Lemma 21.23.7. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E \in D(\mathcal{O})$. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be a subset. Assume
every object of $\mathcal{C}$ has a covering whose members are elements of $\mathcal{B}$, and
for every $V \in \mathcal{B}$ there exist a function $p(V, -) : \mathbf{Z} \to \mathbf{Z}$ and a cofinal system $\text{Cov}_ V$ of coverings of $V$ such that
\[ H^ p(V_ i, H^{m - p}(E)) = 0 \]
for all $\{ V_ i \to V\} \in \text{Cov}_ V$ and all integers $p, m$ satisfying $p > p(V, m)$.
Then the canonical map $E \to R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n} E$ is an isomorphism in $D(\mathcal{O})$.
Proof.
Set $K_ n = \tau _{\geq -n}E$ and $K = R\mathop{\mathrm{lim}}\nolimits K_ n$. The canonical map $E \to K$ comes from the canonical maps $E \to K_ n = \tau _{\geq -n}E$. We have to show that $E \to K$ induces an isomorphism $H^ m(E) \to H^ m(K)$ of cohomology sheaves. In the rest of the proof we fix $m$. If $n \geq -m$, then the map $E \to \tau _{\geq -n}E = K_ n$ induces an isomorphism $H^ m(E) \to H^ m(K_ n)$. To finish the proof it suffices to show that for every $V \in \mathcal{B}$ there exists an integer $n(V) \geq -m$ such that the map $H^ m(K)(V) \to H^ m(K_{n(V)})(V)$ is injective. Namely, then the composition
\[ H^ m(E)(V) \to H^ m(K)(V) \to H^ m(K_{n(V)})(V) \]
is a bijection and the second arrow is injective, hence the first arrow is bijective. By property (1) this will imply $H^ m(E) \to H^ m(K)$ is an isomorphism. Set
\[ n(V) = 1 + \max \{ -m, p(V, m - 1) - m, -1 + p(V, m) - m, -2 + p(V, m + 1) - m\} . \]
so that in any case $n(V) \geq -m$. Claim: the maps
\[ H^{m - 1}(V_ i, K_{n + 1}) \to H^{m - 1}(V_ i, K_ n) \quad \text{and}\quad H^ m(V_ i, K_{n + 1}) \to H^ m(V_ i, K_ n) \]
are isomorphisms for $n \geq n(V)$ and $\{ V_ i \to V\} \in \text{Cov}_ V$. The claim implies conditions (1) and (2) of Lemma 21.23.6 are satisfied and hence implies the desired injectivity. Recall (Derived Categories, Remark 13.12.4) that we have distinguished triangles
\[ H^{-n - 1}(E)[n + 1] \to K_{n + 1} \to K_ n \to H^{-n - 1}(E)[n + 2] \]
Looking at the asssociated long exact cohomology sequence the claim follows if
\[ H^{m + n}(V_ i, H^{-n - 1}(E)),\quad H^{m + n + 1}(V_ i, H^{-n - 1}(E)),\quad H^{m + n + 2}(V_ i, H^{-n - 1}(E)) \]
are zero for $n \geq n(V)$ and $\{ V_ i \to V\} \in \text{Cov}_ V$. This follows from our choice of $n(V)$ and the assumption in the lemma.
$\square$
Comments (2)
Comment #2465 by anonymous on
Comment #2501 by Johan on