Comprehensive Exam - University of South Carolina Mathematics

Road Map Paper Highlights Recent/Future Work

Comprehensive ExamUniversity of South Carolina Mathematics

Jason “Bailey” Heath

October 1, 2021

Jason “Bailey” Heath Comprehensive Exam


Road Map

Question 1

For a field k and n ∈ N, what is the smallest upper bound on therepresentation dimension of all n-dimensional algebraic tori over k?

1 Review highlights from my paper1 Affine group schemes and Hopf algebras2 Characters and diagonalizable group schemes3 Representation dimension4 Etale algebras5 Algebraic tori and character groups

2 Recent/Future Work1 G -lattices and symmetric rank2 1-dimensional and 2-dimensional tori3 What might results look like?



Affine Group Schemes

Definition 1

A functor F is representable if there exists a k-algebra A suchthat F � Homk (A, •). In this case, we say that A represents F .An affine group scheme over k is a representable functor from((k -alg)) to ((Grp)), and an affine group scheme is algebraic ifits representing k-algebra is finitely generated.

Definition 2

If F : ((k -alg)) → ((Grp)) is a functor represented by thek-algebra A and k ′ a field, then Fk′ : ((k ′ -alg)) → ((Grp)) is thefunctor represented by A ⊗ k ′.

Important case: k ′ = ks , the separable closure of k .



Examples of Affine Group Schemes

Example 1

[1, 1.2] The following are important examples of affine groupschemes.

1 The multiplicative group Gm is represented by k[X , 1

X

].

2 The group of nth roots of unity -n is represented byk [X ] /(X n − 1).

3 The general linear group GLn is represented byk

[X11, ...,Xnn,

1det

].

4 The special linear group SLn is represented byk [X11, ...,Xnn] /(det−1).



Yoneda’s Lemma

Lemma 1 (Yoneda’s Lemma)

[1, 1.3] There is a canonical anti-equivalence of categories betweenthe category of representable set functors ((k -alg)) → ((Sets))and the category of finitely generated k-algebras.

The anti-equivalence takes a representable set functor to itsrepresenting k-algebra and a k-algebra to the functor that itrepresents.



Hopf Algebras

By reversing the directions of the arrows in the group axiomdiagrams, Yoneda’s Lemma yields a Hopf algebra correspondingto each affine group scheme and vice versa.

Definition 2

A Hopf algebra A is a k-algebra along with k-algebra maps

comultiplication Δ : A→ A ⊗ A

counit (augmentation) Y : A→ k

coinverse (antipode) S : A→ A

satisfying the group axiom diagrams with the arrows reversed.



Hopf Algebra Structure for Gm

Recall that Gm is represented by k[X , 1

X

]. The Hopf algebra

structure is as follows:

Δ (X ) = X ⊗ X

Y (X ) = 1

S (X ) = 1

X.



Characters

Definition 3

The characters of an affine group scheme G are the group schemehomomorphisms G → Gm.

Definition 4

Given a Hopf algebra A, any b ∈ A× satisfying Δ (b) = b ⊗ b iscalled group-like.

Yoneda’s Lemma gives us the following:

Theorem 1

[1, 2.1] Characters of an affine group scheme G correspond togroup-like elements of the Hopf algebra representing G .



Diagonalizable Group Schemes

Definition 5

A diagonalizable group scheme G is an algebraic affine groupscheme which is represented by the group ring k [M] for somefinitely generated Abelian group M, with Hopf algebra mapsdefined to make the elements of M group-like, i.e., for all m ∈ M,

Δ (m) = m ⊗ m

Y (m) = 1

S (m) = m−1.

Equivalently (when k is a field and G is algebraic) [1, 2.2]:

G is isomorphic to a finite direct product of copies of Gm andvarious -n.

The representing Hopf algebra of G is spanned by group-likeelements.



Why ‘Diagonalizable’?

Definition 6

The Zariski topology of kn is the topology whose closed sets areprecisely the varieties in kn.

Theorem 2

[1, 4.6] Let M ⊆ GLn (k) be a subgroup. The elements of M canbe simultaneously diagonalized if and only if the group schemecorresponding to M (that is, the group scheme represented by

k[M

]) is diagonalizable.



Representation Dimension

Theorem 3

[1, 3.4] Every algebraic affine group scheme over a field isisomorphic to a closed subgroup of GLn for some n ∈ Z+.

Definition 7

Let G be an algebraic affine group scheme. A linear representationG ↩→ GLn is faithful if its kernel is the trivial group scheme. Therepresentation dimension of G , denoted rdim (G ), is the minimaln ∈ Z+ such that there exists a faithful linear representationG ↩→ GLn.

Representation dimension asks, “What is the smallest possible n inthe above theorem?”



Etale Algebras

Definition 8

A finite-dimensional k-algebra A is called an etale k-algebra if itsatisfies the following equivalent conditions:

1 A ⊗ k is reduced.

2 A ⊗ k � km

for some m ∈ N.

3

��Homk

(A, k

)�� = dimk (A).4 A is a product of separable field extensions.

5 A ⊗ ks � kms (where this m is the same as the m in (2)).

6 (if k is perfect) A is reduced.



Etale Algebras and Gal (ks/k)-Actions

Notation:

ks is the separable closure of k .

� := Gal (ks/k) is the absolute Galois group of k .

Theorem 4

[1, 6.3] Let � be the category of etale k-algebras and let � be thecategory of finite sets on which Gal (ks/k) acts continuously.Then, there is a canonical anti-equivalence of the categories � and� given by the following functors:

Homk (•, ks ) : � → �.[Hom( (Sets)) (•, ks )

]�: � → �.



Algebraic Tori

Definition 9

A group scheme G is of multiplicative type if Gks isdiagonalizable.

Definition 10

An (algebraic) torus is an affine group scheme T for which Tks isa finite product of copies of Gm. In particular, if Tks � G

nm, then n

is the dimension of T .

Example 2

Connected algebraic matrix groups are algebraic tori.



Algebraic Tori and Character Groups

Definition 11

Let G be an affine group scheme represented by A and let X bethe group of group-like elements of A ⊗ ks (equivalently, the set ofcharacters of Gks ). Then, the �-module X (i.e., the group Xequipped with the �-action) is the character group of G . Inparticular, � acts on simple tensors via f (a ⊗ _) = a ⊗ f (_) for allf ∈ �.

Theorem 5

[1, 7.3] There is a canonical anti-equivalence between thecategories ℳ of group schemes of multiplicative type and � offinitely generated Abelian groups on which � acts continuously.

X• : ℳ → � takes G to its character group XG .

� →ℳ takes X to the group scheme represented by thek-algebra of fixed elements of ks [X ] under the �-action.



G -Lattices

The previous theorem suggests that the following will be useful inour study of tori:

Definition 12

A lattice L is a free Z-module of finite rank, i.e., L � Zn forn = rank (L). Further, if G is a group, then L is a G -lattice if L isa Z [G ]-module.

Proposition 1

If T is an algebraic torus, then the character group XT of T is a�-lattice.



Symmetric Rank

Definition 13

The symmetric rank of a G -lattice L, denoted symrank (L), is theminimum r ∈ Z+ such that there exists a subset K ⊆ L such that

1 |K | = r ,

2 �K ⊆ K (i.e., K is �-stable), and

3 K generates L as a Z-module.

Theorem 6

If T is an algebraic torus, then

rdim (T ) = symrank (XT ) .



1-Dimensional Tori

Theorem 7

Let T be a 1-dimensional torus over k . Then,

rdim (T ) ={

1 if k is separably quadratically closed

2 otherwise.

Proof.

Since GL1 (Z) = {±1}, there are only two possible �-actions onGL1 (Z) � Z/2Z: (i) the trivial action and (ii) multiplication by −1.Observe that there is an element of � acting as (ii) ⇔ � has asubgroup of order 2 ⇔ k is not separably quadratically closed. �



2-Dimensional Tori

There are two maximal finite subgroups of GL2 (Z) (up toconjugacy):

1

⟨[0 −11 0

],

[−1 00 1

]⟩� D8 (dihedral group of order 8)

symrank(Z2

)= 4, generating set

{±

[10

],±

[01

]}2

⟨[1 −11 0

],

[0 11 0

]⟩� D12 (dihedral group of order 12)

symrank(Z2

)= 6, generating set

{±

[10

],±

[01

],±

[11

]}Thus, if T is a 2-dimensional torus, then rdim (T ) ≤ 6.



Future Work

Results from this research may take some of the following forms:

1 Upper bound for all fields: “If T is an algebraic torus ofdimension r , then rdim (T ) ≤ blah.”

2 Precise answer for specific fields: “If T is an algebraic torus ofdimension r over a field with certain nice conditions, thenrdim (T ) = blah.”

Both approaches require G -lattice theory, the second involves moreGalois theory.



References

Waterhouse, W. C., Introduction to affine group schemes,Graduate Texts in Mathematics, Vol. 66, Springer-Verlag, NewYork-Heidelberg, 1973. ISBN 0-387-90421-2


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Comprehensive Exam - University of South Carolina Mathematics