The Stacks project

Lemma 15.91.6. Let $I$ be an ideal of a ring $A$.

  1. The derived complete $A$-modules form a weak Serre subcategory $\mathcal{C}$ of $\text{Mod}_ A$.

  2. $D_\mathcal {C}(A) \subset D(A)$ is the full subcategory of derived complete objects.

Proof. Part (2) is immediate from Lemma 15.91.1 and the definitions. For part (1), suppose that $M \to N$ is a map of derived complete modules. Denote $K = (M \to N)$ the corresponding object of $D(A)$. Pick $f \in I$. Then $\mathop{\mathrm{Ext}}\nolimits _ A^ n(A_ f, K)$ is zero for all $n$ because $\mathop{\mathrm{Ext}}\nolimits _ A^ n(A_ f, M)$ and $\mathop{\mathrm{Ext}}\nolimits _ A^ n(A_ f, N)$ are zero for all $n$. Hence $K$ is derived complete. By (2) we see that $\mathop{\mathrm{Ker}}(M \to N)$ and $\mathop{\mathrm{Coker}}(M \to N)$ are objects of $\mathcal{C}$. Finally, suppose that $0 \to M_1 \to M_2 \to M_3 \to 0$ is a short exact sequence of $A$-modules and $M_1$, $M_3$ are derived complete. Then it follows from the long exact sequence of $\mathop{\mathrm{Ext}}\nolimits $'s that $M_2$ is derived complete. Thus $\mathcal{C}$ is a weak Serre subcategory by Homology, Lemma 12.10.3. $\square$

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