## Generators and relations

Last update: 20 March 2012

## Generators and relations

Let $S$ be a set. Let $A$ be an algebra.

A free $A-$module on $S$ is a pair $\left({A}^{S},\iota \right)$ where

1. ${A}^{S}$ is an $A-$module
2. $\iota :S\to {A}^{S}$ is a function
such that,
1. if $M$ is an $A-$module with a function $ȷ:S\to M$ then there exists a unique morphism
2. $\stackrel{˜}{ȷ}:{A}^{S}\to M$ such that $\stackrel{˜}{ȷ}\circ \iota =ȷ.$
This is a definition by a universal property.

• A morphism in the category of sets is a function.
• A morphism in the category of categories is a functor.

The forgetful functor is $F: A-modules → Sets M ↦ M$

The free module functor is $Sets → A-modules S ↦ AS$

The universal property for free modules tells us $HomA (AS,M) ≃ HomSets (S,M) ȷ˜ ↤ ȷ$ The free module functor is the left adjoint to the forgetful functor.

Let $M$ be an $A-$module.

A presentation of $M$ is an exact sequence $AT →ρ AS →γ M→0$ where ${A}^{T}$ and ${A}^{S}$ are free modules.

The term exact sequence means that

1. $\mathrm{im}\rho =\mathrm{ker}\gamma$

Let $X$ be a group.

A presentation of $X$ is an exact sequence $R→G→X{1}$ where $R$ and $G$ are free groups.

Examples.

1. $\frac{ℤ}{mℤ}$ is generated by $1$ with relation The cyclic group of order m ${C}_{m}$ is generated by $g$ with relation ${g}^{m}=1.$

Alternatively: $\frac{ℤ}{mℤ}$ is presented by $ℤ → ℤ → ℤmℤ → 0 1 ↦ 1 1 ↦ m$ and ${C}_{m}$ is presented by $G{r} → G{g} → Cm → {1} g ↦ g r ↦ gm$ where ${G}^{S}$ denotes the free group on the set $S.$
2. The dihedral group of order $2m,$ ${G}_{m,m,2}$ is generated by $x$ and $y$ with relations Alternately: $F3 → F2 → Gm,m,2 → {1} g1 ↦ x g2 ↦ y r1 ↦ g12 r2 ↦ g2m r3 ↦ g1g2g1-1g2$ where ${F}_{k}$ denotes a free group on a set with $k$ elements.

## Presentations by exact sequences

A presentation of $E$ is an exact sequence $F(Y) → F(X) → E→0$ where $F\left(Y\right)$ is the free object on the set $Y$ and $F\left(X\right)$ is the free object on the set $X.$

An exact sequence is a sequence of morphisms $⋯→ Ei →γi Ei+1 →γi+1 Ei+2 →γi+2 ⋯$ such that if $i\in ℤ$ then $\mathrm{im}{\gamma }_{i}=\mathrm{ker}{\gamma }_{i+1}.$

Let $G$ be a group and let ${g}_{1},...,{g}_{n} \in G$ and ${r}_{1},{r}_{2},...,{r}_{m} \in F\left\{{x}_{1},...,{x}_{n}\right\}.$

The group $G$ is presented by generators ${g}_{1},...,{g}_{n}$ and relations ${r}_{1},{r}_{2},...,{r}_{m}$ if $F{y1,...,ym} → F{x1,...,xn} → G → {1} xi ↦ gi yj ↦ rj$ is an exact sequence.

Let $V$ be a vector space and let ${b}_{1},...,{b}_{n}\in V.$

The vector space $V$ has basis $\left\{{b}_{1},...,{b}_{n}\right\}$ if ${0} → 𝔽{x1,...,xn} → V → {0} xi ↦ bi$ is an exact sequence.

1. A cyclic group is a group generated by one element.
2. A dihedral group is a group generated by two elements of order two.

### Some "familiar" constructions by generators and relations

1. $𝔽\left[{x}_{1},...,{x}_{n}\right]$ is the free commutative algebra on the set $\left\{{x}_{1},...,{x}_{n}\right\}.$
2. The free group on the set $X=\left\{{x}_{1},...,{x}_{n}\right\}$ is the monoid given by generators $\left\{{x}_{1},...,{x}_{n},{y}_{1},...,{y}_{n}\right\}$ and relations
3. Let $V$ and $W$ be $R-$modules, $V$ a right $R-$module and $W$ a left $R-$module. The tensor product is the abelian group generated by with relations $(z1v1+z2v2) ⊗ w = z1(v1⊗w) + z2(v2⊗w) v ⊗ (z1w1+z2w2) = z1(v⊗w1) + z2(v⊗w2) vr⊗w = v⊗rw$ for
4. $𝔽\left[{x}_{1}^{±1},...,{x}_{n}^{±1}\right]$ is the commutative algebra given by generators ${x}_{1},...,{x}_{n},{y}_{1},...,{y}_{n}$ and relations
5. The free abelian group ${ℤ}^{n}$ is the group given by generators ${x}_{1},...,{x}_{n}$ with relations

## The goal of classical invariant theory

Let $V$ be a vector space with basis $\left\{{x}_{1},...,{x}_{n}\right\}.$ Then $S(V) = 𝔽[x1,...,xn].$ Let ${W}_{0}$ be a subgroup of $GL\left(V\right).$ Since $S\left(V\right)$ is a functor, ${W}_{0}$ acts on $S\left(V\right):$ $w(xi1⋯xik) = (wxi1) (wxi2) ⋯ (wxik)$ for $w\in {W}_{0},$ $1\le {i}_{1}\le \cdots \le {i}_{k}\le n.$

The invariant ring of ${W}_{0}$ is

Find generators and relations for $S{\left(V\right)}^{{W}_{0}}.$

Let $V$ be a free $ℤ-$module with basis $\left\{{x}_{1},...,{x}_{n}\right\}.$ Then $𝔽[V] = 𝔽[x1±1,...,xn±1].$ Let ${W}_{0}$ be a subgroup of $GL\left(V\right)$ (note that $GL\left(V\right)=G{L}_{n}\left(ℤ\right)$), and $wxλ = xwλ$ where ${x}^{\lambda }={x}_{1}^{{\lambda }_{1}}\cdots {x}_{n}^{{\lambda }_{n}}$ if $\lambda ={\lambda }_{1}{\epsilon }_{1}+\cdots +{\lambda }_{n}{\epsilon }_{n}.$

The invariant ring of ${W}_{0}$ is

1. The cyclic groups are
2. The dihedral groups are

## Notes and References

The definition of presentation is found in [Bou, Alg. Ch.10, §1 no.4 (13)]. The definition of cyclic group is in [Bou, Alg.] and the definition of dihedral group is from [Bou, Lie, Ch.IV, §1 Def.2].

References?