Lemma 15.80.2. Let

\[ (A^{-2}_ n \to A^{-1}_ n \to A^0_ n \to A^1_ n) \]

be an inverse system of complexes of abelian groups and denote $A^{-2} \to A^{-1} \to A^0 \to A^1$ its limit. Denote $(H_ n^{-1})$, $(H_ n^0)$ the inverse systems of cohomologies, and denote $H^{-1}$, $H^0$ the cohomologies of $A^{-2} \to A^{-1} \to A^0 \to A^1$. If

$(A^{-2}_ n)$ and $(A^{-1}_ n)$ have vanishing $R^1\mathop{\mathrm{lim}}\nolimits $,

$(H^{-1}_ n)$ has vanishing $R^1\mathop{\mathrm{lim}}\nolimits $,

then $H^0 = \mathop{\mathrm{lim}}\nolimits H_ n^0$.

**Proof.**
Let $K \in D(\textit{Ab}(\mathbf{N}))$ be the object represented by the system of complexes whose $n$th constituent is the complex $A^{-2}_ n \to A^{-1}_ n \to A^0_ n \to A^1_ n$. We will compute $H^0(R\mathop{\mathrm{lim}}\nolimits K)$ using both spectral sequences^{1} of Derived Categories, Lemma 13.21.3. The first has $E_1$-page

\[ \begin{matrix} 0
& 0
& R^1\mathop{\mathrm{lim}}\nolimits A^0_ n
& R^1\mathop{\mathrm{lim}}\nolimits A^1_ n
\\ A^{-2}
& A^{-1}
& A^0
& A^1
\end{matrix} \]

with horizontal differentials and all higher differentials are zero. The second has $E_2$ page

\[ \begin{matrix} R^1\mathop{\mathrm{lim}}\nolimits H^{-2}_ n
& 0
& R^1\mathop{\mathrm{lim}}\nolimits H^0_ n
& R^1 \mathop{\mathrm{lim}}\nolimits H^1_ n
\\ \mathop{\mathrm{lim}}\nolimits H^{-2}_ n
& \mathop{\mathrm{lim}}\nolimits H^{-1}_ n
& \mathop{\mathrm{lim}}\nolimits H^0_ n
& \mathop{\mathrm{lim}}\nolimits H^1_ n
\end{matrix} \]

and degenerates at this point. The result follows.
$\square$

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