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

1. every object of $\mathcal{C}$ has a covering whose members are elements of $\mathcal{B}$, and

2. 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. $\square$

Comment #2465 by anonymous on

Two pedagogical suggestions:

(i) In item (2) of the lemma it would be good to add "for all integers $p$, $m$ satisfying".

(ii) The $m$ in the proof has a different role than the $m$ in the formulation of item (2) in the lemma. I found this a bit confusing. Could you use a different letter in item (2)?

Comment #2501 by on

OK, I followed the first suggestion, but not the second. The reason is that actually roughly the $m = m_p$ in the proof is more or less the $m = m_s$ in statement (2). Namely, in the end we see that we are using condition in (2) for $m_s = m_p - 1$, $m_s = m_p$ and $m_s = m_p + 1$ (in the penultimate sentence of the proof). So they kind of play the same role. Alternatively, you could formulate this lemma one cohomology sheaf at a time and then you'd need the condition of (2) for that cohomological degree and the two adjacent ones. The change is here. Many thanks!

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