# Regular subscheme of a projective limit of schemes

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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Here's a comment when $S$ and $S_i$ are local with rings $(A,m)$ and $(A_i,m_i)$ and all transition maps are local. In particular, $m/m^2 = \lim m_i/m_i^2$ (colimit). Then $Z$ is cut out by a regular sequence $\underline{f} = (f_1,...,f_r)$ that spans an $r$-dimensional subspace of $m/m^2$. Then $\underline{f}$ comes from a fixed $m_j$ and spans an $r$-dimensional subspace in $m_k/m_k^2$ for all $k \geq j$. The corresponding subschemes $Z_k = Z(\underline{f}) \subset S_k$ are then regular, and induce $Z$. Maybe the non-local case follows using the openness of regular locus (under qcqs hyp.) ? – anon Mar 11 '13 at 15:11
Thank you!! It seems that I know how to reduce the general case to the local one. – Mikhail Bondarko Mar 16 '13 at 2:57
I am glad it helped! – anon Mar 17 '13 at 18:06