# Contents

## 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

${𝔸}_{k}:=\mathrm{Spec}\left({F}_{T}\left(*\right)\right)=\mathrm{Spec}k\left[t\right]\phantom{\rule{thinmathspace}{0ex}}.$\mathbb{A}_k := Spec(F_T(*)) = Spec k[t] \,.

Here the free $T$-algebra on a single generator ${F}_{T}\left(*\right)$ is the polynomial algebra $k\left[t\right]\in {\mathrm{Alg}}_{k}$ on a single generator $*=t$ and $\mathrm{Spec}k\left[t\right]$ 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

$\left({𝔸}^{1}-\left\{0\right\}\right):=\mathrm{Spec}k\left[t,{t}^{-1}\right]\phantom{\rule{thinmathspace}{0ex}},$(\mathbb{A}^1 - \{0\}) := Spec k[t,t^{-1}] \,,

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

• whose multiplication operation

$\cdot {𝔾}_{m}×{𝔾}_{m}\to {𝔾}_{m}$\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

$k\left[{t}_{1},{t}_{1}^{-1}\right]{\otimes }_{k}k\left[{t}_{2},{t}_{2}^{-1}\right]←k\left[t,{t}^{-1}\right]$k[t_1,t_1^{-1}] \otimes_k k[t_2, t_2^{-1}] \leftarrow k[t,t^{-1}]

given by $t↦{t}_{1}\cdot {t}_{2}$;

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

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

$t↦{t}^{-1}\phantom{\rule{thinmathspace}{0ex}}.$t \mapsto t^{-1} \,.

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 ${𝔾}_{a}×{𝔾}_{a}\to {𝔾}_{a}$ is dually the ring homomorphism

$k\left[{t}_{1}\right]{\otimes }_{k}k\left[{t}_{2}\right]←k\left[t\right]$k[t_1] \otimes_k k[t_2] \leftarrow k[t]

given by

$t↦{t}_{1}+{t}_{2}\phantom{\rule{thinmathspace}{0ex}};$t \mapsto t_1 + t_2 \,;
• whose unit map is given by

$t↦0\phantom{\rule{thinmathspace}{0ex}};$t \mapsto 0 \,;
• whose inversion map is given by

$t↦-t\phantom{\rule{thinmathspace}{0ex}}.$t \mapsto -t \,.

## Properties

###### Proposition

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

• $ℤ$-gradings on $R$;

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

###### Proof

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

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

is dually the ring homomorphism

$R{\otimes }_{k}ℤ\left[t,{t}^{-1}\right]←R$R \otimes_k \mathbb{Z}[t,t^{-1}] \leftarrow R

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

$r↦r\cdot {t}^{n}\phantom{\rule{thinmathspace}{0ex}}.$r \mapsto r \cdot t^n \,.

The action property

$\begin{array}{ccc}X×{𝔾}_{m}×{𝔾}_{m}& \stackrel{\mathrm{Id}×\cdot }{\to }& X×𝔾\\ {}^{\rho ×\mathrm{Id}}↓& & {↓}^{\rho }\\ X×{𝔾}_{m}& \stackrel{\rho }{\to }& X\end{array}$\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

$r\left({t}_{1}{\right)}^{n}\cdot \left({t}_{2}{\right)}^{n}=r\left({t}_{1}\cdot {t}_{2}{\right)}^{n}$r (t_1)^n \cdot (t_2)^n = r (t_1 \cdot t_2)^n

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

$\left(1{\right)}^{n}=1\phantom{\rule{thinmathspace}{0ex}}.$(1)^n = 1 \,.

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

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

that sends

$r↦\sum _{n\in ℤ}{r}_{n}{t}^{n}\phantom{\rule{thinmathspace}{0ex}}.$r \mapsto \sum_{n \in \mathbb{Z}} r_n t^n \,.

By the action property we have that

$\sum _{n}{r}_{n}\left({t}_{1}{t}_{2}{\right)}^{n}=\sum _{n,k}\left({r}_{n}{\right)}_{k}{t}_{1}^{n}{t}_{2}^{k}\phantom{\rule{thinmathspace}{0ex}}.$\sum_n r_n (t_1 t_2)^n = \sum_{n,k} (r_n)_k t_1^n t_2^k \,.

Hence

$\left({r}_{n}{\right)}_{k}=\left\{\begin{array}{cc}{r}_{n}& \mathrm{if}\phantom{\rule{thickmathspace}{0ex}}n=k\\ 0& \mathrm{otherwise}\end{array}$(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 $ℤ$.

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 $n\in ℕ$

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

is dually the morphism

$k\left[t,{t}_{1},\cdots ,{t}_{n}\right]←k\left[{t}_{1},\cdots ,{t}_{n}\right]$k[t, t_1, \cdots, t_n] \leftarrow k[t_1, \cdots, t_n]

given by

${t}_{i}↦t\cdot {t}_{i}\phantom{\rule{thinmathspace}{0ex}}.$t_i \mapsto t \cdot t_i \,.

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

$𝒪\left({𝔸}^{n}-\left\{0\right\}\right)\simeq k\left[{t}_{1},{t}_{1}^{-1},\cdots ,{t}_{n},{t}_{n}^{-1}\right]\phantom{\rule{thinmathspace}{0ex}}.$\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

$𝔸{P}_{k}^{n}:=\left({𝔸}^{n+1}-\left\{0\right\}\right)/{𝔾}_{m}$\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

${\mathrm{Sh}}_{\infty }\left(C{\right)}_{{𝔸}^{1}}\stackrel{←}{↪}{\mathrm{Sh}}_{\infty }\left(C\right)$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

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

that contract away cartesian factors of the affine line.

Revised on January 5, 2012 19:57:19 by Urs Schreiber (89.204.138.119)