Let $S\cong \varprojlim S_i$, where $S$ and all $S_i$ are separated regular excellent of finite Krull dimension. Let $Z$ be a closed regular subscheme of $S$. As Theorem 8.8.2 of EGA4 shows, $Z$ comes from a closed subscheme $Z_i$ of one of $S_i$. My question is: can we also assume that $Z_i$ is regular? Are there any extra restrictions needed so that the fact will be true? I am actually only interested in equicharacteristic schemes, and the connecting morphisms are affine and dominant.
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