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The Hopf algebra of a split torus and its root-of-unity subgroups

Example

Assume the Axiom of Choice. Let $k$ be a field, let $n \geq 1$ and let $T = G_{m}^{n} = Spec k [t_{1}^{\pm 1}, \dots, t_{n}^{\pm 1}]$ be the split torus of rank $n$ , a product of copies of the multiplicative group scheme of The general linear group scheme and its coordinate ring. Then the coordinate Hopf algebra of $T$ (The coordinate Hopf algebra of an affine group scheme) is given on the Laurent generators by $Δ (t_{i}) = t_{i} \otimes t_{i}, ε (t_{i}) = 1, S (t_{i}) = t_{i}^{- 1} (1 \leq i \leq n),$ extended by multiplicativity; the $t_{i}$ are units, and $T (R) = (R^{\times})^{n}$ with pointwise multiplication for every commutative unital $k$ -algebra $R$ . For integers $r_{1}, \dots, r_{n} \geq 1$ the ideal generated by $t_{1}^{r_{1}} - 1, \dots, t_{n}^{r_{n}} - 1$ is a Hopf ideal (Hopf ideals, kernels and quotients of commutative Hopf algebras) and defines the closed subgroup scheme $μ_{r_{1}} \times \dots \times μ_{r_{n}} \subseteq T$ , where $μ_{r} (R) = {a \in R^{\times} : a^{r} = 1}$ . In characteristic $p > 0$ the scheme $μ_{p}$ has coordinate ring $k [t] / (t - 1)^{p}$ , which is nonreduced, so it is not determined by its $k$ -points.

Verification

Given: AC, a field $k$ , an integer $n \geq 1$ , the torus $T = G_{m}^{n}$ with coordinate ring $A = k [t_{1}^{\pm 1}, \dots, t_{n}^{\pm 1}]$ , integers $r_{1}, \dots, r_{n} \geq 1$ , and the prime $p = char k$ when $p > 0$ .

[F1] For $G_{m} = Spec k [t, t^{- 1}]$ the comorphisms are $Δ (t) = t \otimes t$ , $ε (t) = 1$ , $S (t) = t^{- 1}$ , and $G_{m} (R) = R^{\times}$ . (The general linear group scheme and its coordinate ring)

[F2] Products of affine schemes correspond to tensor products of coordinate rings, $Spec (B \otimes_{k} C) ≅ Spec B \times_{k} Spec C$ , and a product of group schemes is a group scheme with the componentwise group law. (Affine fibre products are spectra of tensor products, Group schemes of finite type over a field)

[F3] A quotient by a Hopf ideal carries a Hopf algebra structure with the quotient map a Hopf morphism, and the induced map of spectra is a closed immersion. For a finitely generated commutative Hopf algebra $B$ , [F5] supplies finite type under the assumed AC; by [F2] and the affine anti-equivalence, the Hopf identities transpose to the group identities, making $Spec B$ a group scheme with structure comorphisms $B \to B \otimes_{k} B$ , $B \to k$ , $B \to B$ . (Hopf ideals, kernels and quotients of commutative Hopf algebras, A surjective ring map induces a closed immersion of affine spectra, Commutative Hopf algebras over a field, Affine schemes are contravariantly equivalent to commutative rings)

[F5] Under AC, affine schemes are quasi-compact, so the spectrum of a finitely generated $k$ -algebra is of finite type by its single affine chart. Both $A$ and $A / I$ are finitely generated, the latter by the images of the Laurent generators. (Every affine scheme is quasi-compact, Subalgebra generated by a subset, algebras of finite type, and module-finite algebras, Locally finite type and finite type morphisms, The Axiom of Choice)

[F4] In characteristic $p > 0$ one has $t^{p} - 1 = (t - 1)^{p}$ in $K [t]$ , and $μ_{p} (K) = {1}$ . (In characteristic $p$ the only $p^{k}$ -th root of unity is $1$ , and $t^{p^{k}} - 1 = (t - 1)^{p^{k}}$ )

1.1F1F2F5

The coordinate ring of $T$ is the $n$ -fold tensor product $⨂_{i = 1}^{n} k [t_{i}, t_{i}^{- 1}] = k [t_{1}^{\pm 1}, \dots, t_{n}^{\pm 1}]$ by [F2]. Its componentwise operations satisfy the group identities, and [F5] supplies finite type; hence $T$ is a group scheme and $A$ is a commutative Hopf algebra with structure maps transposed from the group operations (The coordinate ring of an affine group scheme is a commutative Hopf algebra); the product of the $G_{m}$ structures has comorphisms $Δ (t_{i}) = t_{i} \otimes t_{i}$ , $ε (t_{i}) = 1$ and $S (t_{i}) = t_{i}^{- 1}$ on each Laurent generator, extended multiplicatively. These images are units, so they define $k$ -algebra homomorphisms out of $A$ , and the Hopf axioms hold because they hold componentwise for $k [t_{i}, t_{i}^{- 1}]$ by [F1]. Points satisfy $T (R) = G_{m} (R)^{n} = (R^{\times})^{n}$ with pointwise multiplication.

2.1F3step 1.1algebra

Put $I = (t_{1}^{r_{1}} - 1, \dots, t_{n}^{r_{n}} - 1) \subseteq A$ . For each $i$ one has $Δ (t_{i}^{r_{i}} - 1) = t_{i}^{r_{i}} \otimes t_{i}^{r_{i}} - 1 = (t_{i}^{r_{i}} - 1) \otimes t_{i}^{r_{i}} + 1 \otimes (t_{i}^{r_{i}} - 1) \in I \otimes_{k} A + A \otimes_{k} I$ , also $ε (t_{i}^{r_{i}} - 1) = 1 - 1 = 0$ and $S (t_{i}^{r_{i}} - 1) = t_{i}^{- r_{i}} - 1 = - t_{i}^{- r_{i}} (t_{i}^{r_{i}} - 1) \in I$ ; hence $I$ is a Hopf ideal by [F3], and $A / I$ carries the quotient Hopf algebra structure with $A \to A / I$ a Hopf morphism.

3.1F3F5step 1.1step 2.1algebra

By [F3] the quotient map gives a closed immersion $μ_{r_{1}} \times \dots \times μ_{r_{n}} : = Spec (A / I) ↪ Spec A = T$ , and the quotient Hopf structure gives its group operations. By [F5] it is of finite type, and the inclusion preserves the group operations, hence is a closed subgroup scheme. Its $R$ -points are the algebra maps $A / I \to R$ , equivalently the tuples $(a_{1}, \dots, a_{n})$ with $a_{i} \in R^{\times}$ and $a_{i}^{r_{i}} = 1$ , and the group law is the restriction of the pointwise product of $T (R)$ ; this is exactly $μ_{r_{1}} (R) \times \dots \times μ_{r_{n}} (R)$ with $μ_{r} (R) = {a \in R^{\times} : a^{r} = 1}$ .

4.1F4step 3.1algebra∎

Suppose $char k = p > 0$ and take $n = 1$ , $r_{1} = p$ . By [F4], $t^{p} - 1 = (t - 1)^{p}$ in $k [t]$ , so $k [t, t^{- 1}] / (t^{p} - 1) ≅ k [t] / (t^{p} - 1) ≅ k [s] / (s^{p})$ with $s = t - 1$ ; the class $s$ is a nonzero nilpotent, so $μ_{p}$ is nonreduced. By [F4] again, $μ_{p} (k) = {1}$ , so the $k$ -points are trivial while the coordinate ring $k [s] / (s^{p})$ is not the coordinate ring $k$ of the trivial group scheme; in particular the scheme, and with it the group scheme, is not determined by its $k$ -points. Every computation used the explicit Laurent generators and finitely many relations.

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Sources

J. S. Milne, Algebraic Groups (corrected 2022 printing, Cambridge University Press) (standard reference, not scraped)
J. Swanson (notes), J. Pevtsova (lecturer), Algebraic Groups Lecture Notes, University of Washington, Fall 2014 (standard reference, not scraped)