Introduction to Linear Algebraic Groups II

Introduction to Linear Algebraic Groups II

Table of Contents

1 Projective Algebraic Varieties

k: an algebraically closed field X: a topological space.

Suppose for each non-empty open subset U of X a k-algebra O(U ) is given such that

If V ⊂ U is a non-empty open subset, there is a k-algebra

homomorphism O(U ) → O(V ), called therestriction map.

Let U =S_α∈AUα be an open covering of the open set U .

Suppose that for each α ∈ A we are given fα ∈ O(Uα) such

that fα and fβ restricted to the same function on Uα∩ Uβ.

Then there is f ∈ O(U ) whose restriction to Uα is fα, for all

α ∈ A.

Then O is called a sheaf of k-valued functions on X.

A pair (X, O) of a topological space X and a sheaf of functions O

Example

Let X be an algebraic set and x ∈ X.

A k-valued function f defined on a neighborhood U of x is called

regular at x if there are g, h ∈ k[X] such that h(x) 6= 0 and such that there is an open neighborhood V ⊂ U of x with h(y) 6= 0 and

f (y) = g(y)h(y)−1 for all y ∈ V .

We denote by OX(U ) the k-algebra of functions regular at each

point of U .

From definition, there is a homomorphism φ : k[X] → OX(X).

Proposition

φ : k[X] → OX(X) is a k-algebra isomorphism.

Proof.

Injectivity is obvious. Note that X is compact.

Let (X, O) be a ringed space, let Y be a subset of X with induced topology.

If U is an open subset of Y , let O|Y(U ) consists of the

functions f on U such there exists an open covering

U ⊂S_α∈AUα by open sets of X and for each α ∈ A an

element fα∈ O(Uα) whose restriction to Uα∩ U coincides

with that of f .

Then O|Y is a sheaf of functions on Y and (Y, O|Y) is a

ringed space.

Let (X, OX), (Y, OY) be two ringed spaces and ϕ : X → Y a

continuous map.

If f is a function on an open set V ⊂ Y , ϕ∗

V(f ) := f ◦ ϕ is a

function on the open set ϕ−1(V ) ⊂ X. We say that ϕ is a

morphism of ringed spaces if for each open set V ⊂ Y , we

have ϕ∗

V maps OY(V ) into OX(ϕ−1(V )).

A morphism ϕ : X → Y of ringed spaces is en isomorphismif

ϕ is a homeomorphism of topological spaces and ϕ∗

V is an

Example

If X is a subset of Y , φ the inclusion X → Y and OX := OY|X,

then φ is a morphism of ringed spaces. In this case, we say that

(X, OX) is a ringed subspace of (Y, OY).

When X, Y are affine varieties, the notion of morphism and isomorphism agrees with that defined before.

Aprevariety is a compact ringed space (X, OX) such that any

point of X has an open neighborhood U with the property

that the induced ringed space (U, OX|U) is isomorphic to an

affine variety.

Because X is compact, it can be covered by a finite number of affine varieties.

A prevariety X is Noetherian. Hence X is the union of finite irreducible closed subsets.

If a ring subspace of a prevariety is itself a prevariety, then it is called asub-prevariety.

Let X, Y be prevarieties.

A productof X and Y is a prevariety Z, together with morphisms p : Z → X, q : Z → Y such that for any triple (Z′_{, p}′_{, q}′_{) of prevariety Z}′ _{together with p}′_{: Z}′_{→ X,}

q′_{: Z}′_{→ Y there exists a unique morphism r : Z}′ _{→ Z such}

Proposition

A product of two prevarieties exists, and is unique up to isomorphism.

Proof.

Suppose X =Sm_i=1Ui and Y =Sn_j=1Vj where Ui, Vj are affine

open sets. Then the set theoretic product X × Y is covered by the

product Ui× Vj, which are themselves affine varieties.

A set U ⊂ X × Y is defined to be open if and only if its

intersection with any Ui× Vj is open. This gives a topology on

X × Y .

Let X be a prevariety. Define the diagonal subset

∆X := { (x, x) ∈ X × X | x ∈ X }

and make ∆X into a ringed subspace of X × X.

A prevariety X is called an (algebraic) varietyif ∆X is closed

in X × X.

Example

An affine algebraic variety is a variety. If X and Y are varieties, so is X × Y . A sub-prevariety of a variety is a variety.

If X is an irreducible variety covered by affine open sets Ui,

each Ui is irreducible and each intersection Ui∩ Uj is

non-empty. Then Ui and Uj have the same function field,

which is called the function field k(X) of X.

The dimensiondim(X) is defined to be the transcendental degree of k(X) over k.

If X is reducible and X1, . . . , Xn are its components, then we

define

dim(X) := max

An algebraic group G is a variety

a group

such that the multiplication µ : G × G → G by (x, y) 7→ xy

and the inversion ι : G → G by x 7→ x−1 _{are morphisms of}

varieties.

A mapping ϕ : G → G′ _{ishomomorphism of algebraic groups}

if

ϕ is a morphism of varieties; ϕ is a homomorphism of groups.

A closed subgroup of an algebraic group is an algebraic group.

Define an equivalence on kn+1r{0}: x ∼ y if and only if

there is a ∈ k× _{:= k r {0} such that y = ax.}

Define Pn_{:= (k}n+1_{r{0})/ ∼.}

We write [x] for the equivalence class of x and [x0, . . . , xn] for

For 0 ≤ i ≤ n, put

Ui:= { [x0, . . . , xn] ∈ Pn | xi 6= 0 }.

The map φi: Ui → kn by

φi([x0, . . . , xn]) := (x_x0_i, . . . ,xi−1_xi ,xi+1_xi , . . . ,x_xn_i)

is a bijection. Transport the structure of affine algebraic

variety of kn _{to U}

i.

We define a topology on Pn _{by defining U ⊂ P}n _{to be open if}

A function f defined in a neighborhood of x ∈ Pn _{is regular}

at x if f is regular at x as a function defined locally in Ui for

some i with x ∈ Ui. Because OUi|Ui∩Uj = OUj|Ui∩Uj, the

notion of regular functions is well-defined.

Then Pn _{is a variety and is called theprojective n-space.}

If V := kn+1, then Pn _{is also denoted by P(V ). The}

A projective variety is a closed subset of some Pn_{, together}

with its induced structure of variety.

A quasi-projective varietyis an open subvariety of a projective variety.

An ideal I in S is homogeneous if it is generated by homogeneous polynomials, or, equivalently if

f = f0+ · · · + fk∈ I (where fi is homogeneous of degree i)

implies that fi ∈ I.

If I is a proper homogeneous ideal in S, then if x ∈ kn+1 _{is a}

zero of I, the same is true for all ax, a ∈ k. Hence we can

define a set V∗_{(I) ⊂ P}n _by

V∗(I) := { [x] ∈ Pn| x ∈ Vkn+1(I), x 6= 0 }.

Define φ : Pm_{× P}n_{→ P}mn+m+n _by

φ([x0, . . . , xm], [y0, . . . , yn]) := [xiyj]0≤i≤m, 0≤j≤n.

Proposition

φ defines an isomorphism of Pm_{× P}n _{onto a closed subset of}

Pmn+m+n_.

In fact φ(Pm_{× P}n_{) is the closed subset defined by the}

homogeneous ideal in k[Tij] generated by the elements

TijTi′_j′− T_ij′T_i′_j. Corollary

projection X × Y → Y is a closed mapping, i.e., maps closed sets to closed sets.

Lemma

Let X, Y be varieties.

1 Projective varieties are complete.

2 If X is affine and complete, then X is finite (i.e.,

dim(X) = 0).

3 If ϕ : X → Y is a morphism and X is complete. then ϕ(X) is

closed in Y and is complete.

Proposition

If G is a connected algebraic group whose underlying variety is complete, then G is commutative.

A connected algebraic group whose underlying variety is complete is called an abelian variety.

Theorem (Chevalley)

Suppose k is perfect and G is an algebraic group. Then there exists a unique normal linear algebraic closed subgroup H in G for which G/H is an abelian variety.

A field F isperfect if every algebraic extension of F is

separable.

A field of characteristic 0 is perfect. A finite field is perfect.

k: an algebraically closed field

Suppose first X ⊂ V = kn is an affine variety.

The k-algebra k[ǫ] called the dual numbersis defined to be

k[ǫ] := k[T ]/hT2i = k ⊕ kǫ

with ǫ2 = 0.

The addition and multiplication of k[ǫ] is given by (a + bǫ) + (a′+ b′ǫ) := (a + a′) + (b + b′)ǫ,

For x ∈ X, let Dx denote the set of k-linear mappings k[X] → k such that d(f g) = d(f )g(x) + f (x)d(g) for f, g ∈ k[X]. Dx is a k-vector space by (cd)(f ) := cd(f ), (d + d′_{)(f ) := d(f ) + d}′_{(f ) for c ∈ k, d, d}′ _{∈ D} x and f ∈ k[X].

The tangent bundleof X is defined to be

T (X) := Homk(k[X], k[ǫ]).

The k-algebra homomorphism ψ : k → k[ǫ] by a 7→ a + 0ǫ induces an embedding X → T (X) by η 7→ ψ ◦ η.

The k-algebra homomorphism φ : k[ǫ] → k by a + bǫ 7→ a

induces a projection T (X) → Homk(k[X], k) = X by

For each x ∈ X, the fibre Tx(X) of T (X) → X over x is

called the tangent spaceat x.

An element in Tx(X) is called atangent vector at x.

If ξ ∈ Tx(X), then ξ = ex+ ǫdξ where ex ∈ Homk(k[X], k)

given by ex(f ) := f (x) and dξ is a k-linear mapping

Now

ξ(f ) = f (x) + ǫdξ(f ).

Then

f (x)g(x) + ǫdξ(f g) = ξ(f g) = ξ(f )ξ(g)

= (f (x) + ǫdξ(f ))(g(x) + ǫdξ(g)).

for all f, g ∈ k[X]. Hence

dξ(f g) = dξ(f )g(x) + f (x)dξ(g),

Conversely, each d ∈ Dx determines a tangent vector ex+ ǫd

in Tx(X).

The mapping ξ := ex+ ǫdξ 7→ dξ is a bijection between

Tx(X) and Dx.

Let ϕ : X → Y be a morphism of affine varieties, so we have a

k-algebra homomorphism ϕ∗_{: k[Y ] → k[X], and hence}

dϕ : T (X) → T (Y )

by ξ 7→ ξ ◦ ϕ∗ _{for ξ ∈ T (X).}

Over each x ∈ X the restriction of dϕ to Tx(X) is a k-linear

mapping

(dϕ)x: Tx(X) → Tϕ(x)(Y ).

If ψ : Y → Z is another morphism of affine varieties, we have d(ψ ◦ ϕ) = dψ ◦ dϕ, so that

Now suppose X is a variety covered by affine open sets Ui.

We can check that the tangent bundle T (Ui) patch together

to form a tangent bundleT (X) of X.

T (X) does not depend on the choice of affine open covering.

For each x ∈ X, the tangent space Tx(X) over x is a

Proposition

Let X be an irreducible variety. Then

1 dim(Tx(X)) ≥ dim(X) for all x ∈ X.

2 S(X) := { x ∈ X | dim(Tx(X)) > dim(X) } is a proper

S(X) is called thesingular locus of X. An element

x ∈ X r S(X) is called asmooth point. If S(X) = ∅, then X

is called smooth.

Now let G be a connected linear algebraic group. For each

x ∈ G the left translation λx by x induces an isomorphism

Example

Let k := Fp and G := Ga. Let ϕ : G → G by x 7→ xp.

ϕ is a bijective homomorphism of algebraic groups.

Let ξ ∈ Tx(G) = k. We have (x + ξǫ)p= xp+ ξpǫp= xp. Hence

(dϕ)x: Tx(G) → Txp(G) is given by ξ 7→ 0.

Suppose E is a finitely generated field extension of a field F .

E is said to be separably generatedover F if there exist an

intermediate field E′ _{such that E}′_{/F is a pure transcendental}

extension and E/E′ _{is finite separable extension.}

Suppose X, Y are irreducible varieties. A morphism

ϕ : X → Y isdominant if ϕ(X) is dense in Y . In this case,

ϕ∗ _{induces an embedding of k(Y ) into k(X).}

A dominant morphism ϕ : X → Y is calledseparableif k(X)

is separably generated over ϕ∗_{(k(Y )).}

Proposition

Let ϕ : X → Y be a morphism between irreducible varieties. Suppose x ∈ X, ϕ(x) ∈ Y are smooth points and

(dϕ)x: Tx(X) → Tϕ(x)(Y ) is surjective. Then ϕ is dominant and

separable.

Proposition

Let ϕ : X → Y be a dominant, separable morphism between irreducible varieties. Then there is a nonempty open subsets U of X such that for each x ∈ U the point ϕ(x) is smooth on Y and (dϕ)x: Tx(X) → Tϕ(x)(Y ) is surjective.

Let G be a linear algebraic group.

A k-linear map δ : k[G] → k[G] is called a derivationsif

δ(f g) = δ(f )g + f δ(g) for all f, g ∈ k[G].

For x ∈ G, define λx: k[G] → k[G] by (λxf )(y) = f (x−1y)

for y ∈ G.

Let L(G) denote the space of derivations δ : k[G] → k[G] which are left invariant:

λx◦ δ = δ ◦ λx

for all x ∈ G.

For ξ ∈ Te(G), we have dξ: k[G] → k by

ξ(f ) = f (e) + dξ(f )ǫ.

For ξ ∈ Te(G) and f ∈ k[G], define a function δξf on G by

(δξf )(x) := dξ(λx−1f )

Lemma

δξf is in k[G].

Proof.

Write µ∗_{(f ) =}P

igi⊗ hi for some gi, hi ∈ k[G].

So f (xy) =P_igi(x)hi(y) for all x, y ∈ G.

Then λx−1f =

P

igi(x)hi.

Lemma

δξ is in L(G).

Proof.

It is clear that δξ is a derivation.

δξ is left-invariant:

(λyδξf )(x) = (δξf )(y−1x) = dξ(λ(y−1_x)−1f ) = dξ(λx−1λyf )

Proposition

The mapping ξ 7→ δξ is a k-linear isomorphism of Te(G) onto

L(G).

Proof.

If δ ∈ L(G), define d : k[G] → k by df := (δf )(e).

Then d ∈ De and hence d = dξ for some ξ ∈ Te(G).

For ξ, η ∈ Te(G), define [ξ, η] ∈ Te(G) by

δ_[ξ,η]= [δξ, δη] = δξδη− δηδξ

for all ξ, η ∈ Te(G).

Let g := Te(G) (or L(G)). We have

1 [·, ·] : g × g → g is bilinear in both variables

2 [ξ, ξ] = 0 for all ξ ∈ g

3 [[ξ, η], ζ] + [[η, ζ], ξ] + [[ζ, ξ], η] = 0 for all ξ, η, ζ ∈ g (called theJacobi identity).

Example

G := GLn

g= gl_n

Let ϕ : G → G′ _{be a morphism of linear algebraic groups.}

The induced map dϕ := (dϕ)e: g → g′ is called the differentialof

ϕ (at e).

Lemma

We have dϕ([ξ, η]) = [dϕ(ξ), dϕ(η)] for ξ, η ∈ g, i.e., dϕ is a homomorphism of Lie algebras.

Example

Let G be a linear algebraic group with Lie algebra g. We have

µ : G × G → G by (x, y) 7→ xy and ι : G → G by x 7→ x−1_{. Then}

1 dµ : g × g → g is the map (ξ, η) 7→ ξ + η,

Proposition

Let ϕ : G → G′ _{be a bijective homomorphism of linear algebraic}

groups. Then ϕ is an isomorphism if and only if dϕ : g → g′ _{is an}

isomorphism of Lie algebras.

Proof.

If ϕ is an isomorphism, then dϕ is clearly an isomorphism.

May assume G, G′ _{are connected. It can be shown that k(G) is}

purely inseparable over ϕ∗_(k(G′₎₎

k[G] is separable over ϕ∗_(k(G′_{)) because dϕ is an isomorphism.}

Hence k(G) = ϕ∗_(k(G′_{)), so that ϕ}∗ _{and hence also ϕ is an}

If H is a closed subgroup of G, then the Lie algebra h of H is a Lie subalgebra of g.

Moreover, H is a normal subgroup of G if and only if h is an ideal of g, i.e., [g, h] ⊂ h.

Theorem

Let g be the Lie algebra of a linear algebraic group G. For any ξ ∈ g, there exist unique ξs and ξnin g such that ξ = ξs+ ξn, ξs is

semisimple, ξn is nilpotent.

Proposition

If ϕ : G → G′ _{is a homomorphism of algebraic groups, then}

Let G be a linear algebraic group with Lie algebra g.

For x ∈ G, we have an inner automorphism: Intx: G → G by

g 7→ xgx−1.

Its differential is Adx: g → g.

We have Ad−1x = Adx−1 so that Adx ∈ GL(g).

We have Adxy = Adx◦ Ady and hence we have

Ad : G → GL(g)

Proposition

Let G := GLn. Then Adg(ξ) = gξg−1 for g ∈ GLn and ξ ∈ gln.

Proposition

Let G be a linear algebraic group. The adjoint representation Ad : G → GL(g) is a homomorphism of algebraic groups.