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

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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