The Stacks project

Proposition 95.26.5. Let $S$ be a scheme. Let $X = \text{Spf}(A)$ where $A$ is an an adic Noetherian topological $S$-algebra with ideal of definition $I$, see More on Algebra, Definition 15.36.1 and Formal Spaces, Definition 86.9.9. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ the be category fibred in sets associated to the functor $X$, see Categories, Example 4.38.5. Then $\mathit{QC}(\mathcal{X})$ is canonically equivalent to the category $D_{comp}(A, I)$ of objects of $D(A)$ which are derived complete with respect to $I$.

Proof. Recall that $X = \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Spec}}(A/I^ n)$ as an fppf sheaf. An object of $\mathcal{X}_{affine}$ is the same thing as an affine scheme $U = \mathop{\mathrm{Spec}}(R)$ with a given morphism $f : U \to X$. By Formal Spaces, Lemma 86.9.4 there exists an $n \geq 1$ such that $f$ factors through the monomorphism $\mathop{\mathrm{Spec}}(A/I^ n) \to X$. Consider the full subcategory $\mathcal{C} \subset \mathcal{X}_{affine}$ consisting of the objects $\mathop{\mathrm{Spec}}(A/I^ n) \to X$. By the remarks just made and Differential Graded Sheaves, Lemma 24.34.1 restriction to $\mathcal{C}$ is an exact equivalence $\mathit{QC}(\mathcal{X}) \to \mathit{QC}(\mathcal{C}, \mathcal{O}|_\mathcal {C})$. For simplicity, let us assume that $I^ n \not= I^{n + 1}$ for all $n \geq 1$. Then $(\mathcal{C}, \mathcal{O}|_\mathcal {C})$ is isomorphic as a ringed site to the ringed site $(\mathbf{N}, (A/I^ n))$, see Differential Graded Sheaves, Section 24.35. Hence we conclude by Differential Graded Sheaves, Proposition 24.35.4. $\square$


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