Lemma 32.2.3. Let $I$ be a directed set. Let $(S_ i, f_{ii'})$ be an inverse system of schemes over $I$. Assume all the morphisms $f_{ii'} : S_ i \to S_{i'}$ are affine, Let $S = \mathop{\mathrm{lim}}\nolimits _ i S_ i$. Let $0 \in I$. Suppose that $T$ is a scheme over $S_0$. Then

\[ T \times _{S_0} S = \mathop{\mathrm{lim}}\nolimits _{i \geq 0} T \times _{S_0} S_ i \]

**Proof.**
The right hand side is a scheme by Lemma 32.2.2. The equality is formal, see Categories, Lemma 4.14.10.
$\square$

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