Lemma 13.33.4 (0CRJ)—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. $\square$

The Stacks project

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