nLab
affine line

Idea

For every Lawvere theory $T$ containing the theory of abelian groups Isbell dual sheaf topos over formal duals of $T$-algebras contains a canonical line object ${𝔸}^1$.

For $T$ the theory of commutative rings this is called the affine line .

Definition

Affine line

Let $k$ be a ring, and $T$ the Lawvere theory of associative algebras over $k$, such that the category of algebras over a Lawvere theory $T\mathrm{Alg}={\mathrm{Alg}}_k$ is the category of $k$-algebras.

Definition

The canonical $T$-line object is the affine line

$${𝔸}_k:=\mathrm{Spec}(F_T(*))=\mathrm{Spec}k[t].$$\mathbb{A}_k := Spec(F_T(*)) = Spec k[t] \,.

Here the free $T$-algebra on a single generator $F_T(*)$ is the polynomial algebra $k[t]\in {\mathrm{Alg}}_k$ on a single generator $*=t$ and $\mathrm{Spec}k[t]$ may be regarded as the corresponding object in the opposite category ${\mathrm{Aff}}_k:={\mathrm{Alg}}_k^{\mathrm{op}}$ of affine schemes over $\mathrm{Spec}k$.

Multiplicative group

The multiplicative group object in ${\mathrm{Ring}}^{\mathrm{op}}$ corresponding to the affine line – usually just called the multiplicative group – is the group scheme denoted ${𝔾}_m$

• whose underlying affine scheme is

  $$({𝔸}^1-\{0\}):=\mathrm{Spec}k[t,t^{-1}],$$(\mathbb{A}^1 - \{0\}) := Spec k[t,t^{-1}] \,,

  where $k[t,t^{-1}]$ is the localization of the ring $k[t]$ at the element $t=(t-0)$.

• whose multiplication operation

  $$\cdot {𝔾}_m\times {𝔾}_m\to {𝔾}_m$$\cdot \mathbb{G}_m \times \mathbb{G}_m \to \mathbb{G}_m

  is the morphism in ${\mathrm{Ring}}^{\mathrm{op}}$ corresponding to the morphism in Ring

  $$k[t_1,t_1^{-1}]{\otimes }_{k}k[t_2,t_2^{-1}]\leftarrow k[t,t^{-1}]$$k[t_1,t_1^{-1}] \otimes_k k[t_2, t_2^{-1}] \leftarrow k[t,t^{-1}]

  given by $t\mapsto t_1\cdot t_2$;

• whose unit map $\mathrm{Spec}k\to \mathrm{Spec}k[t,t^{-1}]$ is given by

  $$t\mapsto 1$$t \mapsto 1

• and whose inversion map $\mathrm{Spec}k[t,t^{-1}]\to \mathrm{Spec}[t,t^{-1}]$ is given by

  $$t\mapsto t^{-1}.$$t \mapsto t^{-1} \,.

Additive group

The additive group object in ${\mathrm{Ring}}^{\mathrm{op}}$ corresponding to the affine line – usually just called the additive group – is the group scheme denoted ${𝔾}_a$

• whose underlying object is ${𝔸}^1$ itself;

• whose addition operation ${𝔾}_a\times {𝔾}_a\to {𝔾}_a$ is dually the ring homomorphism

  $$k[t_1]{\otimes }_{k}k[t_2]\leftarrow k[t]$$k[t_1] \otimes_k k[t_2] \leftarrow k[t]

  given by

  $$t\mapsto t_1+t_2;$$t \mapsto t_1 + t_2 \,;

• whose unit map is given by

  $$t\mapsto 0;$$t \mapsto 0 \,;

• whose inversion map is given by

  $$t\mapsto -t.$$t \mapsto -t \,.

Properties

Proposition

Let $R$ be a commutative $k$-algebra. There is a natural isomorphism between

• $\mathbb{Z}$-gradings on $R$;

• ${𝔾}_m$-actions on $\mathrm{Spec}R$.

Proof

For the first direction, let $R$ be a $\mathbb{Z}$-graded commutative algebra. Then $X=\mathrm{Spec}R$ comes with a $𝔾$-action given as follows: the action morphism

$$\rho :X\times {𝔾}_m\to X$$\rho : X \times \mathbb{G}_m \to X

is dually the ring homomorphism

$$R{\otimes }_k\mathbb{Z}[t,t^{-1}]\leftarrow R$$R \otimes_k \mathbb{Z}[t,t^{-1}] \leftarrow R

defined on homogeneous elements $r$ of degree $n$ by

$$r\mapsto r\cdot t^n.$$r \mapsto r \cdot t^n \,.

The action property

$$\begin{array}{ccc}X\times {𝔾}_m\times {𝔾}_m& \stackrel{\mathrm{Id}\times \cdot }{\to }& X\times 𝔾\\ {}^{\rho \times \mathrm{Id}}\downarrow & & {\downarrow }^{\rho }\\ X\times {𝔾}_m& \stackrel{\rho }{\to }& X\end{array}$$\array{ X \times \mathbb{G}_m \times \mathbb{G}_m &\stackrel{Id \times \cdot}{\to}& X \times \mathbb{G} \\ {}^{\mathllap{\rho} \times Id}\downarrow && \downarrow^{\mathrlap{\rho}} \\ X \times \mathbb{G}_m &\stackrel{\rho}{\to}& X }

is equivalently the equation

$$r(t_1{)}^n\cdot (t_2{)}^n=r(t_1\cdot t_2{)}^n$$r (t_1)^n \cdot (t_2)^n = r (t_1 \cdot t_2)^n

for all $n\in \mathbb{Z}$. Similarly the unitality of the action is the equation

$$(1{)}^n=1.$$(1)^n = 1 \,.

Conversely, given an action of ${𝔾}_m$ on $\mathrm{Spec}R$ we have some morphism

$$R[t,t^{-1}]\leftarrow R$$R[t,t^{-1}] \leftarrow R

that sends

$$r\mapsto \sum _{n\in \mathbb{Z}}{r}_n{t}^n.$$r \mapsto \sum_{n \in \mathbb{Z}} r_n t^n \,.

By the action property we have that

$$\sum _n{r}_n(t_1{t}_2{)}^n=\sum _{n,k}(r_n{)}_k{t}_1^n{t}_2^k.$$\sum_n r_n (t_1 t_2)^n = \sum_{n,k} (r_n)_k t_1^n t_2^k \,.

Hence

$$(r_n{)}_k=\{\begin{array}{cc}{r}_n& \mathrm{if}n=k\\ 0& \mathrm{otherwise}\end{array}$$(r_n)_k = \left\{ \array{ r_n & if \; n = k \\ 0 & otherwise } \right.

and so the morphism gives a decomposition of $R$ into pieces labeled by $\mathbb{Z}$.

One sees that these two constructions are inverse to each other.

Examples

Projective space

The diagonal action of the multiplicative group on the product ${𝔸}^n:={\prod }_{i=1\cdots n}{𝔸}^1$ for $n\in \mathbb{N}$

$${𝔸}^n\times {𝔾}_m\to {𝔸}^n$$\mathbb{A}^n \times \mathbb{G}_m \to \mathbb{A}^n

is dually the morphism

$$k[t,t_1,\cdots ,t_n]\leftarrow k[t_1,\cdots ,t_n]$$k[t, t_1, \cdots, t_n] \leftarrow k[t_1, \cdots, t_n]

given by

$$t_i\mapsto t\cdot t_i.$$t_i \mapsto t \cdot t_i \,.

This makes $k[t,\{t_i\}]$ the free graded algebra over $k$ on $n$ generators $t_i$ in degree 1. This in $\mathbb{N}\subset \mathbb{Z}$-graded. What is genuinely $\mathbb{Z}$-graded is

$$𝒪({𝔸}^n-\{0\})\simeq k[t_1,t_1^{-1},\cdots ,t_n,t_n^{-1}].$$\mathcal{O} (\mathbb{A}^n - \{0\}) \simeq k[t_1, t_1^{-1}, \cdots, t_n, t_n^{-1}] \,.

The quotient by the multiplicative group action

$$𝔸P_k^n:=({𝔸}^{n+1}-\{0\})/{𝔾}_m$$\mathbb{A} P^n_k := (\mathbb{A}^{n+1} - \{0\})/\mathbb{G}_m

is the projective space over $k$ of dimension $n$.

${𝔸}^1$-homotopy theory

In A^1 homotopy theory one considers the reflective localizatoin

$${\mathrm{Sh}}_{\mathrm{\infty }}(C{)}_{{𝔸}^1}\overleftarrow{\hookrightarrow }{\mathrm{Sh}}_{\mathrm{\infty }}(C)$$Sh_\infty(C)_{\mathbb{A}^1} \stackrel{\leftarrow}{\hookrightarrow} Sh_\infty(C)

of the (∞,1)-topos of (∞,1)-sheaves over a site $C$ such as the Nisnevich site, at the morphisms of the form

$$p_1:X\times {𝔸}^1\to X$$p_1 : X \times \mathbb{A}^1 \to X

that contract away cartesian factors of the affine line.

Related concepts

• analytic affine line