Remark 13.34.4 (0H9J)—The Stacks project

Remark 13.34.4. Let $\mathcal{D}$ be a triangulated category. Let $(K_ n)$ and $(L_ n)$ be inverse systems of objects of $\mathcal{D}$ with derived limits $K$ and $L$. Suppose we have a morphism $a : (K_ n) \to (L_ n)$ of pro-objects, see Categories, Example 4.22.6. This means we are given maps $a_ n : K_{m(n)} \to L_ n$. We may assume $m(1) < m(2) < m(3) < \ldots $ and that the maps $a_ n$ are compatible with the transition maps in the systems. Then we can consider the maps

\[ a', a'' : \prod K_ n \longrightarrow \prod L_ n \]

defined by the rules $a'((k_ n)) = (a_ n(k_{m(n)}))$ and

\[ a''((k_ n)) = (a_ n(k_{m(n)} + f(k_{m(n) + 1}) + \ldots + f(k_{m(n + 1) - 1}))) \]

where each occurence of $f$ denotes a suitable transition map of the inverse system $(K_ n)$. Then the solid diagram

\[ \xymatrix{ K \ar[r] \ar@{..>}[d] & \prod K_ n \ar[r] \ar[d]^{a'} & \prod K_ n \ar[d]^{a''} \\ L \ar[r] & \prod L_ n \ar[r] & \prod L_ n } \]

is commutative and by TR3 we obtain the dotted arrow producing a morphism of distinguished triangles. We warn the reader that the map $K \to L$ is not unique. We will see later, that if $a$ is a pro-isomorphism, then $K \to L$ is an isomorphism, see More on Algebra, Lemma 15.87.6.

The Stacks project

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