Lie Algebras and Infinitesimal Group Schemes — Examples

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.

Cotangent spaces and dual generators. At the identity of ${\mathbf{G}}_a$ and ${\mathbf{G}}_m$, respectively, the local rings are $k[t{]}_{(t)}$ and $k[t,t^{-1}{]}_{(t-1)}$, with maximal ideals generated by $t$ and $t-1$. Their quotients by the squares of these ideals are $k[u]\mathrm{/}(u^2)$, with $u=t$ or $u=t-1$: any denominator outside the maximal ideal has a nonzero constant term and is invertible in this square-zero quotient. Thus $[t]$ and $[t-1]$ are cotangent bases. In characteristic $p\ge 2$, both infinitesimal coordinate rings are $k[s]\mathrm{/}(s^p)$ by [F1]; every element outside $(s)$ is a unit by a finite geometric sum, so this ring is already local. Its cotangent quotient $(s)\mathrm{/}(s{)}^2$ has basis $[s]$, regardless of whether higher powers survive in the ring. By [F2] the Lie algebras are the duals of these one-dimensional cotangent spaces, generated by the functionals $u^{\ast }$ with $u^{\ast }([u])=1$.

The general linear group. By [F4] the identification $\mathrm{Lie}({\mathrm{GL}}_n)={\mathfrak{g}\mathfrak{l}}_n=M_n(k)$ sends $X$ to $I+\epsilon X$ and the bracket is $[X,Y]=XY-YX$.

Zero brackets and the isomorphisms. By the bracket construction in parts (a)-(d) of The Lie bracket from infinitesimals and the adjoint action, with Choice inherited through its matrix-group and adjoint-representation suppliers, these tangent spaces carry Lie brackets. They are one-dimensional by step 1.1, so [F3] makes each bracket zero. Sending the dual generator $s^{\ast }$ of $\mathrm{Lie}({\alpha }_p)$ to $t^{\ast }$ of $\mathrm{Lie}({\mathbf{G}}_a)$ is therefore a Lie-algebra isomorphism; likewise sending the dual generator for $s=t-1$ to $(t-1{)}^{\ast }$ gives $\mathrm{Lie}({\mu }_p)\cong \mathrm{Lie}({\mathbf{G}}_m)$. Together with step 1.2 this proves (a), (b) and (c).

The subgroup scheme ${\alpha }_p$ is not smooth at its origin. Its coordinate ring is $k[s]\mathrm{/}(s^p)$ with $s\mathrm{\ne }0$ and $s^p=0$ by [F1], so it is not reduced. Every element outside $(s)$ is a unit by a finite geometric sum, so localization at $(s)$ leaves this ring unchanged and the class $s$ remains nonzero. If ${\alpha }_p$ were smooth at the origin, [F3] with the trivial extension $k\mathrm{/}k$ would make the local ring $k[s]\mathrm{/}(s^p{)}_{(s)}$ regular, and [F4] would make that ring an integral domain, contradicting $s\mathrm{\ne }0$ with $s^p=0$; hence ${\alpha }_p$ is not smooth over $k$. The same argument with the same coordinate ring, using $s=t-1$, shows that ${\mu }_p$ is not smooth over $k$.

The ambient groups are smooth. The polynomial ring $k[t]$ is a localisation of a polynomial ring and its Jacobian presentation has no equations, so it is standard smooth over $k$; by [F5] the morphism ${\mathbf{G}}_a\to \mathrm{Spec}k$ is flat with geometrically regular fibres and locally of finite presentation over the Noetherian field $k$, hence smooth by definition. The same presentation with no equations applies to $k[t,t^{-1}]=(k[t]{)}_t$, so ${\mathbf{G}}_m$ is smooth over $k$.

The Lie algebras agree. By [F2] there are isomorphisms of $k$-Lie algebras $\mathrm{Lie}({\alpha }_p)\cong \mathrm{Lie}({\mathbf{G}}_a)$ and $\mathrm{Lie}({\mu }_p)\cong \mathrm{Lie}({\mathbf{G}}_m)$. Combining this with steps 1.1 and 1.2, the nonsmooth finite group scheme ${\alpha }_p$ and the smooth group scheme ${\mathbf{G}}_a$ have isomorphic Lie algebras, and likewise for ${\mu }_p$ and ${\mathbf{G}}_m$; therefore the Lie algebra of a group scheme of finite type over a field does not determine smoothness, and the refuted statement fails. Choice is inherited through the matrix-group and Lie-bracket suppliers [F1, F2], the regular-local-domain supplier [F4], and the standard-smoothness criterion [F5].