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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 \(\operatorname{Spec}k\).