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15.92 The category of derived complete modules

Let $A$ be a ring and let $I$ be an ideal. Denote $\mathcal{C}$ the category of derived complete modules, see Definition 15.91.4. In this section we discuss some properties of this category. In Examples, Section 109.11 we show that $\mathcal{C}$ isn't a Grothendieck abelian category in general.

By Lemma 15.91.6 the category $\mathcal{C}$ is abelian and the inclusion functor $\mathcal{C} \to \text{Mod}_ A$ is exact.

Since $D_{comp}(A) \subset D(A)$ is closed under products (see discussion following Definition 15.91.4) and since products in $D(A)$ are computed on the level of complexes, we see that $\mathcal{C}$ has products which agree with products in $\text{Mod}_ A$. Thus $\mathcal{C}$ in fact has arbitrary limits and the inclusion functor $\mathcal{C} \to \text{Mod}_ A$ commutes with them, see Categories, Lemma 4.14.11.

Assume $I$ is finitely generated. Let ${}^\wedge : D(A) \to D(A)$ denote the derived completion functor of Lemma 15.91.10. Let us show the functor

\[ \text{Mod}_ A \longrightarrow \mathcal{C},\quad M \longmapsto H^0(M^\wedge ) \]

is a left adjoint to the inclusion functor $\mathcal{C} \to \text{Mod}_ A$. Note that $H^ i(M^\wedge ) = 0$ for $i > 0$ for example by Lemma 15.91.20. Hence, if $N$ is a derived complete $A$-module, then we have

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _\mathcal {C}(H^0(M^\wedge ), N) & = \mathop{\mathrm{Hom}}\nolimits _{D_{comp}(A)}(M^\wedge , N)\\ & = \mathop{\mathrm{Hom}}\nolimits _{D(A)}(M, N) \\ & = \mathop{\mathrm{Hom}}\nolimits _ A(M, N) \end{align*}

as desired.

Let $T$ be a preordered set and let $t \mapsto M_ t$ be a system of derived complete $A$-modules, i.e., a system over $T$ in $\mathcal{C}$, see Categories, Section 4.21. Denote $\mathop{\mathrm{colim}}\nolimits _{t \in T} M_ t$ the colimit of the system in $\text{Mod}_ A$. It follows formally from the above that

\[ H^0((\mathop{\mathrm{colim}}\nolimits _{t \in T} M_ t)^\wedge ) \]

is the colimit of the system in $\mathcal{C}$. In this way we see that $\mathcal{C}$ has all colimits. In general the inclusion functor $\mathcal{C} \to \text{Mod}_ A$ will not commute with colimits, see Examples, Section 109.11.

Lemma 15.92.1. Let $A$ be a ring and let $I \subset A$ be an ideal. The category $\mathcal{C}$ of derived complete modules is abelian, has arbitrary limits, and the inclusion functor $F : \mathcal{C} \to \text{Mod}_ A$ is exact and commutes with limits. If $I$ is finitely generated, then $\mathcal{C}$ has arbitrary colimits and $F$ has a left adjoint

Proof. This summarizes the discussion above. $\square$

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