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Algebraic Groups Seminar
1
Andrei Frimu
Talk 4: Cartier Duality
Recall
Assume we have a locally small category C which admits finite products and has a final object ?, i.e. an object
so that for every Z ∈ Ob(C), there exists a unique morphism Z → ?. Note that two morphisms f, g Z → G can
then be identified uniquely with a morphism (f, g) : Z → G × G.
Definition. A commutative group object in C is a pair consisting of an object G ∈ Ob(C) and a morphism
µ : G × G → G such that for every object Z ∈ Ob(C), the map Mor(Z, G) × Mor(Z, G) → Mor(Z, G) sending
(g, g 0 ) 7→ µ ◦ (g, g 0 ) defines a commutative group.
There is an alternate description of commutative group objects which sheds more light on what associativity,
commutativity, existence of identity and inverses means. Before we state the proposition, note that the existence
of products implies the existence of a morphism σ : G × G → G × G interchanging the two factors.
µ
Proposition. An object G ∈ Ob(C) and a morphism G × G −
→ G constitute a commutative group object in C if
and only if the following properties hold:
• Associativity: The following diagram is commutative:
G×G×G
µ×id
/ G×G
µ
id ×µ
G×G
/ G.
µ
• Commutativity: The following diagram is commutative:
/
<G
µ
G×G
σ
µ
G×G
• Identity element: There exists a morphism e : ? → G such that the following diagram commutes
e×id
?×G
pr2
/ G×G
µ
z
G
• Inverses: There exists a morphism i : G → G such that the following diagram commutes
/ G×G
id ×i
G ×O G
µ
diag
/?
G
where e is the morphism from the previous bullet point.
1
e
/G
Algebraic Groups Seminar
2
Andrei Frimu
Talk 4: Cartier Duality
Affine Group Schemes
Let Rings be the category of commutative noetherian rings with 1. The morphisms of this category are ring
homomorphisms preserving the identity. Recall that there is an (anti-)equivalence of categories between Rings
and the category Aff.Sch of affine schemes.
Recall that an object A of Rings together with a morphism R → A constitutes a unitary R-algebra. Note
that since A is a ring and the multiplication A × A → A is R-bilinear, it induces an R-module homomorphism
µ : A ⊗R A → A as follows
A co
A ⊗O R A
µ
⊗
•
A×A
This allows us to (equivalently) view a unitary R-algebra A as an R-module with two homomorphisms of
R-modules
µ
e
R−
→A←
− A ⊗R A,
satisfying µ(a ⊗ a0 ) = µ(a0 ⊗ a) (corresponding to multiplication in A being commutative) and µ(a ⊗ µ(a0 ⊗ a00 )) =
µ(µ(a ⊗ a0 ) ⊗ a00 ) (corresponding to multiplication in A being associative) and µ(e(1) ⊗ a) = a (corresponding to
a · e(1) = a). Denote the category of R-algebras by R -Alg. Then the (anti-)equivalence of categories Rings
and Aff.Sch restricts to an (anti-)equivalence of categories R -Alg and Aff. R -Sch.
Definition. An affine commutative group scheme over Spec R is a commutative group object in the
category of affine schemes over Spec R.
Let G = Spec A be an affine commutative scheme over Spec R. Then in particular A is a unitary R-algebra.
Combining the ’categorical’ definitions for unitary R-algebras and commutative group object, we obtain the
following diagram of R-module homomorphisms:
R
|
µ
?D A
z
A 8 ⊗R A,
e
m
i
The axioms satisfied by all the morphisms above turn A into a Hopf R-algebra.
Definition. A homomorphism of group schemes Φ : G → H over Spec R is a morphism in Aff. R -Sch,
such that the induced morphism Mor(Z, G) → Mor(Z, H) is a homomorphism of groups for all Z ∈ Ob(Aff. R -Sch).
If G = Spec A and H = Spec B, this morphism corresponds to a homomorphism of R-modules φ : B → A such
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Algebraic Groups Seminar
Andrei Frimu
Talk 4: Cartier Duality
that the following diagram commutes:
A
|
R
= D AO
µA
z
A8 ⊗O R A
eA
id
R
mA
iA
φ
|
B
=D B
φ⊗φ
µB
z
B 8 ⊗R B.
mB
eB
iB
Definition. The sum of two homomorphisms Φ, Ψ : G → H is defined by the commutative diagram
G
diag
Φ+Ψ
Ho
3
/ G×G
Φ×Φ
µH
H ×H
µn,k and αp,R
Let’s step aside from the course of the notes to define two objects of interest later. Recall that the multiplicative
def
group over k is Gm,k = Spec k[T , T −1 ]. This is a group scheme, with co-multiplication m : k[T , T −1 ] →
k[T1 , T1−1 ] ⊗k k[T2 , T2−1 ] given by T → T1 ⊗ T2 . Consider the homomorphism n · id : Gm,k → Gm,k induced
by the map (on rings) k[T , T −1 ] → k[T , T −1 ] sending t 7→ tn . We are interested in the fiber produce µn,k =
Gm,k ⊗Gm,k Spec k in the diagram
Gm,k
O
n·id
/ Gm,k
O
/ Spec k
µn,k
where the map Spec k → Gm,k is induced from k[S, S −1 ] → k sending S 7→ 1.
For affine schemes, fiber products correspond to the tensor product of the corresponding rings, implying that
µn,k = Spec A, where A = k[T , T −1 ] ⊗k[S,S −1 ] k. Using T n ⊗ 1 = S(1 ⊗ 1) = 1 ⊗ S = 1 ⊗ 1, one concludes that
A = k[T ]/(T n − 1), hence µn,k = Spec k[T ]/(T n − 1). This is a group scheme with comultiplication m sending,
just like for Gm,k (but in different rings) T 7→ T1 ⊗ T2 .
The other object we introduce is αn,R = Spec R[T ]/(T n ). This is a group scheme with comultiplication
m : R[T ]/(T n ) → R[T1 ]/(T1n ) ⊗R R[T2 ]/(T2n ) given by m(T ) = T1 ⊗ 1 + ⊗T2 .
s
s
Note that over a field of positive characteristic p, with n = ps , we have T p − 1 = (T − 1)p , and hence
s
µps ,k w Spec k[U ]/(U p ) w αps ,k , where we identified U = T − 1. As a particular case, notice that we have
established µp,k w αp,k as schemes over k. By the end of the lecture, we shall see that µp,k and αp,k are not
isomorphic in the category of affine commutative finite flat group schemes over Spec k.
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Algebraic Groups Seminar
4
Andrei Frimu
Talk 4: Cartier Duality
Cartier Duality
Assume now that G = Spec A is a group scheme, finite and flat over R (i.e. that A is a locally free, finitelydef
generated R-module). Let A∗ = HomR (A, R) be the R-dual of A. One can check that if we dualize the diagram
following the definition of an affine commutative group scheme, one obtains the following diagram, under the
identifications R∗ = R and (A ⊗R A)∗ = A∗ ⊗R A∗ :
R
|
e∗
∗
A
=D
∗
m∗
x
A∗ 7 ⊗R A∗ .
µ∗
i∗
The morphisms e∗ , m∗ , µ∗ , ∗ and i∗ satisfy the axioms of a cocommutative Hopf algebra with antipodism, and
def
therefore G∗ = Spec A∗ is an affine commutative finite flat group scheme over Spec R.
Definition. G∗ is called the Cartier dual of G.
If we have homomorphism Φ : G → H of affine commutative finite flat group schemes over Spec R
corresponding to a homomorphism φ : B → A, then φ induces φ∗ : A∗ → B ∗ and thus a homomorphism of
group schemes H ∗ → G∗ . Hence Cartier duality gives a contravariant functor from the category of affine group
schemes to itself.
This functor is additive, i.e. given Ψ, Φ : G → H, then (Ψ + Φ)∗ = Ψ∗ + Φ∗ .
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Constant Group Schemes
Let G be an additive finite abelian group. We want to exhibit a finite commutative group scheme associated
to G. For that we take the disjoint union of |G| copies of the final object Spec R = ∗ in the category of affine
schemes over Spec R. Let GR (standard notation for group schemes) be
GR =
a
Spec R.
g∈G
Notation: Write (Spec R)g = Spec R for the g-component of GR .
Exercise: Why is this a scheme over Spec R? What is the unique morphism to the terminal object Spec R?
Exercise: GR is finite and flat over Spec R.
As written, it is not obvious that this is a commutative group scheme over Spec R. For that, note first that
a
GR × GR w
Spec R.
g,g 0 ∈G
This follows from the fact that the product of two disjoint unions X =
`
Xα and Y =
`
Yβ of R-schemes is the
disjoint union of the products Xα ×R Yβ (EGA, I.3.2.8) and from (Spec R ×R Spec R) w Spec R (EGA Chapter
I.3.2.2).
Define then the morphism µ : G × G → G by sending the component (Spec R)g,g0 of GR × GR to the
(Spec R)g+g0 component of GR .
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Algebraic Groups Seminar
Andrei Frimu
Talk 4: Cartier Duality
Exercise: The scheme GR and the morphism µ define a commutative group scheme over Spec R.
The scheme GR is also called the constant group scheme over R with fiber G.
Lemma. Let
def
RG = {f : G → R| f is a map of sets}.
This is a ring, with addition and multiplication defined componentwise. The zero and the identity are the
constant maps with value 0, respectively 1. Then GR w Spec(RG ).
Proof. This follows by induction from EGA I.3.1.1.
However, as it is necessary in the subsequent discussion, we can describe the isomorphisms between the two
sides in the following way:
As a convenience of notation, let Rg be a copy of the ring R, and assume the g-component of GR is
(Spec R)g = Spec Rg . For every such component we have a morphism Spec Rg → Spec(RG ) induced by the ring
homomorphism RG → Rg sending f to f (g) ∈ R = Rg . This then induces a morphism φ : GR → Spec(RG ).
Conversely, we have a morphism ψ : Spec(RG ) → GR , induced by the ring homomorphisms Rg → RG
sending an element r ∈ R = Rg to the map f : G → R satisfying f (g) = r and f (g 0 ) = 0 for g 6= g 0 .
Exercise: φ and ψ are isomorphisms, inverse to each other.
In particular Gr ×Gr is Spec(RG×G ) and thus the map GR ×GR → GR is induced by the the comultiplication
map m : RG → RG ⊗R RG w RG×G . Let’s explicitly define this map. Let f ∈ RG . From the map on components
GR × GR → GR defined above, one can deduce that m(f ) is the map of RG×G sending (g, g 0 ) to f (g + g 0 ).
The counit RG → R is defined by (f ) = f (0). The coinverse i : RG → RG is, as expected, the map sending
g 7→ f (−g).
Consider now the canonical basis {eg } of RG given by eg : G → R such that eg (g) = 1 and eg (g 0 ) = 0 for
g 0 6= g. Using the above, one can check that we have
µ(eg ⊗ eg0 ) = (eg if g = g 0 ) and (0 otherwise)
(eg ) = (1 if g = 0) and (0 otherwise)
X
e(1) =
eg
g∈G
m(eg ) =
X
eg0 ⊗ eg−g0
g 0 ∈G
i(eg ) = e−g .
To see what the Cartier dual of GR is, let (êg )g∈G be the basis of (RG )∗ dual to the above basis, i.e. êg (g 0 ) = 1
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Algebraic Groups Seminar
Andrei Frimu
Talk 4: Cartier Duality
if g 0 = g and 0 if g 0 6= g. Then the dual maps are given by
µ∗ (êg ) = êg ⊗ êg
∗ (1) = ê0
e∗ (êg ) = 1
m∗ (êg ⊗ êg0 ) = êg+g0
i∗ (êg ) = ê−g .
Proposition. The formulas for m∗ and ∗ show that (RG )∗ is isomorphic to the group R[G] as an R-algebra,
such that e∗ corresponds to the usual augmentation map R[G] → R.
Example 1: G = Z/nZ
def
Denote X = ê1 . Then (RG )∗ w R[G] = R[Z/nZ] = R[X]/(X n − 1). The comultiplication µ∗ : (RG )∗ →
(RG )∗ ⊗R (RG )∗ is µ∗ (X) = X ⊗ X, which we note is the same as what holds for µn,R . Thus
∗
(Z/nZ)R w µn,R .
Dual of αp,R :
Assume R has characteristic p and let A = R[T ]/(T p ). Recall αp,R = Spec A is a group scheme with
comultiplication m(T ) = T ⊗ 1 + 1 ⊗ T . Let ei be the basis (Ti )0≤i<p . Then we have
µ(T i ⊗ T j ) = (T i+j if i + j < p) and (0 otherwise)
(T i ) = (1 if i = 0) and (0 otherwise)
e(1) = T 0
X i
i
m(T ) =
· T i ⊗ T i−j
j
0≤j≤i
i(T i ) = (−1)i T i .
If {ui } is the dual basis of A∗ . Then one can check that the R-linear map A∗ → A sending ui to T i /i! is an
isomorphism of Hopf algebras, and therefore
(αp,R )∗ w αp,R .
We finish the talk with the following
Proposition. Let k be any field with char k = p > 0. The the group schemes Z/pZk , µp,k and αp,k are pairwise
non-isomorphic.
Proof. The group scheme Z/pZ is isomorphic to Spec k p , so it is a reduced scheme, while µp,k = Spec k[T ]/(T p −
k
1) and αp,k = Spec k[T ]/(T p ) are non-reduced, so they are distinct.
As we have seen before µp,k and αp,k are isomorphic as schemes. However as affine group schemes, they are
not, for their Cartier duals are different from the previous two examples.
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Bibliography
[1] Finite Group Schemes,
Richard Pink,
http://www.math.ethz.ch/~pink/ftp/FGS/
CompleteNotes.pdf
[2] Modular Forms and Fermat’s Last Theorem, Chapter V, Finite Group Schemes
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