For every Lawvere theory containing the theory of abelian groups Isbell dual sheaf topos over formal duals of -algebras contains a canonical line object .
For the theory of commutative rings this is called the affine line .
Let be a ring, and the Lawvere theory of associative algebras over , such that the category of algebras over a Lawvere theory is the category of -algebras.
The canonical -line object is the affine line
Here the free -algebra on a single generator is the polynomial algebra on a single generator and may be regarded as the corresponding object in the opposite category of affine schemes over .
The multiplicative group object in corresponding to the affine line – usually just called the multiplicative group – is the group scheme denoted
whose underlying affine scheme is
where is the localization of the ring at the element .
whose multiplication operation
is the morphism in corresponding to the morphism in Ring
given by ;
whose unit map is given by
and whose inversion map is given by
Therefore for any ring a morphism
is equivalently a ring homomorphism
which is equivalently a choice of multiplicatively invertible element in . Therefore
is the group of units of .
The additive group in corresponding to the affine line – usually just called the additive group – is the group scheme denoted
whose underlying object is itself;
whose addition operation is dually the ring homomorphism
whose unit map is given by
whose inversion map is given by
Group of roots of unity
The group of th roots of unity is
This sits inside the multiplicative group via the Kummer sequence
For the first direction, let be a -graded commutative algebra. Then comes with a -action given as follows: the action morphism
is dually the ring homomorphism
defined on homogeneous elements of degree by
The action property
is equivalently the equation
for all . Similarly the unitality of the action is the equation
Conversely, given an action of on we have some morphism
By the action property we have that
and so the morphism gives a decomposition of into pieces labeled by .
One sees that these two constructions are inverse to each other.
Étale homotopy type
(HSS 13, section 1)
Let be a scheme and the big Zariski topos associated to . Denote by (the affine line) the ring object , i.e. the functor represented by the -scheme . Then:
is internally a local ring.
is internally a field in the sense that any nonzero element is invertible.
Internally, any function is a polynomial function, i.e. of the form for some coefficients . More precisely,
Furthermore, these coefficients are uniquely determined.
Since the internal logic is local, we can assume that is affine. The interpretations of the asserted statements using the Kripke–Joyal semantics are:
Let be an -algebra and be elements such that . Then there exists a partition such that in the localized rings , or is invertible.
Let be an -algebra and an element. Assume that any -algebra in which is zero is trivial (fulfills ). Then is invertible in .
Let be an -algebra and be an element. Then there exists a partition such that in the localized rings , is a polynomial with coefficients in .
For the first statement, simply choose , .
For the second statement, consider the -algebra .
The third statement is immediate, localization is not even necessary.
See also at synthetic differential geometry applied to algebraic geometry.
The diagonal action of the multiplicative group on the product for
is dually the morphism
This makes the free graded algebra over on generators in degree 1. This is -graded. What is genuinely -graded is
The quotient by the multiplicative group action
is the projective space over of dimension .
In A^1 homotopy theory one considers the reflective localizatoin
of the (∞,1)-topos of (∞,1)-sheaves over a site 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
- Armin Holschbach, Johannes Schmidt, Jakob Stix, Étale contractible varieties in positive characteristic (arXiv:1310.2784)