Example 10.57.4. Let R be a ring. If S = R[X] with \deg (X) = 1, then the natural map \text{Proj}(S) \to \mathop{\mathrm{Spec}}(R) is a bijection and in fact a homeomorphism. Namely, suppose \mathfrak p \in \text{Proj}(S). Since S_{+} \not\subset \mathfrak p we see that X \not\in \mathfrak p. Thus if aX^ n \in \mathfrak p with a \in R and n > 0, then a \in \mathfrak p. It follows that \mathfrak p = \mathfrak p_0S with \mathfrak p_0 = \mathfrak p \cap R.
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