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Given a ring $A$ and an ideal $I$, the direct system of schemes $\text{Spec}(A/I)\rightarrow \text{Spec}(A/I^2)\rightarrow \ldots$ has a colimit if the category of qcqs-schemes and it is given by $\text{Spec}(\hat{A})$. Here $\hat{A}$ is just a completion of $A$ with respect to $I$. (A proof can be found here) Does this generalize to the non-affine case?

Given a closed sub-scheme $Z$ of $X$, there is the direct system of schemes $Z\rightarrow X_Z^{(1)}\rightarrow X_Z^{(2)}\rightarrow \ldots$. Here $X_Z^{(n)}$ is the $n$-th nilpotent thickening of $Z$ along $X$. Does this system has a colimit in the category of schemes? If so is it given by $\text{Spec}(\hat{A})$ on affine charts?

I am specifically interested in the case $X$ is a projective variety and $Z$ a hyper-surface section.

Edit: In the projective case, is $Proj(\hat{A})$ the colimit? Here $\hat{A}$ means that we just take the inverse limit of homogenous coordinate system and then take proj, very similar to the affine case.

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    $\begingroup$ Note that the affine case actually gives a colimit in schemes (not just qcqs schemes) by Bhatt, Remark 4.6. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Feb 7, 2021 at 19:10
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    $\begingroup$ The answer to the second question is negative because completion does not commute with localization: $\hat{A}[1/f] \neq \widehat{A[1/f]}$. This suggests that the answer to the first question should also be negative. $\endgroup$
    – Piotr Achinger
    Commented Feb 7, 2021 at 20:20
  • $\begingroup$ @Piotr Achinger This might be silly but I'm having hard time understanding necessity of the condition you mentioned, although it does seem to be sufficient. $\endgroup$
    – user127776
    Commented Feb 7, 2021 at 20:56

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