Lemma 15.21.4. Let $R$ be a ring. Let $S = R[T_1, \ldots , T_ n]/J$. Assume $J$ contains elements of the form $P_ i(T_ i)$ with $P_ i(T) = \prod _{j = 1, \ldots , d_ i} (T - \alpha _{ij})$ for some $\alpha _{ij} \in R$. For $\underline{k} = (k_1, \ldots , k_ n)$ with $1 \leq k_ i \leq d_ i$ consider the ring map

$\Phi _{\underline{k}} : R[T_1, \ldots , T_ n] \to R, \quad T_ i \longmapsto \alpha _{ik_ i}$

Set $J_{\underline{k}} = \Phi _{\underline{k}}(J)$. Then the image of $\mathop{\mathrm{Spec}}(S) \to \mathop{\mathrm{Spec}}(R)$ is equal to $V(\bigcap J_{\underline{k}})$.

Proof. This lemma proves itself. Hint: $V(\bigcap J_{\underline{k}}) = \bigcup V(J_{\underline{k}})$. $\square$

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