The Stacks project

Exercise 111.2.4. Let $(I, \geq )$ be a directed set and let $(A_ i, \varphi _{ij})$ be a system of rings over $I$ with colimit $A$. Prove that there is a bijection

\[ \mathop{\mathrm{Spec}}(A) = \{ (\mathfrak p_ i)_{i \in I} \mid \mathfrak p_ i \subset A_ i \text{ and } \mathfrak p_ i = \varphi _{ij}^{-1}(\mathfrak p_ j)\ \forall i \leq j\} \subset \prod \nolimits _{i \in I} \mathop{\mathrm{Spec}}(A_ i) \]

The set on the right hand side of the equality is the limit of the sets $\mathop{\mathrm{Spec}}(A_ i)$. Notation $\mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Spec}}(A_ i)$.

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