## Tag `0D6M`

Chapter 21: Cohomology on Sites > Section 21.22: Derived and homotopy limits

Lemma 21.22.7. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E \in D(\mathcal{O})$. Let $\mathcal{B} \subset \mathop{\rm Ob}\nolimits(\mathcal{C})$ 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 $E \to R\mathop{\rm lim}\nolimits \tau_{\geq -n} E$ is an isomorphism in $D(\mathcal{O})$.

Proof.Set $K_n = \tau_{\geq -n}E$ and $K = R\mathop{\rm lim}\nolimits K_n$. 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 21.22.6 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. $\square$

The code snippet corresponding to this tag is a part of the file `sites-cohomology.tex` and is located in lines 4369–4387 (see updates for more information).

```
\begin{lemma}
\label{lemma-is-limit-per-object}
Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E \in D(\mathcal{O})$.
Let $\mathcal{B} \subset \Ob(\mathcal{C})$ be a subset. Assume
\begin{enumerate}
\item every object of $\mathcal{C}$ has a covering whose members
are elements of $\mathcal{B}$, and
\item 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 $p > p(V, m)$.
\end{enumerate}
Then the canonical map $E \to R\lim \tau_{\geq -n} E$
is an isomorphism in $D(\mathcal{O})$.
\end{lemma}
\begin{proof}
Set $K_n = \tau_{\geq -n}E$ and $K = R\lim K_n$.
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 \ref{lemma-cohomology-derived-limit-injective}
are satisfied and hence implies the desired injectivity.
Recall (Derived Categories, Remark
\ref{derived-remark-truncation-distinguished-triangle})
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.
\end{proof}
```

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