Abelian varieties: 1. Group schemes

This posting follows Matt Stevenson's notes for Math 731 (Fall 2017) at the University of Michigan. Various references available in the notes will not be specified here.

Group schemes. Let $S$ be a scheme. A group scheme over $S$ (also called an $S$-group) is a group object in $\textbf{Sch}_{S},$ the category of $S$-schemes. Concretely, this means an $S$-scheme $G$ such that the functor $$h_{G} = \mathrm{Hom}_{S}(-, G) : \textbf{Sch}_{S}^{\mathrm{op}} \rightarrow \textbf{Set}$$ factors through the forgetful functor $\textbf{Grp} \rightarrow \textbf{Set}.$ We say a group scheme $G$ over $S$ is commutative if $G(T) := h_{G}(T)$ is an abelian group for each $S$-scheme $T.$

We now give examples. We will focus on the affine case $S = \mathrm{Spec}(R)$, although it should be possible to use general $S$ by gluing affine pieces.

Example 1. Consider $$\mathbb{G}_{a} = \mathbb{G}_{a, R} := \mathrm{Spec}(R[t]).$$ For any $R$-scheme $T,$ we have $$\mathbb{G}_{a}(T) = \mathrm{Hom}_{\mathrm{Spec}(R)}(T, \mathbb{G}_{a}) \simeq \Gamma(T, \mathscr{O}_{T}),$$ where the isomorphism is taken in $\textbf{Set},$ given by $$\phi \mapsto \phi^{*}(t).$$ This map is functorial in $T,$ varying in $\textbf{Sch}_{R},$ and the right-hand side gives the functor into $\textbf{Grp},$ where the group structure is given by the addition. We see that $\mathbb{G}_{a}$ is a (commutative) group scheme.

Example 2. Consider $$\mathbb{G}_{m} = \mathbb{G}_{m, R} := \mathrm{Spec}(R[t, t^{-1}]).$$ For any $R$-scheme $T,$ we have $$\mathbb{G}_{m}(T) = \mathrm{Hom}_{\mathrm{Spec}(R)}(T, \mathbb{G}_{m}) \simeq \Gamma(T, \mathscr{O}_{T})^{\times},$$ where the isomorphism is taken in $\textbf{Set},$ given by $$\phi \mapsto \phi^{*}(t).$$ This map is functorial in $T,$ varying in $\textbf{Sch}_{R},$ and the right-hand side gives the functor into $\textbf{Grp},$ where the group structure is given by the multiplication. We see that $\mathbb{G}_{m}$ is a (commutative) group scheme.

Example 3. We define $$\mathrm{GL}_{n} = \mathrm{GL}_{n, R} := \mathrm{Spec}(R[t_{ij}]_{1 \leq i,j \leq n}[1/\mathrm{det}]).$$ For any $R$-scheme $T,$ we have $$\mathrm{GL}_{n}(T) = \mathrm{Hom}_{\mathrm{Spec}(R)}(T, \mathrm{Spec}(R[t_{ij}]_{1 \leq i,j \leq n}[1/\mathrm{det}])) \simeq \mathrm{GL}_{n}(\mathscr{O}_{T}(T)),$$ the general linear group of $n \times n$ matrices over the global sections of $T,$ given by $\phi \mapsto [\phi^{*}(t_{ij})]_{1 \leq i,j \leq n}.$ This is functorial in $T,$ and the right hand-side gives the functor into $\textbf{Grp}$ given by the matrix multiplication. This shows that $\mathrm{GL}_{n}$ is a group scheme.

Example 4. We define $$\mu_{n} = \mu_{n, R} := \mathrm{Spec}(R[t]/(t^{n}-1)).$$ For any $R$-scheme $T,$ we have $$\mu_{n}(T) = \mathrm{Hom}_{\mathrm{Spec}(R)}(T, \mathrm{Spec}(R[t]/(t^{n}-1))) \simeq \{f \in \mathscr{O}_{T}(T)^{\times} : f^{n} = 1\},$$ given by $\phi \mapsto \phi^{*}(\bar{t}),$ which is functorial in $T,$ and the right hand-side gives the functor into $\textbf{Grp}$ given by the matrix multiplication. This shows that $\mu_{n}$ is a (commutative) group scheme. The group structures are coming from multiplications of $\mathscr{O}_{T}(T)^{\times}.$

Remark. The behavior of $\mu_{n, R}$ depends on the characteristic of $R.$

First, consider $R = \mathbb{C}.$ Then the equation $t^{n} = 1$ has $n$ distinct solutions in $\mathbb{C},$ so $$\mu_{n} = \mathrm{Spec}\left(\frac{\mathbb{C}[t]}{(t^{n} - 1)}\right) \simeq \bigsqcup_{z \in \mu_{n}(\mathbb{C})}\mathrm{Spec}(\mathbb{C}) \simeq \mu_{n}(\mathbb{C}),$$ which is in particular smooth over $\mathbb{C}.$ It turns out that all group schemes in characteristic $0$ are smooth, by a theorem due to Oort. (We will see this later.)

On the other hand, when $R = \mathbb{F}_{p}$ for a prime $p,$ then $$\mu_{p} = \mathrm{Spec}(\mathbb{F}_{p}[t]/(t^{p} - 1)) = \mathrm{Spec}(\mathbb{F}_{p}[t]/(t - 1)^{p}),$$ so $\mu_{n}$ is a non-reduced scheme. Moreover, we note that $\mu_{p}$ is not smooth over $\mathbb{F}_{p}$ as the derivative of $(t - 1)^{p}$ in $t$ vanishes. In this case, note that $\mu_{p} = \mu_{p, \mathbb{F}_{p}}$ is a point. Note that for any $\mathbb{F}_{p}$-algebra $A$ that is a domain, there is only one $\mathbb{F}_{p}$-algebra map $\mathbb{F}_{p}[t]/(t-1)^{p} \rightarrow A$ because the only $a \in A$ such that $(a - 1)^{p} = 0$ must be $1$ as $A$ is a domain. Hence, we have $\mu_{p, \mathbb{F}_{p}}(A)$ is a singleton. However, when $A$ is not a domain, this may not be the case. For instance, take $A = \mathbb{F}_{p}[\epsilon]/(\epsilon^{2}).$ We have $$\begin{align*} \mu_{p,\mathbb{F}_{p}}\left( \frac{\mathbb{F}_{p}[\epsilon]}{(\epsilon^{2})} \right) &= \mathrm{Hom}_{\mathrm{Spec}(\mathbb{F}_{p})}\left( \mathrm{Spec}\left( \frac{\mathbb{F}_{p}[\epsilon]}{(\epsilon^{2})} \right), \mu_{p,\mathbb{F}_{p}} \right) \\ &\simeq \{1 + b\bar{\epsilon} \in \mathbb{F}_{p}[\epsilon]/(\epsilon^{2}) : b \in \mathbb{F}_{p}\}\end{align*},$$ given by $\phi \mapsto \phi^{*}(\bar{t}).$ The last set has $p$ elements and $p > 1.$

More conventions/definitions. For convenience, we shall call a group scheme over a base scheme $S$ an $S$-group. Again, an $S$-group is by definition a certain functor $\textbf{Sch}_{S}^{\mathrm{op}} \rightarrow \textbf{Grp},$ and we define an $S$-group map to be a natural transformation between two $S$-groups. With this definition, we see that the $S$-groups form a category. Note that given an $S$-group map $f : G \rightarrow H$ and an $S$-group $T,$ the corresponding map $f_{T} : G(T) \rightarrow H(T)$ is a group homomorphism.