Exercise 111.39.1. Let $R$ be a ring. Let $R \to k$ be a map from $R$ to a field. Let $n \geq 0$. Show that

where $k^*$ acts via scalar multiplication on $k^{n + 1}$. From now on we denote $(x_0 : \ldots : x_ n)$ the morphism $\mathop{\mathrm{Spec}}(k) \to \mathbf{P}^ n_ k$ corresponding to the equivalence class of the element $(x_0, \ldots , x_ n) \in k^{n + 1} \setminus \{ 0\} $.

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