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I am currently reading basics of deformation theory from Hartshorne's book on Deformation theory. I understood how the n'th infinitesimal neighbourhood of diagonal is defined for classical schemes. My question is

How one defines these infinitesimal neighbourhoods of derived schemes.

The classical definition sees not correct because the ideal of a closed immersion of derived scheme is contained in $\pi_0\mathcal{O}_X$. It does not contain any derived information. I looked in Lurie's DAG- XII where he defined the completion of a spectral Deligne Mumford stack completely using functor of points. Since the completion is defined in derived set up so there should be similar kind of objects in derived geometry which captures the idea of n'th infinitesimal neighbourhoods. Any suitable referance is also welcone.

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    $\begingroup$ For $f : X \rightarrow Y$ a map of derived stacks, you define the formal neighborhood of f to be the pullback $Y\times_{Y_{\mathrm{dR}}} X_{\mathrm{dR}}$, where $X_{\mathrm{dR}}$ is the de Rham stack of $X$. For nice stacks, you can fix $X_\mathrm{dR}(A) := X(A^\mathrm{red})$. When $X$ is locally almost of finite presentation over a field, this has been studied by Gaistgory and Rozenblyum in their book "A study in derived algebraic geometry". $\endgroup$ Commented Mar 25 at 16:37

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When $f: X \to Y$ is a map of prestacks locally almost of finite type admitting deformation theory, Gaitsgory and Rozenblyum gave a definition of the (derived) $n$th infinitesimal neighborhood of $X$ in $Y$. See:

A study in derived algebraic geometry, Volume II, Chapter 9, §5.

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    $\begingroup$ For a construction outside char 0, see 3.15 in arxiv.org/abs/2511.19412 $\endgroup$ Commented Mar 27 at 5:20

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