0} is a division algebra (or a skew ¬eld ) if every non-zero element in A is

invertible.

1.A. Fundamental theorems. For the convenience of further reference, we

summarize without proofs some basic results from the theory of central simple

algebras. The structure of these algebras is determined by the following well-known

theorem of Wedderburn:

(1.1) Theorem (Wedderburn). For an algebra A over a ¬eld F , the following

conditions are equivalent:

(1) A is central simple.

(2) The canonical map A —F Aop ’ EndF (A) which associates to a — bop the linear

map x ’ axb is an isomorphism.

(3) There is a ¬eld K containing F such that AK is isomorphic to a matrix algebra

over K, i.e., AK Mn (K) for some n.

(4) If „¦ is an algebraically closed ¬eld containing F ,

A„¦ Mn („¦) for some n.

4 I. INVOLUTIONS AND HERMITIAN FORMS

(5) There is a ¬nite dimensional central division algebra D over F and an integer r

such that A Mr (D).

Moreover, if these conditions hold, all the simple left (or right) A-modules are

isomorphic, and the division algebra D is uniquely determined up to an algebra

isomorphism as D = EndA (M ) for any simple left A-module M .

References: See for instance Scharlau [?, Chapter 8] or Draxl [?, §3].

The ¬elds K for which condition (??) holds are called splitting ¬elds of A.

Accordingly, the algebra A is called split if it is isomorphic to a matrix algebra

Mn (F ) (or to EndF (V ) for some vector space V over F ).

Since the dimension of an algebra does not change under an extension of scalars,

it follows from the above theorem that the dimension of every central simple algebra

is a square: dimF A = n2 if AK Mn (K) for some extension K/F . The integer n is

called the degree of A and is denoted by deg A. The degree of the division algebra D

in condition (??) is called the index of A (or sometimes the Schur index of A) and

denoted by ind A. Alternately, the index of A can be de¬ned by the relation

deg A ind A = dimF M

where M is any simple left module over A. This relation readily follows from the

fact that if A Mr (D), then Dr is a simple left module over A.

We rephrase the implication (??) ’ (??) in Wedderburn™s theorem:

(1.2) Corollary. Every central simple F -algebra A has the form

A EndD (V )

for some (¬nite dimensional ) central division F -algebra D and some ¬nite-dimen-

sional right vector space V over D. The F -algebra D is uniquely determined by A

up to isomorphism, V is a simple left A-module and deg A = deg D dimD V .

In view of the uniqueness (up to isomorphism) of the division algebra D (or,

equivalently, of the simple left A-module M ), we may formulate the following de¬-

nition:

(1.3) De¬nition. Finite dimensional central simple algebras A, B over a ¬eld F

are called Brauer-equivalent if the F -algebras of endomorphisms of any simple left

A-module M and any simple left B-module N are isomorphic:

EndA (M ) EndB (N ).

Equivalently, A and B are Brauer-equivalent if and only if M (A) Mm (B)

for some integers , m.

Clearly, every central simple algebra is Brauer-equivalent to one and only one

division algebra (up to isomorphism). If A and B are Brauer-equivalent central

simple algebras, then ind A = ind B; moreover, A B if and only if deg A = deg B.

The tensor product endows the set of Brauer equivalence classes of central

simple algebras over F with the structure of an abelian group, denoted Br(F ) and

called the Brauer group of F . The unit element in this group is the class of F

which is also the class of all the matrix algebras over F . The inverse of the class of

a central simple algebra A is the class of the opposite algebra Aop , as part (??) of

Wedderburn™s theorem shows.

Uniqueness (up to isomorphism) of simple left modules over central simple

algebras leads to the following two fundamental results:

§1. CENTRAL SIMPLE ALGEBRAS 5

(1.4) Theorem (Skolem-Noether). Let A be a central simple F -algebra and let

B ‚ A be a simple subalgebra. Every F -algebra homomorphism ρ : B ’ A extends

to an inner automorphism of A: there exists a ∈ A— such that ρ(b) = aba’1 for all

b ∈ B. In particular, every F -algebra automorphism of A is inner.

References: Scharlau [?, Theorem 8.4.2], Draxl [?, §7] or Pierce [?, §12.6].

The centralizer CA B of a subalgebra B ‚ A is, by de¬nition, the set of elements

in A which commute with every element in B.

(1.5) Theorem (Double centralizer). Let A be a central simple F -algebra and let

B ‚ A be a simple subalgebra with center K ⊃ F . The centralizer C A B is a simple

subalgebra of A with center K which satis¬es

dimF A = dimF B · dimF CA B CA CA B = B.

and

If K = F , then multiplication in A de¬nes a canonical isomorphism A = B—F CA B.

References: Scharlau [?, Theorem 8.4.5], Draxl [?, §7] or Pierce [?, §12.7].

Let „¦ denote an algebraic closure of F . Under scalar extension to „¦, every

central simple F -algebra A of degree n becomes isomorphic to Mn („¦). We may

therefore ¬x an F -algebra embedding A ’ Mn („¦) and view every element a ∈ A

as a matrix in Mn („¦). Its characteristic polynomial has coe¬cients in F and is

independent of the embedding of A in Mn („¦) (see Scharlau [?, Ch. 8, §5], Draxl [?,

§22], Reiner [?, §9] or Pierce [?, §16.1]); it is called the reduced characteristic

polynomial of A and is denoted

PrdA,a (X) = X n ’ s1 (a)X n’1 + s2 (a)X n’2 ’ · · · + (’1)n sn (a).

(1.6)

The reduced trace and reduced norm of a are denoted TrdA (a) and NrdA (a) (or

simply Trd(a) and Nrd(a)):

TrdA (a) = s1 (a), NrdA (a) = sn (a).

We also write

(1.7) SrdA (a) = s2 (a).

(1.8) Proposition. The bilinear form TA : A — A ’ F de¬ned by

TA (x, y) = TrdA (xy) for x, y ∈ A

is nonsingular.

Proof : The result is easily checked in the split case and follows in the general case

by scalar extension to a splitting ¬eld. (See Reiner [?, Theorem 9.9]).

1.B. One-sided ideals in central simple algebras. A fundamental result

of the Wedderburn theory of central simple algebras is that all the ¬nitely generated

left (resp. right) modules over a central simple F -algebra A decompose into direct

sums of simple left (resp. right) modules (see Scharlau [?, p. 283]). Moreover, as

already pointed out in (??), the simple left (resp. right) modules are all isomorphic.

If A = Mr (D) for some integer r and some central division algebra D, then D r is

a simple left A-module (via matrix multiplication, writing the elements of D r as

column vectors). Therefore, every ¬nitely generated left A-module M is isomorphic

to a direct sum of copies of D r :

(Dr )s

M for some integer s,

6 I. INVOLUTIONS AND HERMITIAN FORMS

hence

dimF M = rs dimF D = s deg A ind A.

More precisely, we may represent the elements in M by r — s-matrices with entries

in D:

M Mr,s (D)

so that the action of A = Mr (D) on M is the matrix multiplication.

(1.9) De¬nition. The reduced dimension of the left A-module M is de¬ned by

dimF M

rdimA M = .

deg A

The reduced dimension rdimA M will be simply denoted by rdim M when the al-

gebra A is clear from the context. Observe from the preceding relation that the re-

duced dimension of a ¬nitely generated left A-module is always a multiple of ind A.

Moreover, every left A-module M of reduced dimension s ind A is isomorphic to

Mr,s (D), hence the reduced dimension classi¬es left A-modules up to isomorphism.

The preceding discussion of course applies also to right A-modules; writing the

elements of Dr as row vectors, matrix multiplication also endows D r with a right

A-module structure, and D r is then a simple right A-module. Every right module

of reduced dimension s ind A over A = Mr (D) is isomorphic to Ms,r (D).

(1.10) Proposition. Every left module of ¬nite type M over a central simple F -

algebra A has a natural structure of right module over E = EndA (M ), so that

M is an A-E-bimodule. If M = {0}, the algebra E is central simple over F and

Brauer-equivalent to A; moreover,

deg E = rdimA M, rdimE M = deg A,

and

A = EndE (M ).

Conversely, if A and E are Brauer-equivalent central simple algebras over F , then

there is an A-E-bimodule M = {0} such that A = EndE (M ), E = EndA (M ),

rdimA (M ) = deg E and rdimE (M ) = deg A.

Proof : The ¬rst statement is clear. (Recall that endomorphisms of left modules

are written on the right of the arguments.) Suppose that A = Mr (D) for some

integer r and some central division algebra D. Then D r is a simple left A-module,

hence D EndA (Dr ) and M (Dr )s for some s. Therefore,

Ms EndA (Dr )

EndA (M ) Ms (D).

This shows that E is central simple and Brauer-equivalent to A. Moreover, deg E =

s deg D = rdimA M , hence

rs dim D

rdimE M = = r deg D = deg A.

s deg D

Since M is an A-E-bimodule, we have a natural embedding A ’ EndE (M ). Com-

puting the degree of EndE (M ) as we computed deg EndA (M ) above, we get

deg EndE (M ) = deg A,

hence this natural embedding is surjective.

§1. CENTRAL SIMPLE ALGEBRAS 7

For the converse, suppose that A and E are Brauer-equivalent central simple

F -algebras. We may assume that

A = Mr (D) and E = Ms (D)

for some central division F -algebra D and some integers r and s. Let M = Mr,s (D)

be the set of r — s-matrices over D. Matrix multiplication endows M with an A-

E-bimodule structure, so that we have natural embeddings

(1.11) A ’ EndE (M ) and E ’ EndA (M ).

Since dimF M = rs dimF D, it is readily computed that rdimE M = deg A and

rdimA M = deg E. The ¬rst part of the proposition then yields

deg EndA (M ) = rdimA M = deg E and deg EndE (M ) = rdimE M = deg A,

hence the natural embeddings (??) are surjective.

Ideals and subspaces. Suppose now that A = EndD (V ) for some central

division algebra D over F and some ¬nite dimensional right vector space V over D.

We aim to get an explicit description of the one-sided ideals in A in terms of

subspaces of V .

Let U ‚ V be a subspace. Composing every linear map from V to U with the

inclusion U ’ V , we identify HomD (V, U ) with a subspace of A = EndD (V ):

HomD (V, U ) = { f ∈ EndD (V ) | im f ‚ U }.

This space clearly is a right ideal in A, of reduced dimension

rdim HomD (V, U ) = dimD U deg D.

Similarly, composing every linear map from the quotient space V /U to V with

the canonical map V ’ V /U , we may identify HomD (V /U, V ) with a subspace of

A = EndD (V ):

HomD (V /U, V ) = { f ∈ EndD (V ) | ker f ⊃ U }.

This space is clearly a left ideal in A, of reduced dimension

rdim HomD (V /U, V ) = dimD (V /U ) deg D.

(1.12) Proposition. The map U ’ HomD (V, U ) de¬nes a one-to-one correspon-

dence between subspaces of dimension d in V and right ideals of reduced dimen-

sion d ind A in A = EndD (V ). Similarly, the map U ’ HomD (V /U, V ) de¬nes a

one-to-one correspondence between subspaces of dimension d in V and left ideals

of reduced dimension deg A ’ d ind A in A. Moreover, there are canonical isomor-

phisms of F -algebras:

EndA HomD (V, U ) EndD (U ) EndA HomD (V /U, V ) EndD (V /U ).

and

Proof : The last statement is clear: multiplication on the left de¬nes an F -algebra

homomorphism EndD (U ) ’ EndA HomD (V, U ) and multiplication on the right

de¬nes an F -algebra homomorphism

EndD (V /U ) ’ EndA HomD (V /U, V ) .

Since rdim HomD (V, U ) = dimD U deg D, we have

deg EndA HomD (V, U ) = dimD U deg D = deg EndD (U ),

8 I. INVOLUTIONS AND HERMITIAN FORMS

so the homomorphism EndD (U ) ’ EndA HomD (V, U ) is an isomorphism. Simi-

larly, the homomorphism EndD (V /U ) ’ EndA HomD (V /U, V ) is an isomorphism

by dimension count.

For the ¬rst part, it su¬ces to show that every right (resp. left) ideal in A has

the form HomD (V, U ) (resp. HomD (V /U, V )) for some subspace U ‚ V . This is

proved for instance in Baer [?, §5.2].

(1.13) Corollary. For every left (resp. right) ideal I ‚ A there exists an idempo-

tent e ∈ A such that I = Ae (resp. I = eA). Multiplication on the right (resp. left)

induces a surjective homomorphism of right (resp. left) EndA (I)-modules:

ρ : I ’ EndA (I)

which yields an isomorphism: eAe EndA (I).

Proof : If I = HomD (V /U, V ) (resp. HomD (V, U )), choose a complementary sub-

space U in V , so that V = U • U , and take for e the projection on U parallel to U

(resp. the projection on U parallel to U ). We then have I = Ae (resp. I = eA).

For simplicity of notation, we prove the rest only in the case of a left ideal I.

Then EndA (I) acts on I on the right. For x ∈ I, de¬ne ρ(x) ∈ EndA (I) by

y ρ(x) = yx.

For f ∈ EndA (I) we have

f

(yx)f = yxf = y ρ(x ) ,

hence

ρ(xf ) = ρ(x) —¦ f,

which means that ρ is a homomorphism of right EndA (I)-modules. In order to see

that ρ is onto, pick an idempotent e ∈ A such that I = Ae. For every y ∈ I we

have y = ye; it follows that every f ∈ EndA (I) is of the form f = ρ(ef ), since for

every y ∈ I,

f

y f = (ye)f = yef = y ρ(e ) .

Therefore, ρ is surjective.

To complete the proof, we show that the restriction of ρ to eAe is an isomor-

∼

phism eAe ’ EndA (I). It is readily veri¬ed that this restriction is an F -algebra

’

homomorphism. Moreover, for every x ∈ I one has ρ(x) = ρ(ex) since y = ye for

every y ∈ I. Therefore, the restriction of ρ to eAe is also surjective onto End A (I).

Finally, if ρ(ex) = 0, then in particular

eρ(ex) = ex = 0,

so ρ is injective on eAe.

Annihilators. For every left ideal I in a central simple algebra A over a ¬eld F ,

the annihilator I 0 is de¬ned by

I 0 = { x ∈ A | Ix = {0} }.

This set is clearly a right ideal. Similarly, for every right ideal I, the annihilator I 0

is de¬ned by

I 0 = { x ∈ A | xI = {0} };

it is a left ideal in A.

§1. CENTRAL SIMPLE ALGEBRAS 9

(1.14) Proposition. For every left or right ideal I ‚ A, rdim I + rdim I 0 = deg A

and I 00 = I.

Proof : Let A = EndD (V ). For any subspace U ‚ V it follows from the de¬nition

of the annihilator that

HomD (V, U )0 = HomD (V /U, V ) HomD (V /U, V )0 = HomD (V, U ).

and

Since every left (resp. right) ideal I ‚ A has the form I = HomD (V /U, V ) (resp.

I = HomD (V, U )), the proposition follows.

Now, let J ‚ A be a right ideal of reduced dimension k and let B ‚ A be the

idealizer of J:

B = { a ∈ A | aJ ‚ J }.

This set is a subalgebra of A containing J as a two-sided ideal. It follows from the

de¬nition of J 0 that J 0 b ‚ J 0 for all b ∈ B and that J 0 ‚ B. Therefore, (??) shows

that the map ρ : B ’ EndA (J 0 ) de¬ned by multiplication on the right is surjective.

∼

Its kernel is J 00 = J, hence it induces an isomorphism B/J ’ EndA (J 0 ).

’

For every right ideal I ‚ A containing J, let

˜

I = ρ(I © B).

˜

(1.15) Proposition. The map I ’ I de¬nes a one-to-one correspondence between

right ideals of reduced dimension r in A which contain J and right ideals of reduced

dimension r ’ k in EndA (J 0 ). If A = EndD (V ) and J = HomD (V, U ) for some

subspace U ‚ V of dimension r/ ind A, then for I = HomD (V, W ) with W ⊃ U , we

have under the natural isomorphism EndA (J 0 ) = EndD (V /U ) of (??) that

˜

I = HomD (V /U, W/U ).

Proof : In view of (??), the second part implies the ¬rst, since the map W ’ W/U

de¬nes a one-to-one correspondence between subspaces of dimension r/ ind A in V

which contain U and subspaces of dimension (r ’ k)/ ind A in V /U .

Suppose that A = EndD (V ) and J = HomD (V, U ), hence J 0 = HomD (V /U, V )

and B = { f ∈ A | f (U ) ‚ U }. Every f ∈ B induces a linear map f ∈ EndD (V /U ),

and the homomorphism ρ : B ’ EndA (J 0 ) = EndD (V /U ) maps f to f since for

g ∈ J 0 we have

g ρ(f ) = g —¦ f = g —¦ f .

For I = HomD (V, W ) with W ⊃ U , it follows that

˜

I = { f | f ∈ I and f (U ) ‚ U } ‚ HomD (V /U, W/U ).

The converse inclusion is clear, since using bases of U , W and V it is easily seen

that every linear map h ∈ HomD (V /U, W/U ) is of the form h = f for some f ∈

HomD (V, W ) such that f (U ) ‚ U .

1.C. Severi-Brauer varieties. Let A be a central simple algebra of degree n

over a ¬eld F and let r be an integer, 1 ¤ r ¤ n. Consider the Grassmannian

Gr(rn, A) of rn-dimensional subspaces in A. The Pl¨cker embedding identi¬es

u

Gr(rn, A) with a closed subvariety of the projective space on the rn-th exterior

power of A (see Harris [?, Example 6.6, p. 64]):

rn

Gr(rn, A) ‚ P( A).

10 I. INVOLUTIONS AND HERMITIAN FORMS

The rn-dimensional subspace U ‚ A corresponding to a non-zero rn-vector u1 §

rn

· · · § urn ∈ A is

U = { x ∈ A | u1 § · · · § urn § x = 0 } = u1 F + · · · + urn F.

Among the rn-dimensional subspaces in A, the right ideals of reduced dimension r

are the subspaces which are preserved under multiplication on the right by the

elements of A. Such ideals may fail to exist: for instance, if A is a division algebra,

it does not contain any nontrivial ideal; on the other hand, if A Mn (F ), then it

contains right ideals of every reduced dimension r = 0, . . . , n. Since every central

simple F -algebra becomes isomorphic to a matrix algebra over some scalar extension

of F , this situation is best understood from an algebraic geometry viewpoint: it is

comparable to the case of varieties de¬ned over some base ¬eld F which have no

rational point over F but acquire points over suitable extensions of F .

To make this viewpoint precise, consider an arbitrary basis (ei )1¤i¤n2 of A.

rn

The rn-dimensional subspace represented by an rn-vector u1 § · · · § urn ∈ A

is a right ideal of reduced dimension r if and only if it is preserved under right

multiplication by e1 , . . . , en2 , i.e.,

for i = 1, . . . , n2 ,

u1 ei § · · · § urn ei ∈ u1 § · · · § urn F

or, equivalently,

u1 ei § · · · § urn ei § uj = 0 for i = 1, . . . , n2 and j = 1, . . . , rn.

This condition translates to a set of equations on the coordinates of the rn-vector

u1 § · · · § urn , hence the right ideals of reduced dimension r in A form a closed

subvariety of Gr(rn, A).

(1.16) De¬nition. The (generalized ) Severi-Brauer variety SBr (A) is the vari-

ety of right ideals of reduced dimension r in A. It is a closed subvariety of the

Grassmannian:

SBr (A) ‚ Gr(rn, A).

For r = 1, we write simply SB(A) = SB1 (A). This is the (usual) Severi-Brauer

variety of A, ¬rst de¬ned by F. Chˆtelet [?].

a

(1.17) Proposition. The Severi-Brauer variety SBr (A) has a rational point over

an extension K of F if and only if the index ind AK divides r. In particular, SB(A)

has a rational point over K if and only if K splits A.

Proof : From the de¬nition, it follows that SBr (A) has a rational point over K if

and only if AK contains a right ideal of reduced dimension r. Since the reduced

dimension of any ¬nitely generated right AK -module is a multiple of ind AK , it

follows that ind AK divides r if SBr (A) has a rational point over K. Conversely,

suppose r = m ind AK for some integer m and let AK Mt (D) for some division

algebra D and some integer t. The set of matrices in Mt (D) whose t ’ m last rows

are zero is a right ideal of reduced dimension r, hence SBr (A) has a rational point

over K.

The following theorem shows that Severi-Brauer varieties are twisted forms of

Grassmannians:

(1.18) Theorem. For A = EndF (V ), there is a natural isomorphism

SBr (A) Gr(r, V ).

§1. CENTRAL SIMPLE ALGEBRAS 11

In particular, for r = 1,

P(V ).

SB(A)

Proof : Let V — = HomF (V, F ) be the dual of V . Under the natural isomorphism

A = EndF (V ) V —F V — , multiplication is given by

(v — φ) · (w — ψ) = (v — ψ)φ(w).

By (??), the right ideals of reduced dimension r in A are of the form HomF (V, U ) =

U — V — where U is an r-dimensional subspace in V .

We will show that the correspondence U ” U — V — between r-dimensional

subspaces in V and right ideals of reduced dimension r in A induces an isomorphism

of varieties Gr(r, V ) SBr (A).

For any vector space W of dimension n, there is a morphism Gr(r, V ) ’

Gr(rn, V —W ) which maps an r-dimensional subspace U ‚ V to U —W ‚ V —W . In

the particular case where W = V — we thus get a morphism ¦ : Gr(r, V ) ’ SBr (A)

which maps U to U — V — .

In order to show that ¦ is an isomorphism, we consider the following a¬ne

covering of Gr(r, V ): for each subspace S ‚ V of dimension n ’ r, we denote by US

the set of complementary subspaces:

US = { U ‚ V | U • S = V }.

The set US is an a¬ne open subset of Gr(r, V ); more precisely, if U0 is a ¬xed

complementary subspace of S, there is an isomorphism:

∼

HomF (U0 , S) ’ US

’

which maps f ∈ HomF (U0 , S) to U = { x + f (x) | x ∈ U0 } (see Harris [?, p. 65]).

Similarly, we may also consider US—V — ‚ Gr(rn, A). The image of the restriction

of ¦ to US is

{ U — V — ‚ V — V — | (U — V — ) • (S — V — ) = V — V — } = US—V — © SBr (A).

Moreover, there is a commutative diagram:

¦|U

’ ’S

US ’’’ US—V —

¦ ¦

¦ ¦

φ

HomF (U0 , S) ’ ’ ’ HomF (U0 — V — , S — V — )

’’

where φ(f ) = f — IdV — . Since φ is linear and injective, it is an isomorphism of

varieties between HomF (U0 , S) and its image. Therefore, the restriction of ¦ to US