The Stacks project

Lemma 15.78.2. Let $R$ be a ring. Let $K \in D(R)$ be an object such that for every countable set of objects $E_ n \in D(R)$ the canonical map

\[ \bigoplus \mathop{\mathrm{Hom}}\nolimits _{D(R)}(K, E_ n) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{D(R)}(K, \bigoplus E_ n) \]

is a bijection. Then, given any system $L_ n^\bullet $ of complexes over $\mathbf{N}$ we have that

\[ \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _{D(R)}(K, L^\bullet _ n) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{D(R)}(K, L^\bullet ) \]

is a bijection, where $L^\bullet $ is the termwise colimit, i.e., $L^ m = \mathop{\mathrm{colim}}\nolimits L_ n^ m$ for all $m \in \mathbf{Z}$.

Proof. Consider the short exact sequence of complexes

\[ 0 \to \bigoplus L_ n^\bullet \to \bigoplus L_ n^\bullet \to L^\bullet \to 0 \]

where the first map is given by $1 - t_ n$ in degree $n$ where $t_ n : L_ n^\bullet \to L_{n + 1}^\bullet $ is the transition map. By Derived Categories, Lemma 13.12.1 this is a distinguished triangle in $D(R)$. Apply the homological functor $\mathop{\mathrm{Hom}}\nolimits _{D(R)}(K, -)$, see Derived Categories, Lemma 13.4.2. Thus a long exact cohomology sequence

\[ \xymatrix{ & \ldots \ar[r] & \mathop{\mathrm{Hom}}\nolimits _{D(R)}(K, \mathop{\mathrm{colim}}\nolimits L^\bullet _ n[-1]) \ar[lld] \\ \mathop{\mathrm{Hom}}\nolimits _{D(R)}(K, \bigoplus L^\bullet _ n) \ar[r] & \mathop{\mathrm{Hom}}\nolimits _{D(R)}(K, \bigoplus L^\bullet _ n) \ar[r] & \mathop{\mathrm{Hom}}\nolimits _{D(R)}(K, \mathop{\mathrm{colim}}\nolimits L^\bullet _ n) \ar[lld] \\ \mathop{\mathrm{Hom}}\nolimits _{D(R)}(K, \bigoplus L^\bullet _ n[1]) \ar[r] & \ldots } \]

Since we have assumed that $\mathop{\mathrm{Hom}}\nolimits _{D(R)}(K, \bigoplus L^\bullet _ n)$ is equal to $\bigoplus \mathop{\mathrm{Hom}}\nolimits _{D(R)}(K, L^\bullet _ n)$ we see that the first map on every row of the diagram is injective (by the explicit description of this map as the sum of the maps induced by $1 - t_ n$). Hence we conclude that $\mathop{\mathrm{Hom}}\nolimits _{D(R)}(K, \mathop{\mathrm{colim}}\nolimits L^\bullet _ n)$ is the cokernel of the first map of the middle row in the diagram above which is what we had to show. $\square$

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