Roots of Unity Group Scheme \(\mu_n\)

Let \(n\) be a non-negative integer and \(S\) a scheme. The \(n\)-th roots of unity group scheme over \(S\), denoted \(\mu_{n,S}\), is defined as the kernel of the \(n\)-th power map on \(\mathbb{G}_m\):

\[ \mu_{n,S} \;=\; \ker\bigl(\mathbb{G}_{m,S} \to \mathbb{G}_{m,S}\bigr) \quad (\text{the } n\text{-th power map}) \;=\; S \,\times_{\operatorname{Spec} \mathbb{Z}}\, \operatorname{Spec}\bigl(\mathbb{Z}[T]/(T^n - 1)\bigr). \]

Its coordinate Hopf algebra is the quotient \(\mathbb{Z}[T]/(T^n - 1)\), which is canonically isomorphic to the additive group algebra of the cyclic group of order \(n\):

\[ \mathbb{Z}[T]/(T^n - 1) \;\cong\; \mathbb{Z}[\mathbb{Z}/n] \qquad (T^k \mapsto \mathrm{single}\, k\, 1). \]

The Hopf structure, transported via this isomorphism, gives the group-like Hopf structure on basis elements:

\[ \Delta(\mathrm{single}\, k\, 1) = \mathrm{single}\, k\, 1 \otimes \mathrm{single}\, k\, 1, \qquad \varepsilon(\mathrm{single}\, k\, 1) = 1, \qquad S(\mathrm{single}\, k\, 1) = \mathrm{single}\, (-k)\, 1. \]

For \(n > 0\), \(\mu_n\) is finite (of order \(n\)) and is an algebraic group over \(S\). For \(n = 0\), the convention \(\mathbb{Z}/0 = \mathbb{Z}\) yields \(\mu_0 = \mathbb{G}_m\) (the no-constraint case).

In characteristic coprime to \(n\), \(\mu_n\) is étale (a finite étale group scheme of order \(n\)). In characteristic \(p \mid n\), \(\mu_n\) acquires infinitesimal pieces (e.g., \(\mu_p\) over \(\mathbb{F}_p\) is the non-étale "Frobenius kernel"), but it is always an algebraic group in the modern Milne / SGA 3 convention.