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\begin{document}

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\section*{affine line}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{geometry}{}\paragraph*{{Geometry}}\label{geometry}

[[!include higher geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{affine_line}{Affine line}\dotfill \pageref*{affine_line} \linebreak
\noindent\hyperlink{MultiplicativeGroup}{Multiplicative group}\dotfill \pageref*{MultiplicativeGroup} \linebreak
\noindent\hyperlink{AdditiveGroup}{Additive group}\dotfill \pageref*{AdditiveGroup} \linebreak
\noindent\hyperlink{group_of_roots_of_unity}{Group of roots of unity}\dotfill \pageref*{group_of_roots_of_unity} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{Grading}{Grading}\dotfill \pageref*{Grading} \linebreak
\noindent\hyperlink{tale_homotopy_type}{\'E{}tale homotopy type}\dotfill \pageref*{tale_homotopy_type} \linebreak
\noindent\hyperlink{InternalFormulation}{Internal formulation}\dotfill \pageref*{InternalFormulation} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{projective_space}{Projective space}\dotfill \pageref*{projective_space} \linebreak
\noindent\hyperlink{homotopy_theory}{$\mathbb{A}^1$-homotopy theory}\dotfill \pageref*{homotopy_theory} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

For every [[Lawvere theory]] $T$ containing the theory of [[abelian group]]s [[Isbell duality|Isbell dual]] [[sheaf topos]] over formal duals of $T$-algebras contains a canonical [[line object]] $\mathbb{A}^1$.

For $T$ the theory of commutative [[ring]]s this is called the \emph{affine line} .

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{affine_line}{}\subsubsection*{{Affine line}}\label{affine_line}

Let $k$ be a [[ring]], and $T$ the [[Lawvere theory]] of [[associative algebra]]s over $k$, such that the category of [[algebras over a Lawvere theory]] $T Alg = Alg_k$ is the [[category]] of $k$-algebras.

\begin{defn}
\label{}\hypertarget{}{}
The canonical $T$-[[line object]] is the \textbf{affine line}

\begin{displaymath}
\mathbb{A}_k := Spec(F_T(*))  = Spec (k[t])
  \,.
\end{displaymath}
\end{defn}
Here the [[free functor|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 scheme]]s over $Spec  k$.

\hypertarget{MultiplicativeGroup}{}\subsubsection*{{Multiplicative group}}\label{MultiplicativeGroup}

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

\begin{itemize}%
\item whose underlying affine scheme is

\begin{displaymath}
(\mathbb{A}^1 - \{0\}) := Spec \left(k[t,t^{-1}]\right)
  \,,
\end{displaymath}
where $k[t,t^{-1}]$ is the [[localization]] of the ring $k[t]$ at the element $t = (t-0)$.


\item whose multiplication operation

\begin{displaymath}
\cdot \mathbb{G}_m \times \mathbb{G}_m \to \mathbb{G}_m
\end{displaymath}
is the morphism in $Ring^{op}$ corresponding to the morphism in [[Ring]]

\begin{displaymath}
k[t_1,t_1^{-1}] \otimes_k k[t_2, t_2^{-1}] \leftarrow k[t,t^{-1}]
\end{displaymath}
given by $t \mapsto t_1 \cdot t_2$;


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

\begin{displaymath}
t \mapsto 1
\end{displaymath}

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

\begin{displaymath}
t \mapsto t^{-1}
  \,.
\end{displaymath}


\end{itemize}
Therefore for $R$ any [[ring]] a morphism

\begin{displaymath}
Spec R \longrightarrow \mathbb{G}_m
\end{displaymath}
is equivalently a ring homomorphism

\begin{displaymath}
R \leftarrow k[t,t^{-1}]
\end{displaymath}
which is equivalently a choice of multiplicatively invertible element in $R$. Therefore

\begin{displaymath}
Hom(Spec R , \mathbb{G}_m) \simeq R^\times = GL_1(R)
\end{displaymath}
is the [[group of units]] of $R$.

\hypertarget{AdditiveGroup}{}\subsubsection*{{Additive group}}\label{AdditiveGroup}

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

\begin{itemize}%
\item whose underlying object is $\mathbb{A}^1$ itself;


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

\begin{displaymath}
k[t_1] \otimes_k k[t_2] \leftarrow k[t]
\end{displaymath}
given by

\begin{displaymath}
t \mapsto t_1 + t_2
  \,;
\end{displaymath}

\item whose unit map is given by

\begin{displaymath}
t \mapsto 0
  \,;
\end{displaymath}

\item whose inversion map is given by

\begin{displaymath}
t \mapsto -t
  \,.
\end{displaymath}


\end{itemize}
\hypertarget{group_of_roots_of_unity}{}\subsubsection*{{Group of roots of unity}}\label{group_of_roots_of_unity}

The group of $n$th [[roots of unity]] is

\begin{displaymath}
\mu_n = Spec(k[t](t^n -1))
  \,.
\end{displaymath}
This sits inside the [[multiplicative group]] via the [[Kummer sequence]]

\begin{displaymath}
\mu_n \longrightarrow \mathbb{G}_m \stackrel{(-)^n}{\longrightarrow}\mathbb{G}_m
 \,.
\end{displaymath}
\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

\hypertarget{Grading}{}\subsubsection*{{Grading}}\label{Grading}

\begin{prop}
\label{}\hypertarget{}{}
Let $R$ be a commutative $k$-algebra. There is a [[natural isomorphism]] between

\begin{itemize}%
\item $\mathbb{Z}$-[[graded algebra|gradings]] on $R$;


\item $\mathbb{G}_m$-[[action]]s on $Spec R$.



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

\begin{displaymath}
\rho :  X \times \mathbb{G}_m \to X
\end{displaymath}
is dually the ring homomorphism

\begin{displaymath}
R \otimes_k \mathbb{Z}[t,t^{-1}] \leftarrow R
\end{displaymath}
defined on homogeneous elements $r$ of degree $n$ by

\begin{displaymath}
r \mapsto r \cdot t^n
  \,.
\end{displaymath}
The action property

\begin{displaymath}
\itexarray{
     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
  }
\end{displaymath}
is equivalently the equation

\begin{displaymath}
r (t_1)^n \cdot (t_2)^n = r (t_1 \cdot t_2)^n
\end{displaymath}
for all $n \in \mathbb{Z}$. Similarly the [[unitality]] of the action is the equation

\begin{displaymath}
(1)^n = 1
  \,.
\end{displaymath}
Conversely, given an action of $\mathbb{G}_m$ on $Spec R$ we have some morphism

\begin{displaymath}
R[t,t^{-1}] \leftarrow R
\end{displaymath}
that sends

\begin{displaymath}
r \mapsto \sum_{n \in \mathbb{Z}} r_n t^n
  \,.
\end{displaymath}
By the action property we have that

\begin{displaymath}
\sum_n r_n (t_1 t_2)^n = \sum_{n,k} (r_n)_k t_1^n t_2^k
  \,.
\end{displaymath}
Hence

\begin{displaymath}
(r_n)_k = 
  \left\{
    \itexarray{
       r_n & if \; n = k
       \\
       0 & otherwise
    }
  \right.
\end{displaymath}
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.

\end{proof}
\hypertarget{tale_homotopy_type}{}\subsubsection*{{\'E{}tale homotopy type}}\label{tale_homotopy_type}

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

\end{example}
(\hyperlink{HSS13}{HSS 13, section 1})

\hypertarget{InternalFormulation}{}\subsubsection*{{Internal formulation}}\label{InternalFormulation}

\begin{prop}
\label{}\hypertarget{}{}
Let $X$ be a [[scheme]] and $Sh(Sch/X)$ the [[Zariski site|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:

\begin{itemize}%
\item $\mathbb{A}^1$ is [[internal logic|internally]] a [[local ring]].


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


\item 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,

\begin{displaymath}
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.
\end{displaymath}
Furthermore, these coefficients are uniquely determined.



\end{itemize}
\end{prop}
\begin{proof}
Since the internal logic is local, we can assume that $X = Spec(R)$ is affine. The interpretations of the asserted statements using the [[Zariski site\#KripkeJoyal|Kripke?Joyal semantics]] are:

\begin{itemize}%
\item 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.


\item 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 $S$.


\item 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 $S[s_i^{-1}]$.



\end{itemize}
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.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
Since the big Zariski topos is [[cocomplete category|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]].

\end{remark}
See also at \emph{[[synthetic differential geometry applied to algebraic geometry]]}.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{projective_space}{}\subsubsection*{{Projective space}}\label{projective_space}

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

\begin{displaymath}
\mathbb{A}^n \times \mathbb{G}_m \to \mathbb{A}^n
\end{displaymath}
is dually the morphism

\begin{displaymath}
k[t, t_1, \cdots, t_n] \leftarrow k[t_1, \cdots, t_n]
\end{displaymath}
given by

\begin{displaymath}
t_i \mapsto t \cdot t_i
  \,.
\end{displaymath}
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

\begin{displaymath}
\mathcal{O} (\mathbb{A}^n - \{0\}) \simeq k[t_1, t_1^{-1}, \cdots, t_n, t_n^{-1}]
  \,.
\end{displaymath}
The quotient by the multiplicative group action

\begin{displaymath}
\mathbb{A} P^n_k := (\mathbb{A}^{n+1} - \{0\})/\mathbb{G}_m
\end{displaymath}
is the [[projective space]] over $k$ of [[dimension]] $n$.

\hypertarget{homotopy_theory}{}\subsubsection*{{$\mathbb{A}^1$-homotopy theory}}\label{homotopy_theory}

In [[A1-homotopy theory|A{\tt \symbol{94}}1 homotopy theory]] one considers the [[reflective sub-(∞,1)-category|reflective]] [[localization of an (∞,1)-category|localizatoin]]

\begin{displaymath}
Sh_\infty(C)_{\mathbb{A}^1}
  \stackrel{\leftarrow}{\hookrightarrow}
  Sh_\infty(C)
\end{displaymath}
of the [[(∞,1)-topos]] of [[(∞,1)-sheaves]] over a [[site]] $C$ such as the [[Nisnevich site]], at the morphisms of the form

\begin{displaymath}
p_1 : X \times \mathbb{A}^1 \to X
\end{displaymath}
that contract away cartesian factors of the affine line.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[analytic affine line]]


\item [[spectral affine line]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Discussion of [[étale homotopy type]] is in

\begin{itemize}%
\item Armin Holschbach, Johannes Schmidt, Jakob Stix, \emph{\'E{}tale contractible varieties in positive characteristic} (\href{http://arxiv.org/abs/1310.2784}{arXiv:1310.2784})

\end{itemize}
[[!redirects affine lines]]



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