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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asked Mar 11, 2013 at 8:42
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2$\begingroup$ 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.) ? $\endgroup$– anonCommented Mar 11, 2013 at 15:11
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$\begingroup$ Thank you!! It seems that I know how to reduce the general case to the local one. $\endgroup$– Mikhail BondarkoCommented Mar 16, 2013 at 2:57
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$\begingroup$ I am glad it helped! $\endgroup$– anonCommented Mar 17, 2013 at 18:06
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