Lemma 69.4.4. Let $S$ be a scheme. Let $I$ be a directed set. Let $(X_ i, f_{i'i})$ be an inverse systems over $I$ of algebraic spaces over $S$. If $X_ i$ is reduced for all $i$, then $X$ is reduced.

Proof. Observe that $\mathop{\mathrm{lim}}\nolimits X_ i$ exists by Lemma 69.4.1. Pick $0 \in I$ and choose an affine scheme $V_0$ and an étale morphism $U_0 \to X_0$. Then the affine schemes $U_ i = X_ i \times _{X_0} U_0$ are reduced. Hence $U = X \times _{X_0} U_0$ is a reduced affine scheme as a limit of reduced affine schemes: a filtered colimit of reduced rings is reduced. Since the étale morphisms $U \to X$ form an étale covering of $X$ as we vary our choice of $U_0 \to X_0$ we see that the lemma is true. $\square$

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