Remark 15.87.10. Let $A$ be a ring. Let $(E_ n)$ be an inverse system of objects of $D(A)$. We've seen above that a derived limit $R\mathop{\mathrm{lim}}\nolimits E_ n$ exists. Thus for every object $K$ of $D(A)$ also the derived limit $R\mathop{\mathrm{lim}}\nolimits ( K \otimes _ A^\mathbf {L} E_ n )$ exists. It turns out that we can construct these derived limits functorially in $K$ and obtain an exact functor

$R\mathop{\mathrm{lim}}\nolimits (- \otimes _ A^\mathbf {L} E_ n) : D(A) \longrightarrow D(A)$

of triangulated categories. Namely, we first lift $(E_ n)$ to an object $E$ of $D(\mathbf{N}, A)$, see Lemma 15.87.5. (The functor will depend on the choice of this lift.) Next, observe that there is a “diagonal” or “constant” functor

$\Delta : D(A) \longrightarrow D(\mathbf{N}, A)$

mapping the complex $K^\bullet$ to the constant inverse system of complexes with value $K^\bullet$. Then we simply define

$R\mathop{\mathrm{lim}}\nolimits (K \otimes _ A^\mathbf {L} E_ n) = R\mathop{\mathrm{lim}}\nolimits (\Delta (K)\otimes ^\mathbf {L} E)$

where on the right hand side we use the functor $R\mathop{\mathrm{lim}}\nolimits$ of Lemma 15.87.1 and the functor $- \otimes ^\mathbf {L} -$ of Lemma 15.87.9.

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