The Stacks project

Lemma 21.30.4. In Situation 21.30.1 if $(V_ n)$ holds, then for $X$ in $\mathcal{C}$ and $L \in D(\mathcal{C}_{\tau '}/X)$ with $H^ i(L) = 0$ for $i < 0$ and $H^ i(L)$ in $\mathcal{A}'_ X$ for $0 \leq i \leq n$ we have $H^ n_{\tau '}(X, L) = H^ n_\tau (X, \epsilon _ X^{-1}L)$.

Proof. By Lemma 21.20.5 we have $H^ n_\tau (X, \epsilon _ X^{-1}L) = H^ n_{\tau '}(X, R\epsilon _{X, *}\epsilon _ X^{-1}L)$. There is a spectral sequence

\[ E_2^{p, q} = R^ p\epsilon _{X, *}\epsilon _ X^{-1}H^ q(L) \]

converging to $H^{p + q}(R\epsilon _{X, *}\epsilon _ X^{-1}L)$. By $(V_ n)$ we have the vanishing of $E_2^{p, q}$ for $0 < p \leq n$ and $0 \leq q \leq n$. Thus $E_2^{0, q} = \epsilon _{X, *}\epsilon _ X^{-1}H^ q(L) = H^ q(L)$ are the only nonzero terms $E_2^{p, q}$ with $p + q \leq n$. It follows that the map

\[ L \longrightarrow R\epsilon _{X, *}\epsilon _ X^{-1}L \]

is an isomorphism in degrees $\leq n$ (small detail omitted). Hence we find that $H^ i_{\tau '}(X, L) = H^ i_{\tau '}(X, R\epsilon _{X, *}\epsilon _ X^{-1}L)$ for $i \leq n$. Thus the lemma is proved. $\square$

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