Proposition 61.21.1. Let $\mathcal{C}$ be a site. Assume $\mathcal{C}$ has enough weakly contractible objects. Let $\Lambda $ be a Noetherian ring. Let $I \subset \Lambda $ be an ideal.

The category of derived complete sheaves $\Lambda $-modules is a weak Serre subcategory of $\textit{Mod}(\mathcal{C}, \Lambda )$.

A sheaf $\mathcal{F}$ of $\Lambda $-modules satisfies $\mathcal{F} = \mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F}$ if and only if $\mathcal{F}$ is derived complete and $\bigcap I^ n\mathcal{F} = 0$.

The sheaf $\underline{\Lambda }^\wedge $ is derived complete.

If $\ldots \to \mathcal{F}_3 \to \mathcal{F}_2 \to \mathcal{F}_1$ is an inverse system of derived complete sheaves of $\Lambda $-modules, then $\mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n$ is derived complete.

An object $K \in D(\mathcal{C}, \Lambda )$ is derived complete if and only if each cohomology sheaf $H^ p(K)$ is derived complete.

An object $K \in D_{comp}(\mathcal{C}, \Lambda )$ is bounded above if and only if $K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I}$ is bounded above.

An object $K \in D_{comp}(\mathcal{C}, \Lambda )$ is bounded if $K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I}$ has finite tor dimension.

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