Contents

# 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 $\mathbb{A}^1$.

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 $T Alg = Alg_k$ is the category of $k$-algebras.

###### Definition

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

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

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

### Multiplicative group

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

• whose underlying affine scheme is

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

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

• whose multiplication operation

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

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

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

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

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

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

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

Therefore for $R$ any ring a morphism

$Spec R \longrightarrow \mathbb{G}_m$

is equivalently a ring homomorphism

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

which is equivalently a choice of multiplicatively invertible element in $R$. Therefore

$Hom(Spec R , \mathbb{G}_m) \simeq R^\times = GL_1(R)$

is the group of units of $R$.

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

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

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

$k[t_1] \otimes_k k[t_2] \leftarrow k[t]$

given by

$t \mapsto t_1 + t_2 \,;$
• whose unit map is given by

$t \mapsto 0 \,;$
• whose inversion map is given by

$t \mapsto -t \,.$

### Group of roots of unity

The group of $n$th roots of unity is

$\mu_n = Spec(k[t](t^n -1)) \,.$

This sits inside the multiplicative group via the Kummer sequence

$\mu_n \longrightarrow \mathbb{G}_m \stackrel{(-)^n}{\longrightarrow}\mathbb{G}_m \,.$

## Properties

###### Proposition

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

• $\mathbb{Z}$-gradings on $R$;

• $\mathbb{G}_m$-actions on $Spec R$.

###### Proof

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

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

is dually the ring homomorphism

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

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

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

The action property

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

is equivalently the equation

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

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

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

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

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

that sends

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

By the action property we have that

$\sum_n r_n (t_1 t_2)^n = \sum_{n,k} (r_n)_k t_1^n t_2^k \,.$

Hence

$(r_n)_k = \left\{ \array{ r_n & if \; n = k \\ 0 & otherwise } \right.$

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.

### Étale homotopy type

###### Example

For $k$ a field of characteristic 0, then the affine line $\mathbb{A}^1_k$ has a contractible étale homotopy type . This is no longer the case in positive characteristic.

### Internal formulation

###### Proposition

Let $X$ be a scheme and $Sh(Sch/X)$ the big Zariski topos associated to $X$. Denote by $\mathbb{A}^1$ (the affine line) the ring object $T \mapsto \Gamma(T,\mathcal{O}_T)$, i.e. the functor represented by the $X$-scheme $\mathbb{A}^1_X \coloneqq X \times Spec(\mathbb{Z}[t])$. Then:

• $\mathbb{A}^1$ is internally a local ring.

• $\mathbb{A}^1$ is internally a field in the sense that any nonzero element is invertible.

• Internally, any function $f : \mathbb{A}^1 \to \mathbb{A}^1$ is a polynomial function, i.e. of the form $f(x) = \sum_i a_i x^i$ for some coefficients $a_i : \mathbb{A}^1$. More precisely,

$Sh(Sch/X) \models \forall f : [\mathbb{A}^1,\mathbb{A}^1]. \bigvee_{n \in \mathbb{N}} \exists a_0,\ldots,a_n : \mathbb{A}^1. \forall x : \mathbb{A}^1. f(x) = \sum_i a_i x^i.$

Furthermore, these coefficients are uniquely determined.

###### Proof

Since the internal logic is local, we can assume that $X = Spec(R)$ is affine. The interpretations of the asserted statements using the Kripke?Joyal semantics are:

• Let $S$ be an $R$-algebra and $f, g \in S$ be elements such that $f + g = 1$. Then there exists a partition $1 = \sum_i s_i \in S$ such that in the localized rings $S[s_i^{-1}]$, $f$ or $g$ is invertible.

• Let $S$ be an $R$-algebra and $f \in S$ an element. Assume that any $S$-algebra $T$ in which $f$ is zero is trivial (fulfills $1 = 0 \in T$). Then $f$ is invertible in $S$.

• Let $S$ be an $R$-algebra and $f \in [\mathbb{A}^1,\mathbb{A}^1](S) = S[T]$ be an element. Then there exists a partition $1 = \sum_i s_i \in S$ such that in the localized rings $S[s_i^{-1}]$, $f$ is a polynomial with coefficients in $S[s_i^{-1}]$.

For the first statement, simply choose $s_1 \coloneqq f$, $s_2 \coloneqq g$.

For the second statement, consider the $S$-algebra $T \coloneqq S/(f)$.

The third statement is immediate, localization is not even necessary.

###### Remark

Since the big Zariski topos is cocomplete (being a Grothendieck topos), one can also get rid of the external disjunction and refer to the object $\mathbb{A}^1[X]$ of internal polynomials: The canonical ring homomorphism $\mathbb{A}^1[X] \to [\mathbb{A}^1,\mathbb{A}^1]$ (given by evaluation) is an isomorphism.

## Examples

### Projective space

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

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

is dually the morphism

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

given by

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

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

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

The quotient by the multiplicative group action

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

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

### $\mathbb{A}^1$-homotopy theory

In A^1 homotopy theory one considers the reflective localizatoin

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

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

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

that contract away cartesian factors of the affine line.

Discussion of étale homotopy type is in

• Armin Holschbach, Johannes Schmidt, Jakob Stix, Étale contractible varieties in positive characteristic (arXiv:1310.2784)