The Stacks project

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. $\square$

Comments (2)

Comment #2465 by anonymous on

Two pedagogical suggestions:

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

(ii) The in the proof has a different role than the 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 in the proof is more or less the in statement (2). Namely, in the end we see that we are using condition in (2) for , and (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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