61.21 Derived completion and weakly contractible objects
We continue the discussion in Section 61.20. In this section we will see how the existence of weakly contractible objects simplifies the study of derived complete modules.
Let $\mathcal{C}$ be a site. Let $\Lambda $ be a Noetherian ring. Let $I \subset \Lambda $ be an ideal. Although the general theory concerning $D_{comp}(\mathcal{C}, \Lambda )$ is quite satisfactory it is hard to explicitly give examples of derived complete complexes. We know that
every object $M$ of $D(\mathcal{C}, \Lambda /I^ n)$ restricts to a derived complete object of $D(\mathcal{C}, \Lambda )$, and
for every $K \in D(\mathcal{C}, \Lambda )$ the derived completion $K^\wedge = R\mathop{\mathrm{lim}}\nolimits (K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I^ n})$ is derived complete.
The first type of objects are trivially complete and perhaps not interesting. The problem with (2) is that derived completion in general is somewhat mysterious, even in case $K = \underline{\Lambda }$. Namely, by definition of homotopy limits there is a distinguished triangle
\[ R\mathop{\mathrm{lim}}\nolimits (\underline{\Lambda /I^ n}) \to \prod \underline{\Lambda /I^ n} \to \prod \underline{\Lambda /I^ n} \to R\mathop{\mathrm{lim}}\nolimits (\underline{\Lambda /I^ n})[1] \]
in $D(\mathcal{C}, \Lambda )$ where the products are in $D(\mathcal{C}, \Lambda )$. These are computed by taking products of injective resolutions (Injectives, Lemma 19.13.4), so we see that the sheaf $H^ p(\prod \underline{\Lambda /I^ n})$ is the sheafification of the presheaf
\[ U \longmapsto \prod H^ p(U, \Lambda /I^ n). \]
As an explicit example, if $X = \mathop{\mathrm{Spec}}(\mathbf{C}[t, t^{-1}])$, $\mathcal{C} = X_{\acute{e}tale}$, $\Lambda = \mathbf{Z}$, $I = (2)$, and $p = 1$, then we get the sheafification of the presheaf
\[ U \mapsto \prod H^1(U_{\acute{e}tale}, \mathbf{Z}/2^ n\mathbf{Z}) \]
for $U$ étale over $X$. Note that $H^1(X_{\acute{e}tale}, \mathbf{Z}/m\mathbf{Z})$ is cyclic of order $m$ with generator $\alpha _ m$ given by the finite étale $\mathbf{Z}/m\mathbf{Z}$-covering given by the equation $t = s^ m$ (see Étale Cohomology, Section 59.6). Then the section
\[ \alpha = (\alpha _{2^ n}) \in \prod H^1(X_{\acute{e}tale}, \mathbf{Z}/2^ n\mathbf{Z}) \]
of the presheaf above does not restrict to zero on any nonempty étale scheme over $X$, whence the sheaf associated to the presheaf is not zero.
However, on the pro-étale site this phenomenon does not occur. The reason is that we have enough (quasi-compact) weakly contractible objects. In the following proposition we collect some results about derived completion in the Noetherian constant case for sites having enough weakly contractible objects (see Sites, Definition 7.40.2).
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.
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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