Similar to affine schemes, we say thatG is anaffine subgroup of HifG(R) is a subgroup of H(R) over all R-points. We call it an affine closed subgroup when G is also a closed sub- scheme ofH. We say that a homomorphism of affine group Φ :G → His aclosed embedding
2.5. CLOSED SUBGROUPS AND QUOTIENT GROUPS 27
(resp. monomorphism). Since the image of a group homomorphism is a subgroup of the target group, as in affine schemes, the notion of closed subgroup and closed embedding are essentially the same: a closed embedding is an isomorphism onto a closed subgroup of the target affine group.
For affine schemes, we know that the notion of closed embedding is strictly stronger than monomorphism. But for affine group schemes, these two notions are equivalent when working over fields. To see this, we need a technical theorem about faithfully flat Hopf Algebra over fields, which we will not prove but provide reference for interested readers:
Theorem 2.5.1. Let A⊆B be Hopf algebras over a field k. Then A is faithfully flat over
B.
Proof. [Wat12, pg. 109]
Proposition 2.5.2. Let Φ : G → H be an affine group homomorphism over k with k a field. ThenΦ is a closed embedding if and only if it is a monomorphism.
Proof. We have shown previously that being a monomorphism is a necessary condition. To show its sufficiency, consider the factorisation of Φ#:
k[H]Φ#(k[H]),→i k[G].
By Theorem 2.5.1, we get that i is also faithfully flat. Letting H0 = Spec(Φ#(k[H])), we
have the factorisation of Φ into
G→ Hi# 0 Φe → H.
Note that given any morphisms of affine groups Θ1,Θ2 :G0 → G withi#◦Θ1 =i#◦Θ2, we
get that
e
Φ◦(i#◦Θ1) =Φe◦(i#◦Θ2)
Φ◦Θ1 = Φ◦Θ2
∴Θ1 = Θ2,
with the last equality following from the hypothesis that Φ is a monomorphism. Thus i#
is a monomorphism. By Proposition 1.7.6, we get that i# is an isomorphism and so is i,
therefore Φ# is surjective.
To each homomorphism of affine groups Φ, we can define a functorN :Algk →Grpsto beN(R) = ker Φ(R). It is obvious by definition that it is a group functor. Withu:k→R
making R ak-algebra, we have
ker(Φ(R)) ={ϕ∈ G(R) : Φ(R)(ϕ) =ϕ◦Φ#=u◦H},
i.e. it consists of elementsϕ:k[G]→R making the following diagram commutative:
k[H] k[G]
k R.
Φ#
H ϕ
Since the counitH:k[H]→kis ak-algebra homomorphism, it must be surjective and
therefore k ∼= k[H]/IH, with IH denoting the kernel of H. The ideal IH is called the
augmented ideal of k[H]. So elements ϕ of ker(Φ(R)) are exactly elements with ϕ◦Φ# :
k[H]→R factoring through k∼=k[H]/IH. But this is true if and only if Φ#(IH)⊆kerϕ, which is equivalent tohΦ#(I
H)ik[G]⊆kerϕ. Thus we have that
N(R) = ker(Φ(R)) ={ϕ∈ G(R) :hΦ#(IH)ik[G]⊆kerϕ}
≡ {ϕe∈Homk−Alg(k[G]/hΦ#(IH)ik[G], R)}
= Spec(k[G]/hΦ#(IH)ik[G], R).
This shows that N can be represented by k[G]/hΦ#(I
H)ik[G] and is therefore an affine
group. With this we defineN to be the kernel of Φ. The following proposition relates the concept of kernel and injectivity.
Proposition 2.5.3. If Φ : G → H is a closed embedding, then its kernel N is the trivial affine group. These are equivalent when working over fields.
Proof. Note the N is trivial if and only if N(R) is trivial over all R-points, i.e. Φ is a monomorphism. Thus the first statement is just a restatement of proposition 1.6.2, whereas the second is equivalent to Proposition 2.5.2.
Motivated by Proposition 2.5.3, over fields, we shall equivalently say that closed embed- dings areinjective.
It is only natural now to also have the notion of surjectivity. As we know for groups, the notion of surjectivity is simply surjectivity as sets. Indeed, one might suggest that a surjective affine group homomorphism is one that is surjective on all R-points. But this condition is very strong and would exclude many cases of interests. As we will see, when working over fields, we can have a weaker notion of surjectivity that still coincides with the usual concept of surjectivity in groups. From now on, all affine groups will be over a field.
We say that a homomorphism of affine groups G → His surjective if its corresponding algebra map k[H]→ k[G] is injective. Since we are working over fields, this is equivalent tok[H]→k[G] being faithfully flat (cf. Theorem 2.5.1). With this definition, we have the following criterion for isomorphism as we have for groups:
Proposition 2.5.4. A homomorphism of affine groups G → H is an isomorphism if and only if it is both injective and surjective.
Proof. This just follows from the fact thatk[H]→k[G] is both surjective and injective. In groups, we also have that any homomorphism G→ H can be decompose intoG
H0,→H, where we callH0 the image ofG→H. This is also true for affine groups:
Proposition 2.5.5. Any homomorphism of affine groups G → H can be decomposed into
G → H0→ H with G → H0 surjective and H0→ H injective.
Proof. On Hopf algebras we can decompose k[H]→ k[G] into k[H] B ,→ k[G], with B
2.5. CLOSED SUBGROUPS AND QUOTIENT GROUPS 29
With this we shall also call H0 theimage of the homomorphismG → H.
Let N ⊆ G be a normal subgroup. We know that for every G → H0 with kernel containingN, the map factors uniquely throughG/N:
N G G/N
H0.
∃! 0
The next proposition provides us with a similar universal property for surjective affine group homomorphisms:
Proposition 2.5.6. LetG → H be a surjective affine group homomorphism with kernelN. Then every homomorphism G → H0 with kernel containing N factors uniquely throughH:
N G H
H0
∃! 0
Proof. LetG ∼= Spec(A),H ∼= Spec(B) and H0∼= Spec(B0). For each pairg, g0 ∈ G(R) such that their images agree inH(R), we have thatgg0−1 ∈ N(R). But this means thatgg0−1 is in the kernel of G(R)→ H0(R) and therefore the image of gand g0 also agree inH0(R). So
on the corresponding algebras, we get that the two compositions
B0 →A i1 ⇒ i2 A⊗BA (g,g0) −−−→R
agree, with the two maps i1, i2 : A ⇒ A⊗BA the canonical inclusions a 7→ a⊗1 and
a7→1⊗arespectively. Since this works for all R, in particular it works for R=A⊗BA. The image ofi1andi2 obviously agree inH(A⊗BA). With (g, g0) = (i1, i2) = id, the above
compositions reduce to
B0 →A⇒A⊗BA.
This means that the image of B0 in A lies in the kernel of the map θ= i1−i2. But with
B →Afaithfully flat, by Proposition 1.7.5 we have the exact sequence
B→A−→θ A⊗BA.
So the image of B0 inAlies in B and therefore B0 →A factorise uniquely as:
A B
which gives us
G H
H0
as desired. The factorisation is unique as shown on the algebras.
Corollary 2.5.7. If G → H and G → H0 are both surjective and their kernels agree, then
H ∼=H0.
Proof. Just note that this is a universal property, which captures objects uniquely up to isomorphism.
Motivated by this, given a surjective affine group homomorphismG → Hwith kernel N we shall callH the quotient of G by N and denote it asG/N. With the notion image and
kernel of a homomorphism as discussed, we can define an exact sequence of affine groups to be a sequence · · ·−−−→ Gϕi−2 i−1 ϕi−1 −−−→ Gi ϕi −→ Gi+1 ϕi+1 −−−→ · · ·
such that ker(ϕj) = im(ϕj−1) for eachj. In particular, the following sequence
1→ N → G → H →1
is ashort exact sequence. We will also sometimes use,→ to indicate an injective morphism and to indicate a surjective morphism.
If a short exact sequence has a sectionH → G, then it is said to be split. Note that in this case, on eachR-points we have an exact sequence of groups
1→ N(R)→ G(R)→ H(R)→1
that splits, and thusG(R) is a semi-direct product of groupsN(R)oH(R). We shall then
say that G is a semi-direct product of N and H, denoted as N oH. Note that if G was abelian, thenN oHis just the product N × H.
Remark 2.5.8. We remind the reader that having an exact sequence of affine groups 1 → N −→ Gψ −→ H →ϕ 1 does not always imply that the sequence of groups at each R-points 1→ N(R)−−−→ G(ψ(R) R)−−−→ H(ϕ(R) R) →1 is exact. This is because the notion of a surjective affine group homomorphism is usually weaker than the notion of surjectivity at all R- points. Despite so, being a splitting exact sequence does give us that because having a sections:H → G means that on each R-point ϕ(R)◦s(R) = id, which shows that ϕ(R) is surjective.