Say we have a derived scheme over an algebraically closed field $X/k$, viewed as a functor $X : \operatorname{Aff}_k^{\operatorname{op}} \to \infty\operatorname{-Grpd}$ and we know its formal neighborhood around every geometric point $X^{\land}_x$ (e.g. $\operatorname{char}k=0$ and we know how to compute the corresponding tangent complex) as well as its restriction to discrete rings $X_0 := X\mid \operatorname{Aff}_k^{\text{cl}}$. Will $X$ be determined uniquely by this data, and if so - is there an explicit way of computing its values on a general derived affine scheme? In the situation I have in mind $X_0$ is a classical scheme (i.e. $X(\pi_0(R)) \in \operatorname{Sets}$ and does not have higher homotopy groups).
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3$\begingroup$ That data suffices to detect equivalences, in the sense that a morphism of derived schemes preserving those must be an equivalence. However, it clearly doesn't determine the derived scheme. Just take a scheme $Y$ and non-isomorphic vector bundles $E,F$ on $Y$ of the same rank. Then the derived schemes $E[-1]:= \mathrm{RSpec Symm}_{Y}(E^*[1]) $ and $F[-1]$ are not equivalent, but have underived truncation $Y$ and equivalent derived formal completions at all geometric points. $\endgroup$– Jon PridhamCommented Feb 26 at 17:03
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$\begingroup$ Oh I see, thank you! $\endgroup$– E. KOWCommented Feb 26 at 17:41
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