The Stacks project

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.

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

  2. 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 $\bigcap I^ n\mathcal{F} = 0$.

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

  4. 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.

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

  6. 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.

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

Proof. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be a subset such that every $U \in \mathcal{B}$ is weakly contractible and every object of $\mathcal{C}$ has a covering by elements of $\mathcal{B}$. We will use the results of Cohomology on Sites, Lemma 21.51.1 and Proposition 21.51.2 without further mention.

Recall that $R\mathop{\mathrm{lim}}\nolimits $ commutes with $R\Gamma (U, -)$, see Injectives, Lemma 19.13.6. Let $f \in I$. Recall that $T(K, f)$ is the homotopy limit of the system

\[ \ldots \xrightarrow {f} K \xrightarrow {f} K \xrightarrow {f} K \]

in $D(\mathcal{C}, \Lambda )$. Thus

\[ R\Gamma (U, T(K, f)) = T(R\Gamma (U, K), f). \]

Since we can test isomorphisms of maps between objects of $D(\mathcal{C}, \Lambda )$ by evaluating at $U \in \mathcal{B}$ we conclude an object $K$ of $D(\mathcal{C}, \Lambda )$ is derived complete if and only if for every $U \in \mathcal{B}$ the object $R\Gamma (U, K)$ is derived complete as an object of $D(\Lambda )$.

The remark above implies that items (1), (5) follow from the corresponding results for modules over rings, see More on Algebra, Lemmas 15.91.1 and 15.91.6. In the same way (2) can be deduced from More on Algebra, Proposition 15.91.5 as $(I^ n\mathcal{F})(U) = I^ n \cdot \mathcal{F}(U)$ for $U \in \mathcal{B}$ (by exactness of evaluating at $U$).

Proof of (4). The homotopy limit $R\mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n$ is in $D_{comp}(X, \Lambda )$ (see discussion following Algebraic and Formal Geometry, Definition 52.6.4). By part (5) just proved we conclude that $\mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n = H^0(R\mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n)$ is derived complete. Part (3) is a special case of (4).

Proof of (6) and (7). Follows from Lemma 61.20.1 and Cohomology on Sites, Lemma 21.46.9 and the computation of homotopy limits in Cohomology on Sites, Proposition 21.51.2. $\square$

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