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Let $X$ be a quasi separated, quasi compact scheme, $Z$ be a closed subscheme on $X$. We denote by $\hat{X}$ the formal scheme of $X$ along $Z$.

My questions are the following.

(1) How to define the stable infinity category $\text{perf}(\hat{X})$ of perfect complexes on $\hat{X}$. (2) Assume $x$ is proper over an affine scheme $S$. Is there the Grothendieck existence theorem for perfect complexes on $\hat{X}$? i.e, any perfect complex on $\hat{X}$ comes from a perfect complex on $X_n$?

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    $\begingroup$ Re (1): perfect complexes are defined on any ringed space [Tag 08CL] or ringed topos [Tag 08G4]. For (2), I only know a reference [Tag 0DIA] in the setting of proper schemes over a complete local ring (and it's not explicitly about perfect complexes, only pseudo-coherent). But I'm far from an expert, so I hope someone else can say more. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Oct 20, 2023 at 14:05

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