Lemma 15.86.9. Let $(K_ n)$ be an inverse system of objects of $D(\textit{Ab})$. Then there exists an object $M = (M_ n^\bullet )$ of $D(\textit{Ab}(\mathbf{N}))$ and isomorphisms $M_ n^\bullet \to K_ n$ in $D(\textit{Ab})$ such that the diagrams
commute in $D(\textit{Ab})$.
Proof. Namely, let $M_1^\bullet $ be a complex of abelian groups representing $K_1$. Suppose we have constructed $M_ e^\bullet \to M_{e - 1}^\bullet \to \ldots \to M_1^\bullet $ and maps $\psi _ i : M_ i^\bullet \to K_ i$ such that the diagrams in the statement of the lemma commute for all $n < e$. Then we consider the diagram
in $D(\textit{Ab})$. By the definition of morphisms in $D(\textit{Ab})$ we can find a complex $M_{n + 1}^\bullet $ of abelian groups, an isomorphism $M_{n + 1}^\bullet \to K_{n + 1}$ in $D(\textit{Ab})$, and a morphism of complexes $M_{n + 1}^\bullet \to M_ n^\bullet $ representing the composition
in $D(\textit{Ab})$. Thus the lemma holds by induction. $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)