higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
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 .
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.
The canonical $T$-line object is the affine line
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 $Spec k$.
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
where $k[t,t^{-1}]$ is the localization of the ring $k[t]$ at the element $t = (t-0)$.
whose multiplication operation
is the morphism in $Ring^{op}$ corresponding to the morphism in Ring
given by $t \mapsto t_1 \cdot t_2$;
whose unit map $Spec k \to Spec k[t,t^{-1}]$ is given by
and whose inversion map $Spec k[t,t^{-1}] \to Spec[t,t^{-1}]$ is given by
Therefore for $R$ any ring a morphism
is equivalently a ring homomorphism
which is equivalently a choice of multiplicatively invertible element in $R$. Therefore
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
given by
whose unit map is given by
whose inversion map is given by
The group of $n$th roots of unity is
This sits inside the multiplicative group via the Kummer sequence
Let $R$ be a commutative $k$-algebra. There is a natural isomorphism between
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
is dually the ring homomorphism
defined on homogeneous elements $r$ of degree $n$ by
The action property
is equivalently the equation
for all $n \in \mathbb{Z}$. Similarly the unitality of the action is the equation
Conversely, given an action of $\mathbb{G}_m$ on $Spec R$ we have some morphism
that sends
By the action property we have that
Hence
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.
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.
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,
Furthermore, these coefficients are uniquely determined.
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.
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.
See also at synthetic differential geometry applied to algebraic geometry.
The diagonal action of the multiplicative group on the product $\mathbb{A}^n := \prod_{i = 1 \cdots n} \mathbb{A}^1$ for $n \in \mathbb{N}$
is dually the morphism
given by
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
The quotient by the multiplicative group action
is the projective space over $k$ of dimension $n$.
In A^1 homotopy theory one considers the reflective localizatoin
of the (∞,1)-topos of (∞,1)-sheaves over a site $C$ such as the Nisnevich site, at the morphisms of the form
that contract away cartesian factors of the affine line.
Discussion of étale homotopy type is in