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affine line

Contents

Idea

For every Lawvere theory TT containing the theory of abelian groups Isbell dual sheaf topos over formal duals of TT-algebras contains a canonical line object 𝔸 1\mathbb{A}^1.

For TT the theory of commutative rings this is called the affine line .

Definition

Affine line

Let kk be a ring, and TT the Lawvere theory of associative algebras over kk, such that the category of algebras over a Lawvere theory TAlg=Alg kT Alg = Alg_k is the category of kk-algebras.

Definition

The canonical TT-line object is the affine line

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

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

Multiplicative group

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

  • whose underlying affine scheme is

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

    where k[t,t 1]k[t,t^{-1}] is the localization of the ring k[t]k[t] at the element t=(t0)t = (t-0).

  • whose multiplication operation

    𝔾 m×𝔾 m𝔾 m \cdot \mathbb{G}_m \times \mathbb{G}_m \to \mathbb{G}_m

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

    k[t 1,t 1 1] kk[t 2,t 2 1]k[t,t 1] k[t_1,t_1^{-1}] \otimes_k k[t_2, t_2^{-1}] \leftarrow k[t,t^{-1}]

    given by tt 1t 2t \mapsto t_1 \cdot t_2;

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

    t1 t \mapsto 1
  • and whose inversion map Speck[t,t 1]Spec[t,t 1]Spec k[t,t^{-1}] \to Spec[t,t^{-1}] is given by

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

Therefore for RR any ring a morphism

SpecR𝔾 m Spec R \longrightarrow \mathbb{G}_m

is equivalently a ring homomorphism

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

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

Hom(SpecR,𝔾 m)R ×=GL 1(R) Hom(Spec R , \mathbb{G}_m) \simeq R^\times = GL_1(R)

is the group of units of RR.

Additive group

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

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

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

    k[t 1] kk[t 2]k[t] k[t_1] \otimes_k k[t_2] \leftarrow k[t]

    given by

    tt 1+t 2; t \mapsto t_1 + t_2 \,;
  • whose unit map is given by

    t0; t \mapsto 0 \,;
  • whose inversion map is given by

    tt. t \mapsto -t \,.

Group of roots of unity

The group of nnth roots of unity is

μ n=Spec(k[t](t n1)). \mu_n = Spec(k[t](t^n -1)) \,.

This sits inside the multiplicative group via the Kummer sequence

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

Properties

Grading

Proposition

Let RR be a commutative kk-algebra. There is a natural isomorphism between

  • \mathbb{Z}-gradings on RR;

  • 𝔾 m\mathbb{G}_m-actions on SpecRSpec R.

Proof

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

ρ:X×𝔾 mX \rho : X \times \mathbb{G}_m \to X

is dually the ring homomorphism

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

defined on homogeneous elements rr of degree nn by

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

The action property

X×𝔾 m×𝔾 m Id× X×𝔾 ρ×Id ρ X×𝔾 m ρ X \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(t 2) n=r(t 1t 2) n r (t_1)^n \cdot (t_2)^n = r (t_1 \cdot t_2)^n

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

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

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

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

that sends

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

By the action property we have that

nr n(t 1t 2) n= n,k(r n) kt 1 nt 2 k. \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={r n ifn=k 0 otherwise (r_n)_k = \left\{ \array{ r_n & if \; n = k \\ 0 & otherwise } \right.

and so the morphism gives a decomposition of RR into pieces labeled by \mathbb{Z}.

One sees that these two constructions are inverse to each other.

Étale homotopy type

Example

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

(HSS 13, section 1)

Internal formulation

Proposition

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

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

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

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

    Sh(Sch/X)f:[𝔸 1,𝔸 1]. na 0,,a n:𝔸 1.x:𝔸 1.f(x)= ia ix i. 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)X = Spec(R) is affine. The interpretations of the asserted statements using the Kripke–Joyal semantics are:

  • Let SS be an RR-algebra and f,gSf, g \in S be elements such that f+g=1f + g = 1. Then there exists a partition 1= is iS1 = \sum_i s_i \in S such that in the localized rings S[s i 1]S[s_i^{-1}], ff or gg is invertible.

  • Let SS be an RR-algebra and fSf \in S an element. Assume that any SS-algebra TT in which ff is zero is trivial (fulfills 1=0T1 = 0 \in T). Then ff is invertible in SS.

  • Let SS be an RR-algebra and f[𝔸 1,𝔸 1](S)=S[T]f \in [\mathbb{A}^1,\mathbb{A}^1](S) = S[T] be an element. Then there exists a partition 1= is iS1 = \sum_i s_i \in S such that in the localized rings S[s i 1]S[s_i^{-1}], ff is a polynomial with coefficients in S[s i 1]S[s_i^{-1}].

For the first statement, simply choose s 1fs_1 \coloneqq f, s 2gs_2 \coloneqq g.

For the second statement, consider the SS-algebra TS/(f)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 𝔸 1[X]\mathbb{A}^1[X] of internal polynomials: The canonical ring homomorphism 𝔸 1[X][𝔸 1,𝔸 1]\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.

Examples

Projective space

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

𝔸 n×𝔾 m𝔸 n \mathbb{A}^n \times \mathbb{G}_m \to \mathbb{A}^n

is dually the morphism

k[t,t 1,,t n]k[t 1,,t n] k[t, t_1, \cdots, t_n] \leftarrow k[t_1, \cdots, t_n]

given by

t itt i. t_i \mapsto t \cdot t_i \,.

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

𝒪(𝔸 n{0})k[t 1,t 1 1,,t n,t n 1]. \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

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

is the projective space over kk of dimension nn.

𝔸 1\mathbb{A}^1-homotopy theory

In A^1 homotopy theory one considers the reflective localizatoin

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

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

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

that contract away cartesian factors of the affine line.

References

Discussion of étale homotopy type is in

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

Revised on November 20, 2014 17:29:07 by Ingo Blechschmidt (137.250.162.16)