is an isomorphism ¦|US : US ’ US—V — © SBr (A). Since the open sets US form a

’

covering of Gr(r, V ), it follows that ¦ is an isomorphism.

Although Severi-Brauer varieties are de¬ned in terms of right ideals, they can

also be used to derive information on left ideals. Indeed, if J is a left ideal in a

central simple algebra A, then the set

J op = { j op ∈ Aop | j ∈ J }

is a right ideal in the opposite algebra Aop . Therefore, the variety of left ideals

of reduced dimension r in A can be identi¬ed with SBr (Aop ). We combine this

observation with the annihilator construction (see §??) to get the following result:

12 I. INVOLUTIONS AND HERMITIAN FORMS

(1.19) Proposition. For any central simple algebra A of degree n, there is a

canonical isomorphism

∼

± : SBr (A) ’ SBn’r (Aop )

’

which maps a right ideal I ‚ A of reduced dimension r to (I 0 )op .

Proof : In order to prove that ± is an isomorphism, we may extend scalars to a

splitting ¬eld of A. We may therefore assume that A = EndF (V ) for some n-

dimensional vector space V . Then Aop = EndF (V — ) under the identi¬cation f op =

f t for f ∈ EndF (V ). By (??), we may further identify

SBn’r (Aop ) = Gr(n ’ r, V — ).

SBr (A) = Gr(r, V ),

Under these identi¬cations, the map ± : Gr(r, V ) ’ Gr(n ’ r, V — ) carries every

r-dimensional subspace U ‚ V to U 0 = { • ∈ V — | •(U ) = {0} }.

To show that ± is an isomorphism of varieties, we restrict it to the a¬ne open

sets US de¬ned in the proof of Theorem (??): let S be an (n ’ r)-dimensional

subspace in V and

US = { U ‚ V | U • S = V } ‚ Gr(r, V ).

Let U0 ‚ V be such that U0 • S = V , so that US HomF (U0 , S). We also have

0 0 — 0 0

U0 • S = V , US 0 HomF (U0 , S ), and the map ± restricts to ±|US : US ’ US 0 .

It therefore induces a map ± which makes the following diagram commute:

±|U

’ ’S

US ’’’ US 0

¦ ¦

¦ ¦

±

HomF (U0 , S) ’ ’ ’ HomF (U0 , S 0 ).

0

’’

We now proceed to show that ± is an isomorphism of (a¬ne) varieties.

0

Every linear form in U0 restricts to a linear form on S, and since V = U0 •S we

0

S — . Similarly, S 0 U0 , so HomF (U0 , S 0 )

— 0

thus get a natural isomorphism U0

HomF (S — , U0 ). Under this identi¬cation, a direct calculation shows that the map ±

—

carries f ∈ HomF (U0 , S) to ’f t ∈ HomF (S — , U0 ) = HomF (U0 , S 0 ). It is therefore

— 0

an isomorphism of varieties. Since the open sets US cover Gr(r, V ), it follows that

± is an isomorphism.

If V is a vector space of dimension n over a ¬eld F and U ‚ V is a subspace of

dimension k, then for r = k, . . . , n the Grassmannian Gr(r ’ k, V /U ) embeds into

Gr(r, V ) by mapping every subspace W ‚ V /U to the subspace W ⊃ U such that

W/U = W . The image of Gr(r ’ k, V /U ) in Gr(r, V ) is the sub-Grassmannian of

r-dimensional subspaces in V which contain U (see Harris [?, p. 66]). There is an

analogous notion for Severi-Brauer varieties:

(1.20) Proposition. Let A be a central simple F -algebra and let J ‚ A be a right

ideal of reduced dimension k (i.e., a rational point of SBk (A)). The 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 ) set up in (??) de¬nes an

embedding:

SBr’k EndA (J 0 ) ’ SBr (A).

The image of SBr’k EndA (J 0 ) in SBr (A) is the variety of right ideals of reduced

dimension r in A which contain J.

§2. INVOLUTIONS 13

Proof : It su¬ces to prove the proposition over a scalar extension. We may therefore

assume that A is split, i.e., that A = EndF (V ). Let then J = HomF (V, U ) for some

subspace U ‚ V of dimension k. We have J 0 = HomF (V /U, V ) and (??) shows

that there is a canonical isomorphism EndA (J 0 ) = EndF (V /U ). Theorem (??)

then yields canonical isomorphisms SBr (A) = Gr(r, V ) and SBr’k EndA (J 0 ) =

Gr(r ’ k, V /U ). Moreover, from (??) it follows that the map SBr’k EndA (J 0 ) ’

SBr (A) corresponds under these identi¬cations to the embedding Gr(r’k, V /U ) ’

Gr(r, V ) described above.

§2. Involutions

An involution on a central simple algebra A over a ¬eld F is a map σ : A ’ A

subject to the following conditions:

(a) σ(x + y) = σ(x) + σ(y) for x, y ∈ A.

(b) σ(xy) = σ(y)σ(x) for x, y ∈ A.

(c) σ 2 (x) = x for x ∈ A.

Note that the map σ is not required to be F -linear. However, it is easily checked

that the center F (= F · 1) is preserved under σ. The restriction of σ to F is

therefore an automorphism which is either the identity or of order 2. Involutions

which leave the center elementwise invariant are called involutions of the ¬rst kind.

Involutions whose restriction to the center is an automorphism of order 2 are called

involutions of the second kind.

This section presents the basic de¬nitions and properties of central simple alge-

bras with involution. Involutions of the ¬rst kind are considered ¬rst. As observed

in the introduction to this chapter, they are adjoint to nonsingular symmetric or

skew-symmetric bilinear forms in the split case. Involutions of the ¬rst kind are

correspondingly divided into two types: the orthogonal and the symplectic types.

We show in (??) how to characterize these types by properties of the symmetric ele-

ments. Involutions of the second kind, also called unitary, are treated next. Various

examples are provided in (??)“(??).

2.A. Involutions of the ¬rst kind. Throughout this subsection, A denotes

a central simple algebra over a ¬eld F of arbitrary characteristic, and σ is an

involution of the ¬rst kind on A. Our basic object of study is the couple (A, σ); from

this point of view, a homomorphism of algebras with involution f : (A, σ) ’ (A , σ )

is an F -algebra homomorphism f : A ’ A such that σ —¦ f = f —¦ σ. Our main tool

is the extension of scalars: if L is any ¬eld containing F , the involution σ extends

to an involution of the ¬rst kind σL = σ — IdL on AL = A —F L. In particular, if

L is a splitting ¬eld of A, we may identify AL = EndL (V ) for some vector space V

over L of dimension n = deg A. As observed in the introduction to this chapter,

the involution σL is then the adjoint involution σb with respect to some nonsingular

symmetric or skew-symmetric bilinear form b on V . By means of a basis of V , we

may further identify V with Ln , hence also A with Mn (L). For any matrix m, let

mt denote the transpose of m. If g ∈ GLn (L) denotes the Gram matrix of b with

respect to the chosen basis, then

b(x, y) = xt · g · y

where x, y are considered as column matrices and g t = g if b is symmetric, g t = ’g

if b is skew-symmetric. The involution σL is then identi¬ed with the involution σg

14 I. INVOLUTIONS AND HERMITIAN FORMS

de¬ned by

σg (m) = g ’1 · mt · g for m ∈ Mn (L).

For future reference, we summarize our conclusions:

(2.1) Proposition. Let (A, σ) be a central simple F -algebra of degree n with in-

volution of the ¬rst kind and let L be a splitting ¬eld of A. Let V be an L-vector

space of dimension n. There is a nonsingular symmetric or skew-symmetric bilin-

ear form b on V and an invertible matrix g ∈ GLn (L) such that g t = g if b is

symmetric and g t = ’g if b is skew-symmetric, and

(AL , σL ) EndL (V ), σb Mn (L), σg .

As a ¬rst application, we have the following result:

(2.2) Corollary. For all a ∈ A, the elements a and σ(a) have the same reduced

characteristic polynomial. In particular, TrdA σ(a) = TrdA (a) and NrdA σ(a) =

NrdA (a).

Proof : For all m ∈ Mn (L), g ∈ GLn (L), the matrix g ’1 · mt · g has the same

characteristic polynomial as m.

Of course, in (??), neither the form b nor the matrix g (nor even the splitting

¬eld L) is determined uniquely by the involution σ; some of their properties re¬‚ect

properties of σ, however. As a ¬rst example, we show in (??) below that two types

of involutions of the ¬rst kind can be distinguished which correspond to symmetric

and to alternating1 forms. This distinction is made on the basis of properties of

symmetric elements which we de¬ne next.

In a central simple F -algebra A with involution of the ¬rst kind σ, the sets of

symmetric, skew-symmetric, symmetrized and alternating elements in A are de¬ned

as follows:

Sym(A, σ) = { a ∈ A | σ(a) = a },

Skew(A, σ) = { a ∈ A | σ(a) = ’a },

Symd(A, σ) = { a + σ(a) | a ∈ A },

Alt(A, σ) = { a ’ σ(a) | a ∈ A }.

If char F = 2, then Symd(A, σ) = Sym(A, σ), Alt(A, σ) = Skew(A, σ) and A =

1

Sym(A, σ) • Skew(A, σ) since every element a ∈ A decomposes as a = 2 a +

1

σ(a) + 2 a ’ σ(a) . If char F = 2, then Symd(A, σ) = Alt(A, σ) ‚ Skew(A, σ) =

Sym(A, σ), and (??) below shows that this inclusion is strict.

(2.3) Lemma. Let n = deg A; then dim Sym(A, σ) + dim Alt(A, σ) = n2 . More-

over, Alt(A, σ) is the orthogonal space of Sym(A, σ) for the bilinear form T A on A

induced by the reduced trace:

Alt(A, σ) = { a ∈ A | TrdA (as) = 0 for s ∈ Sym(A, σ) }.

Similarly, dim Skew(A, σ)+dim Symd(A, σ) = n2 , and Symd(A, σ) is the orthogonal

space of Skew(A, σ) for the bilinear form TA .

1 If

char F = 2, every skew-symmetric bilinear form is alternating; if char F = 2, the notions

of symmetric and skew-symmetric bilinear forms coincide, but the notion of alternating form is

more restrictive.

§2. INVOLUTIONS 15

Proof : The ¬rst relation comes from the fact that Alt(A, σ) is the image of the

linear endomorphism Id ’ σ of A, whose kernel is Sym(A, σ). If a ∈ Alt(A, σ), then

a = x ’ σ(x) for some x ∈ A, hence for s ∈ Sym(A, σ),

TrdA (as) = TrdA (xs) ’ TrdA σ(x)s = TrdA (xs) ’ TrdA σ(sx) .

Corollary (??) shows that the right side vanishes, hence the inclusion

Alt(A, σ) ‚ { a ∈ A | TrdA (as) = 0 for s ∈ Sym(A, σ) }.

Dimension count shows that this inclusion is an equality since TA is nonsingular

(see (??)).

The statements involving Symd(A, σ) readily follow, either by mimicking the

arguments above, or by using the fact that in characteristic di¬erent from 2,

Symd(A, σ) = Sym(A, σ) and Alt(A, σ) = Skew(A, σ), and, in characteristic 2,

Symd(A, σ) = Alt(A, σ) and Skew(A, σ) = Sym(A, σ).

We next determine the dimensions of Sym(A, σ) and Skew(A, σ) (and therefore

also of Symd(A, σ) and Alt(A, σ)).

Consider ¬rst the split case, assuming that A = EndF (V ) for some vector

space V over F . As observed in the introduction to this chapter, every involution

of the ¬rst kind σ on A is the adjoint involution with respect to a nonsingular

symmetric or skew-symmetric bilinear form b on V which is uniquely determined

by σ up to a factor in F — .

(2.4) Lemma. Let σ = σb be the adjoint involution on A = EndF (V ) with respect

to the nonsingular symmetric or skew-symmetric bilinear form b on V , and let

n = dimF V .

(1) If b is symmetric, then dimF Sym(A, σ) = n(n + 1)/2.

(2) If b is skew-symmetric, then dimF Skew(A, σ) = n(n + 1)/2.

(3) If char F = 2, then b is alternating if and only if tr(f ) = 0 for all f ∈

Sym(A, σ). In this case, n is necessarily even.

Proof : As in (??), we use a basis of V to identify (A, σ) with Mn (F ), σg , where

g ∈ GLn (F ) satis¬es g t = g if b is symmetric and g t = ’g if b is skew-symmetric.

For m ∈ Mn (F ), the relation gm = (gm)t is equivalent to σg (m) = m if g t = g and

to σg (m) = ’m if g t = ’g. Therefore,

Sym(A, σ) if b is symmetric,

g ’1 · Sym Mn (F ), t =

Skew(A, σ) if b is skew-symmetric.

The ¬rst two parts then follow from the fact that the space Sym Mn (F ), t of n — n

symmetric matrices (with respect to the transpose) has dimension n(n + 1)/2.

Suppose now that char F = 2. If b is not alternating, then b(v, v) = 0 for some

v ∈ V . Consider the map f : V ’ V de¬ned by

f (x) = vb(v, x)b(v, v)’1 for x ∈ V .

Since b is symmetric we have

b f (x), y = b(v, y)b(v, x)b(v, v)’1 = b x, f (y) for x, y ∈ V ,

hence σ(f ) = f . Since f is an idempotent in A, its trace is the dimension of its

image:

tr(f ) = dim im f = 1.

16 I. INVOLUTIONS AND HERMITIAN FORMS

Therefore, if the trace of every symmetric element in A is zero, then b is alternating.

Conversely, suppose b is alternating; it follows that n is even, since every al-

ternating form on a vector space of odd dimension is singular. Let (ei )1¤i¤n be

a symplectic basis of V , in the sense that b(e2i’1 , e2i ) = 1, b(e2i , e2i+1 ) = 0 and

b(ei , ej ) = 0 if |i ’ j| > 1. Let f ∈ Sym(A, σ); for j = 1, . . . , n let

n

f (ej ) = ei aij for some aij ∈ F ,

i=1

n

so that tr(f ) = aii . For i = 1, . . . , n/2 we have

i=1

b f (e2i’1 ), e2i = a2i’1,2i’1 and b e2i’1 , f (e2i ) = a2i,2i ;

since σ(f ) = f , it follows that a2i’1,2i’1 = a2i,2i for i = 1, . . . , n/2, hence

n/2

tr(f ) = 2 a2i,2i = 0.

i=1

We now return to the general case where A is an arbitrary central simple F -

algebra and σ is an involution of the ¬rst kind on A. Let n = deg A and let L be a

splitting ¬eld of A. Consider an isomorphism as in (??):

(AL , σL ) EndL (V ), σb .

This isomorphism carries Sym(AL , σL ) = Sym(A, σ)—F L to Sym EndL (V ), σb and

Skew(AL , σL ) to Skew EndL (V ), σb . Since extension of scalars does not change

dimensions, (??) shows

(a) dimF Sym(A, σ) = n(n + 1)/2 if b is symmetric;

(b) dimF Skew(A, σ) = n(n + 1)/2 if b is skew-symmetric.

These two cases coincide if char F = 2 but are mutually exclusive if char F = 2;

indeed, in this case A = Sym(A, σ)•Skew(A, σ), hence the dimensions of Sym(A, σ)

and Skew(A, σ) add up to n2 .

Since the reduced trace of A corresponds to the trace of endomorphisms under

the isomorphism AL EndL (V ), we have TrdA (s) = 0 for all s ∈ Sym(A, σ) if

and only if tr(f ) = 0 for all f ∈ Sym EndL (V ), σb , and Lemma (??) shows that,

when char F = 2, this condition holds if and only if b is alternating. Therefore, in

arbitrary characteristic, the property of b being symmetric or skew-symmetric or

alternating depends only on the involution and not on the choice of L nor of b. We

may thus set the following de¬nition:

(2.5) De¬nition. An involution σ of the ¬rst kind is said to be of symplectic type

(or simply symplectic) if for any splitting ¬eld L and any isomorphism (AL , σL )

EndL (V ), σb , the bilinear form b is alternating; otherwise it is called of orthogonal

type (or simply orthogonal ). In the latter case, for any splitting ¬eld L and any

isomorphism (AL , σL ) EndL (V ), σb , the bilinear form b is symmetric (and

nonalternating).

The preceding discussion yields an alternate characterization of orthogonal and

symplectic involutions:

(2.6) Proposition. Let (A, σ) be a central simple F -algebra of degree n with in-

volution of the ¬rst kind.

§2. INVOLUTIONS 17

(1) Suppose that char F = 2, hence Symd(A, σ) = Sym(A, σ) and Alt(A, σ) =

Skew(A, σ). If σ is of orthogonal type, then

n(n+1) n(n’1)

dimF Sym(A, σ) = dimF Skew(A, σ) = .

and

2 2

If σ is of symplectic type, then

n(n’1) n(n+1)

dimF Sym(A, σ) = dimF Skew(A, σ) = .

and

2 2

Moreover, in this case n is necessarily even.

(2) Suppose that char F = 2, hence Sym(A, σ) = Skew(A, σ) and Symd(A, σ) =

Alt(A, σ); then

n(n+1) n(n’1)

dimF Sym(A, σ) = dimF Alt(A, σ) = .

and

2 2

The involution σ is of symplectic type if and only if TrdA Sym(A, σ) = {0}, which

holds if and only if 1 ∈ Alt(A, σ). In this case n is necessarily even.

Proof : The only statement which has not been observed before is that, if char F =

2, the reduced trace of every symmetric element is 0 if and only if 1 ∈ Alt(A, σ).

This follows from the characterization of alternating elements in (??).

Given an involution of the ¬rst kind on a central simple algebra A, all the other

involutions of the ¬rst kind on A can be obtained by the following proposition:

(2.7) Proposition. Let A be a central simple algebra over a ¬eld F and let σ be

an involution of the ¬rst kind on A.

(1) For each unit u ∈ A— such that σ(u) = ±u, the map Int(u) —¦ σ is an involution

of the ¬rst kind on A.

(2) Conversely, for every involution σ of the ¬rst kind on A, there exists some

u ∈ A— , uniquely determined up to a factor in F — , such that

σ = Int(u) —¦ σ σ(u) = ±u.

and

We then have

u · Sym(A, σ) = Sym(A, σ) · u’1 if σ(u) = u

Sym(A, σ ) =

u · Skew(A, σ) = Skew(A, σ) · u’1 if σ(u) = ’u

and

u · Skew(A, σ) = Skew(A, σ) · u’1 if σ(u) = u

Skew(A, σ ) =

u · Sym(A, σ) = Sym(A, σ) · u’1 if σ(u) = ’u.

If σ(u) = u, then Alt(A, σ ) = u · Alt(A, σ) = Alt(A, σ) · u’1 .

(3) Suppose that σ = Int(u) —¦ σ where u ∈ A— is such that u = ±u. If char F = 2,

then σ and σ are of the same type if and only if σ(u) = u. If char F = 2, the

involution σ is symplectic if and only if u ∈ Alt(A, σ).

2

Proof : A computation shows that Int(u) —¦ σ = Int uσ(u)’1 , proving (??).

If σ is an involution of the ¬rst kind on A, then σ —¦ σ is an automorphism

of A which leaves F elementwise invariant. The Skolem-Noether theorem then

yields an element u ∈ A— , uniquely determined up to a factor in F — , such that

2

σ —¦ σ = Int(u), hence σ = Int(u) —¦ σ. It follows that σ = Int uσ(u)’1 , hence

2

the relation σ = IdA yields

σ(u) = »u for some » ∈ F — .

18 I. INVOLUTIONS AND HERMITIAN FORMS

Applying σ to both sides of this relation and substituting »u for σ(u) in the resulting

equation, we get u = »2 u, hence » = ±1. If σ(u) = u, then for all x ∈ A,

x ’ σ (x) = u · u’1 x ’ σ(u’1 x) = xu ’ σ(xu) · u’1 .

This proves Alt(A, σ ) = u · Alt(A, σ) = Alt(A, σ) · u’1 . The relations between

Sym(A, σ ), Skew(A, σ ) and Sym(A, σ), Skew(A, σ) follow by straightforward com-

putations, completing the proof of (??).

If char F = 2, the involutions σ and σ have the same type if and only if

Sym(A, σ) and Sym(A, σ ) have the same dimension. Part (??) shows that this

condition holds if and only if σ(u) = u. If char F = 2, the involution σ is symplectic

if and only if TrdA (s ) = 0 for all s ∈ Sym(A, σ ). In view of (??), this condition

may be rephrased as

TrdA (us) = 0 for s ∈ Sym(A, σ).

Lemma (??) shows that this condition holds if and only if u ∈ Alt(A, σ).

(2.8) Corollary. Let A be a central simple F -algebra with an involution σ of the

¬rst kind.

(1) If deg A is odd, then A is split and σ is necessarily of orthogonal type. Moreover,

the space Alt(A, σ) contains no invertible elements.

(2) If deg A is even, then the index of A is a power of 2 and A has involutions of

both types. Whatever the type of σ, the space Alt(A, σ) contains invertible elements

and the space Sym(A, σ) contains invertible elements which are not in Alt(A, σ).

Proof : De¬ne a homomorphism of F -algebras

σ— : A —F A ’ EndF (A)

by σ— (a — b)(x) = axσ(b) for a, b, x ∈ A. This homomorphism is injective since

A —F A is simple and surjective by dimension count, hence it is an isomorphism.

Therefore, A —F A splits2 , and the exponent of A is 1 or 2. Since the index and

the exponent of a central simple algebra have the same prime factors (see Draxl [?,

Theorem 11, p. 66]), it follows that the index of A, ind A, is a power of 2. In

particular, if deg A is odd, then A is split. In this case, Proposition (??) shows

that every involution of the ¬rst kind has orthogonal type. If Alt(A, σ) contains an

invertible element u, then Int(u)—¦σ has symplectic type, by (??); this is impossible.

Suppose henceforth that the degree of A is even. If A is split, then it has

involutions of both types, since a vector space of even dimension carries nonsingular

alternating bilinear forms as well as nonsingular symmetric, nonalternating bilinear

forms. Let σ be an involution of the ¬rst kind on A. In order to show that Alt(A, σ)

contains invertible elements, we consider separately the case where char F = 2. If

char F = 2, consider an involution σ whose type is di¬erent from the type of σ.

Proposition (??) yields an invertible element u ∈ A such that σ = Int(u) —¦ σ

and σ(u) = ’u. Note also that 1 is an invertible element which is symmetric

but not alternating. If char F = 2, consider a symplectic involution σ and an

orthogonal involution σ . Again, (??) yields invertible elements u, v ∈ A— such

that σ = Int(u) —¦ σ and σ = Int(v) —¦ σ, and shows that u ∈ Alt(A, σ) and

v ∈ Sym(A, σ) Alt(A, σ).

2 Alternately, Aop by mapping a ∈ A to

the involution σ yields an isomorphism A

op op, hence the Brauer class [A] of A satis¬es [A] = [A]’1 .

∈A

σ(a)

§2. INVOLUTIONS 19

Assume next that A is not split. The base ¬eld F is then in¬nite, since the

Brauer group of a ¬nite ¬eld is trivial (see for instance Scharlau [?, Corollary 8.6.3]).

Since invertible elements s are characterized by NrdA (s) = 0 where NrdA is the

reduced norm in A, the set of invertible alternating elements is a Zariski-open subset

of Alt(A, σ). Our discussion above of the split case shows that this open subset is

nonempty over an algebraic closure. Since F is in¬nite, rational points are dense,

hence this open set has a rational point. Similarly, the set of invertible symmetric

elements is a dense Zariski-open subset in Sym(A, σ), hence it is not contained in

the closed subset Sym(A, σ)©Alt(A, σ). Therefore, there exist invertible symmetric

elements which are not alternating.

If u ∈ Alt(A, σ) is invertible, then Int(u)—¦σ is an involution of the type opposite

to σ if char F = 2, and is a symplectic involution if char F = 2. If char F = 2 and

v ∈ Sym(A, σ) is invertible but not alternating, then Int(v) —¦ σ is an orthogonal

involution.

The existence of involutions of both types on central simple algebras of even

degree with involution can also be derived from the proof of (??) below.

The following proposition highlights a special feature of symplectic involutions:

(2.9) Proposition. Let A be a central simple F -algebra with involution σ of sym-

plectic type. The reduced characteristic polynomial of every element in Symd(A, σ)

is a square. In particular, NrdA (s) is a square in F for all s ∈ Symd(A, σ).

Proof : Let K be a Galois extension of F which splits A and let s ∈ Symd(A, σ).

It su¬ces to show that the reduced characteristic polynomial PrdA,s (X) ∈ F [X]

is a square in K[X], for then its monic square root is invariant under the action

of the Galois group Gal(K/F ), hence it is in F [X]. Extending scalars from F to