Lemma 13.33.9 (094A)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 13: Derived Categories
Section 13.33: Derived colimits
Lemma 13.33.9 (cite)

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Lemma 13.33.9. Let $\mathcal{D}$ be a triangulated category with countable direct sums. Let $K \in \mathcal{D}$ be an object such that for every countable set of objects $E_ n \in \mathcal{D}$ the canonical map

$\bigoplus \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(K, E_ n) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(K, \bigoplus E_ n)$

is a bijection. Then, given any system $L_ n$ of $\mathcal{D}$ over $\mathbf{N}$ whose derived colimit $L = \text{hocolim} L_ n$ exists we have that

$\mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(K, L_ n) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(K, L)$

is a bijection.

Proof. Consider the defining distinguished triangle

$\bigoplus L_ n \to \bigoplus L_ n \to L \to \bigoplus L_ n[1]$

Apply the cohomological functor $\mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(K, -)$ (see Lemma 13.4.2). By elementary considerations concerning colimits of abelian groups we get the result. $\square$

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