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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

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