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.
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