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The Stacks project

Lemma 13.33.4. Let \mathcal{D} be a triangulated category. Let (K_ n, f_ n) be a system of objects of \mathcal{D}. Let n_1 < n_2 < n_3 < \ldots be a sequence of integers. Assume \bigoplus K_ n and \bigoplus K_{n_ i} exist. Then there exists an isomorphism \text{hocolim} K_{n_ i} \to \text{hocolim} K_ n such that

\xymatrix{ K_{n_ i} \ar[r] \ar[d]_{\text{id}} & \text{hocolim} K_{n_ i} \ar[d] \\ K_{n_ i} \ar[r] & \text{hocolim} K_ n }

commutes for all i.

Proof. Let g_ i : K_{n_ i} \to K_{n_{i + 1}} be the composition f_{n_{i + 1} - 1} \circ \ldots \circ f_{n_ i}. We construct commutative diagrams

\vcenter { \xymatrix{ \bigoplus \nolimits _ i K_{n_ i} \ar[r]_{1 - g_ i} \ar[d]_ b & \bigoplus \nolimits _ i K_{n_ i} \ar[d]^ a \\ \bigoplus \nolimits _ n K_ n \ar[r]^{1 - f_ n} & \bigoplus \nolimits _ n K_ n } } \quad \text{and}\quad \vcenter { \xymatrix{ \bigoplus \nolimits _ n K_ n \ar[r]_{1 - f_ n} \ar[d]_ d & \bigoplus \nolimits _ n K_ n \ar[d]^ c \\ \bigoplus \nolimits _ i K_{n_ i} \ar[r]^{1 - g_ i} & \bigoplus \nolimits _ i K_{n_ i} } }

as follows. Let a_ i = a|_{K_{n_ i}} be the inclusion of K_{n_ i} into the direct sum. In other words, a is the natural inclusion. Let b_ i = b|_{K_{n_ i}} be the map

K_{n_ i} \xrightarrow {1,\ f_{n_ i},\ f_{n_ i + 1} \circ f_{n_ i}, \ \ldots ,\ f_{n_{i + 1} - 2} \circ \ldots \circ f_{n_ i}} K_{n_ i} \oplus K_{n_ i + 1} \oplus \ldots \oplus K_{n_{i + 1} - 1}

If n_{i - 1} < j \leq n_ i, then we let c_ j = c|_{K_ j} be the map

K_ j \xrightarrow {f_{n_ i - 1} \circ \ldots \circ f_ j} K_{n_ i}

We let d_ j = d|_{K_ j} be zero if j \not= n_ i for any i and we let d_{n_ i} be the natural inclusion of K_{n_ i} into the direct sum. In other words, d is the natural projection. By TR3 these diagrams define morphisms

\varphi : \text{hocolim} K_{n_ i} \to \text{hocolim} K_ n \quad \text{and}\quad \psi : \text{hocolim} K_ n \to \text{hocolim} K_{n_ i}

Since c \circ a and d \circ b are the identity maps we see that \varphi \circ \psi is an isomorphism by Lemma 13.4.3. The other way around we get the morphisms a \circ c and b \circ d. Consider the morphism h = (h_ j) : \bigoplus K_ n \to \bigoplus K_ n given by the rule: for n_{i - 1} < j < n_ i we set

h_ j : K_ j \xrightarrow {1,\ f_ j,\ f_{j + 1} \circ f_ j, \ \ldots ,\ f_{n_ i - 1} \circ \ldots \circ f_ j} K_ j \oplus \ldots \oplus K_{n_ i}

Then the reader verifies that (1 - f) \circ h = \text{id} - a \circ c and h \circ (1 - f) = \text{id} - b \circ d. This means that \text{id} - \psi \circ \varphi has square zero by Lemma 13.4.5 (small argument omitted). In other words, \psi \circ \varphi differs from the identity by a nilpotent endomorphism, hence is an isomorphism. Thus \varphi and \psi are isomorphisms as desired. \square


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