The Stacks project

Remark 15.86.6. With assumptions as in Lemma 15.86.5. A priori there are many isomorphism classes of objects $M$ of $D(\textit{Mod}(\mathbf{N}, (A_ n)))$ which give rise to the system $(K_ n, \varphi _ n)$ of the lemma. For each such $M$ we can consider the complex $R\mathop{\mathrm{lim}}\nolimits M \in D(A)$ where $A = \mathop{\mathrm{lim}}\nolimits A_ n$. By Lemma 15.86.3 we see that $R\mathop{\mathrm{lim}}\nolimits M$ is a derived limit of the inverse system $(K_ n)$ of $D(A)$. Hence we see that the isomorphism class of $R\mathop{\mathrm{lim}}\nolimits M$ in $D(A)$ is independent of the choices made in constructing $M$. In particular, we may apply results on $R\mathop{\mathrm{lim}}\nolimits $ proved in this section to derived limits of inverse systems in $D(A)$. For example, for every $p \in \mathbf{Z}$ there is a canonical short exact sequence

\[ 0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{p - 1}(K_ n) \to H^ p(R\mathop{\mathrm{lim}}\nolimits K_ n) \to \mathop{\mathrm{lim}}\nolimits H^ p(K_ n) \to 0 \]

because we may apply Lemma 15.86.3 to $M$. This can also been seen directly, without invoking the existence of $M$, by applying the argument of the proof of Lemma 15.86.3 to the (defining) distinguished triangle $R\mathop{\mathrm{lim}}\nolimits K_ n \to \prod K_ n \to \prod K_ n \to (R\mathop{\mathrm{lim}}\nolimits K_ n)[1]$ of the derived limit.


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