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Let $X$ be an affine integral normal scheme, $Z\subset X$ a constructible closed subscheme of codimension $\geq 2$. We can write $X$ as a limit of schemes $(X_{\lambda})_{\lambda\in \Lambda}$ of finite type over $\mathbb{Z}$, and by [EGAIV3, 8.3.11], there is $\lambda_0\in \Lambda$ and a constructible closed $Z_{\lambda_0}\subset X_{\lambda_0}$ whose preimage is $Z$.

My question is: is it possible to descend the codimension of $Z$? Precisely, is there a $\lambda_1\in \Lambda$ such that $Z_{\lambda_1}\subset X_{\lambda_1}$ is constructible closed of codimension $\geq 2$ whose preimage is $Z$? In fact, I only care about descending the codimension at a fixed point $x\in Z$, so localizations are permitted.

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    $\begingroup$ Just making sure: $\text{codim}(Z, X) = \inf \text{codim}(Z', X)$ where $Z'$ runs over the irreducible closed subschemes of $Z$. $\endgroup$
    – Johan
    Commented Dec 17, 2021 at 23:11
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    $\begingroup$ No this isn't true: take a local domain with exactly three prime ideals $(0) \subset \mathfrak p \subset \mathfrak m$. (By Hochster such a ring exists.) Take $t \in \mathfrak m$, $t \not \in \mathfrak p$. Then write $A = \colim A_i$ as a filtered colimit with $A_i \subset A$ of finite type over $\mathbf{Z}$ with $t \in A_i$ for all $i$. Then $V(t)$ has codim 1 in the spectrum of $A_i$ but $V(t)$ has codimen $2$ in the spectrum of $A$. $\endgroup$
    – Johan
    Commented Dec 17, 2021 at 23:22
  • $\begingroup$ @Johan, thank you very much, Johan. I now feel that descending codim is not easy because of your example. For the ring $A$ you mentioned, I consider it as a valuation ring of rank two (it is a normal local domain). But I still don't know how you make sure that all $\mathbb{Z}$-finite type subrings of $A$ satisfy that $V(t)$ has codim=1. I appreciate it if you could explain a little or give a reference. $\endgroup$
    – Takagi Benseki
    Commented Dec 18, 2021 at 8:43
  • $\begingroup$ @TakagiBenseki You did not write that your directed system of rings is the system of all finite type $\mathbb{Z}$-subalgebras, so Johan is free to choose a different directed system. At any rate, the directed system of finite type $\mathbb{Z}$-subalgebras that contain $t$ is cofinal in the directed system of all finite type $\mathbb{Z}$-subalgebras. $\endgroup$
    – Jason Starr
    Commented Dec 18, 2021 at 14:00
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    $\begingroup$ Krull hauptidealsatz is always a fun result to use. $\endgroup$
    – Johan
    Commented Dec 18, 2021 at 18:02

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