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  • scheme

functor of points

For schemes:

Examples

• $F(R) = R^n$ is ${\mathbf{A}}^n$ represented by $k[x_1,x_2,\cdots, x_n]$.
• $F(R) = R^{\times}$ is ${\mathbf{G}}_{m}$, represented by $k[x,x^{-1}]$.
  • This is the diagonal group scheme for ${\mathbf{Z}}$.
• $F(R) = (R, +)$ is ${\mathbf{G}}_{a}$, represented by $k[x]$.
• $F(R) = \left\{{x\in R {~\mathrel{\Big\vert}~}x^n=1}\right\}$ is $\mu_{n}$, represented by $k[x]/(x^n-1)$.
  • This is the diagonal group scheme for $C_n$.
• $F(R) = {}_{k} \mathsf{Alg} (k, R)$ is $\operatorname{Spec}k$.