higher geometry / derived geometry
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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 localization
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
Last revised on October 22, 2022 at 16:50:39. See the history of this page for a list of all contributions to it.