The Stacks project

Lemma 21.23.1. Let $\mathcal{C}$ be a site. Let $K$ be an object of $D(\mathcal{C} \times \mathbf{N})$. Set $K_ n = i_ n^{-1}K$ as above. Then

\[ R\mathop{\mathrm{lim}}\nolimits K \cong R\mathop{\mathrm{lim}}\nolimits K_ n \]

in $D(\mathcal{C})$.

Proof. To calculate $R\mathop{\mathrm{lim}}\nolimits $ on an object $K$ of $D(\mathcal{C} \times \mathbf{N})$ we choose a K-injective representative $\mathcal{I}^\bullet $ whose terms are injective objects of $\textit{Ab}(\mathcal{C} \times \mathbf{N})$, see Injectives, Theorem 19.12.6. We may and do think of $\mathcal{I}^\bullet $ as an inverse system of complexes $(\mathcal{I}_ n^\bullet )$ and then we see that

\[ R\mathop{\mathrm{lim}}\nolimits K = \mathop{\mathrm{lim}}\nolimits \mathcal{I}_ n^\bullet \]

where the right hand side is the termwise inverse limit.

Let $\mathcal{J} = (\mathcal{J}_ n)$ be an injective object of $\textit{Ab}(\mathcal{C} \times \mathbf{N})$. The morphisms $(U, n) \to (U, n + 1)$ are monomorphisms of $\mathcal{C} \times \mathbf{N}$, hence $\mathcal{J}(U, n + 1) \to \mathcal{J}(U, n)$ is surjective (Lemma 21.12.6). It follows that $\mathcal{J}_{n + 1} \to \mathcal{J}_ n$ is surjective as a map of presheaves.

Note that the functor $i_ n^{-1}$ has an exact left adjoint $i_{n, !}$. Namely, $i_{n, !}\mathcal{F}$ is the inverse system $\ldots 0 \to 0 \to \mathcal{F} \to \ldots \to \mathcal{F}$. Thus the complexes $i_ n^{-1}\mathcal{I}^\bullet = \mathcal{I}_ n^\bullet $ are K-injective by Derived Categories, Lemma 13.31.9.

Because we chose our K-injective complex to have injective terms we conclude that

\[ 0 \to \mathop{\mathrm{lim}}\nolimits \mathcal{I}_ n^\bullet \to \prod \mathcal{I}_ n^\bullet \to \prod \mathcal{I}_ n^\bullet \to 0 \]

is a short exact sequence of complexes of abelian sheaves as it is a short exact sequence of complexes of abelian presheaves. Moreover, the products in the middle and the right represent the products in $D(\mathcal{C})$, see Injectives, Lemma 19.13.4 and its proof (this is where we use that $\mathcal{I}_ n^\bullet $ is K-injective). Thus $R\mathop{\mathrm{lim}}\nolimits K$ is a homotopy limit of the inverse system $(K_ n)$ by definition of homotopy limits in triangulated categories. $\square$

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