Additive and infinitesimal group schemes

The additive group. On the generator $t$ of $k[t]$, the assignments $\mathrm{\Delta }(t)=t\otimes 1+1\otimes t$, $\epsilon (t)=0$, $S(t)=-t$ are algebra maps satisfying the Hopf identities of [F2]: both iterated comultiplications give $t\otimes 1\otimes 1+1\otimes t\otimes 1+1\otimes 1\otimes t$, the two counit composites give $t$, and the two antipode composites give $S(t)\cdot 1+1\cdot t=-t+t=0=\epsilon (t)\cdot 1$. Thus $k[t]$ is a commutative Hopf algebra, so by [F3] ${\mathbf{G}}_a=\mathrm{Spec}k[t]$ is a group scheme, of finite type since $k[t]$ is finitely generated over $k$ by [F4]. For a commutative $k$-algebra $R$, ${\mathbf{G}}_a(R)={\mathrm{Hom}}_k(k[t],R)\cong R$ via $t\mapsto a$, and the group law induced by $\mathrm{\Delta }$ sends the pair $(a,b)$ to the homomorphism with $t\mapsto a\cdot 1+1\cdot b=a+b$, so ${\mathbf{G}}_a(R)=(R,+)$; since $k[t]$ is an integral domain it is reduced.

The multiplicative group. On $k[t,t^{-1}]$ the assignments $\mathrm{\Delta }(t)=t\otimes t$, $\epsilon (t)=1$, $S(t)=t^{-1}$ are algebra maps satisfying the Hopf identities: $\mathrm{\Delta }(t)$ and $t$ are units with the stated inverses, both iterated comultiplications give $t\otimes t\otimes t$, the counit composites give $t\cdot 1=t$, and the antipode composites give $t\cdot t^{-1}=1=\epsilon (t)$. Hence $k[t,t^{-1}]$ is a commutative Hopf algebra and ${\mathbf{G}}_m=\mathrm{Spec}k[t,t^{-1}]$ is the group scheme with ${\mathbf{G}}_m(R)=R^{\times }$ and group law multiplication, agreeing with [F5]; $k[t,t^{-1}]$ is a domain, so ${\mathbf{G}}_m$ is reduced.

The infinitesimal schemes. Let $\mathrm{char}k=p>0$. The quotient algebras $k[t]\mathrm{/}(t^p)$ and $k[t,t^{-1}]\mathrm{/}(t^p-1)$ are finitely generated over $k$, by the images of $t$ and of $t,t^{-1}$ respectively, so [F3] applies once their Hopf structures are established. The quotient map $k[t]\to k[t]\mathrm{/}(t^p)$ is a morphism of Hopf algebras: in $k[t]\mathrm{/}(t^p)\otimes k[t]\mathrm{/}(t^p)$ one has $\mathrm{\Delta }(t{)}^p=(t\otimes 1+1\otimes t{)}^p=t^p\otimes 1+1\otimes t^p=0$ because the intermediate binomial coefficients are divisible by $p$, so $\mathrm{\Delta }$ descends; likewise $\epsilon (t^p)=0$ and $S(t{)}^p=(-t{)}^p=-t^p=0$, so $(t^p)$ is a Hopf ideal and [F6] gives $k[t]\mathrm{/}(t^p)$ a quotient Hopf algebra structure with ${\alpha }_p=\mathrm{Spec}k[t]\mathrm{/}(t^p)$ a group scheme, the closed immersion ${\alpha }_p\hookrightarrow {\mathbf{G}}_a$ of [F7] being a morphism of group schemes. Similarly, in $k[t,t^{-1}]\mathrm{/}(t^p-1)$ one has $\mathrm{\Delta }(t{)}^p=t^p\otimes t^p=1$, $\epsilon (t^p)=1$ and $S(t{)}^p=(t^{-1}{)}^p=(t^p{)}^{-1}=1$, so $(t^p-1)$ is a Hopf ideal and ${\mu }_p=\mathrm{Spec}k[t,t^{-1}]\mathrm{/}(t^p-1)$ is a group scheme with a closed-immersion morphism ${\mu }_p\hookrightarrow {\mathbf{G}}_m$ of group schemes. Evaluating on a commutative $k$-algebra $R$, a homomorphism $k[t]\mathrm{/}(t^p)\to R$ is the same as an element $a$, the image of $t$, with $a^p=0$, and a homomorphism $k[t,t^{-1}]\mathrm{/}(t^p-1)\to R$ is the same as a unit $u$ with $u^p=1$; hence ${\alpha }_p(R)=\{a\in R:a^p=0\}$ and ${\mu }_p(R)=\{u\in R^{\times }:u^p=1\}$.

Length and nonreducedness. The ring $k[t]\mathrm{/}(t^p)$ has $k$-basis $1,t,\dots ,t^{p-1}$, so it is a finite $k$-algebra of dimension $p$ and $t\mathrm{\ne }0$ is nilpotent; the substitution $t=1+s$ identifies $k[t,t^{-1}]\mathrm{/}(t^p-1)\cong k[s]\mathrm{/}(s^p)$ because $(1+s{)}^p-1=s^p$ in characteristic $p$ and $t=1+s$ is a unit of $k[s]\mathrm{/}(s^p)$ with inverse $1-s+s^2-\cdots +(-s{)}^{p-1}$; thus ${\mu }_p$ also has coordinate ring of length $p$ with nonzero nilpotent $s=t-1$.

Closed subgroup schemes. By step 1.3 the addition formula of step 1.1 restricts on the quotient to the addition of the subset ${\alpha }_p(R)\subseteq (R,+)$: it is closed under addition and negation because $(a+b{)}^p=a^p+b^p=0$ and $(-a{)}^p=-a^p=0$ in characteristic $p$, and it contains $0$; so ${\alpha }_p(R)$ is a subgroup of $(R,+)$ and the closed immersion ${\alpha }_p\hookrightarrow {\mathbf{G}}_a$ is a morphism of group schemes, making ${\alpha }_p$ a closed subgroup scheme of ${\mathbf{G}}_a$. Likewise the multiplication formula of step 1.2 restricts to the subset ${\mu }_p(R)\subseteq R^{\times }$, which is closed under multiplication and inversion and contains $1$, so ${\mu }_p$ is a closed subgroup scheme of ${\mathbf{G}}_m$.

Conclusion. Steps 1.1 and 1.2 produce ${\mathbf{G}}_a$ and ${\mathbf{G}}_m$ with their stated coordinate Hopf algebras, points and reducedness; step 1.3 produces the quotient Hopf algebra structures and the closed immersions defining ${\alpha }_p$ and ${\mu }_p$ together with their point descriptions; step 1.4 computes both coordinate rings as $k[s]\mathrm{/}(s^p)$, giving length $p$ and nonreducedness; and step 2.1 identifies the induced group laws on the point sets, so that ${\alpha }_p$ and ${\mu }_p$ are closed subgroup schemes.