The Stacks project

Lemma 21.30.8. In Situation 21.30.1 we have that $(V_ n)$ is true for all $n$. Moreover:

  1. For $X$ in $\mathcal{C}$ and $K' \in D^+_{\mathcal{A}'_ X}(\mathcal{C}_{\tau '}/X)$ the map $K' \to R\epsilon _{X, *}(\epsilon _ X^{-1}K')$ is an isomorphism.

  2. For $f : X \to Y$ in $\mathcal{P}$ and $K' \in D^+_{\mathcal{A}'_ X}(\mathcal{C}_{\tau '}/X)$ we have $Rf_{\tau ', *}K' \in D^+_{\mathcal{A}'_ X}(\mathcal{C}_{\tau '}/Y)$ and $\epsilon _ Y^{-1}(Rf_{\tau ', *}K') = Rf_{\tau , *}(\epsilon _ X^{-1}K')$.

Proof. Observe that $(V_0)$ holds as it is the empty condition. Then we get $(V_ n)$ for all $n$ by Lemma 21.30.7.

Proof of (1). The object $K = \epsilon _ X^{-1}K'$ has cohomology sheaves $H^ i(K) = \epsilon _ X^{-1}H^ i(K')$ in $\mathcal{A}_ X$. Hence the spectral sequence

\[ E_2^{p, q} = R^ p\epsilon _{X, *} H^ q(K) \Rightarrow H^{p + q}(R\epsilon _{X, *}K) \]

degenerates by $(V_ n)$ for all $n$ and we find

\[ H^ n(R\epsilon _{X, *}K) = \epsilon _{X, *}H^ n(K) = \epsilon _{X, *}\epsilon _ X^{-1}H^ i(K') = H^ i(K'). \]

again because $H^ i(K')$ is in $\mathcal{A}'_ X$. Thus the canonical map $K' \to R\epsilon _{X, *}(\epsilon _ X^{-1}K')$ is an isomorphism.

Proof of (2). Using the spectral sequence

\[ E_2^{p, q} = R^ pf_{\tau ', *}H^ q(K') \Rightarrow R^{p + q}f_{\tau ', *}K' \]

the fact that $R^ pf_{\tau ', *}H^ q(K')$ is in $\mathcal{A}'_ Y$ by (4), the fact that $\mathcal{A}'_ Y$ is a weak Serre subcategory of $\textit{Ab}(\mathcal{C}_{\tau '}/Y)$, and Homology, Lemma 12.24.11 we conclude that $Rf_{\tau ', *}K' \in D^+_{\mathcal{A}'_ X}(\mathcal{C}_{\tau '}/X)$. To finish the proof we have to show the base change map

\[ \epsilon _ Y^{-1}(Rf_{\tau ', *}K') \longrightarrow Rf_{\tau , *}(\epsilon _ X^{-1}K') \]

is an isomorphism. Comparing the spectral sequence above to the spectral sequence

\[ E_2^{p, q} = R^ pf_{\tau , *}H^ q(\epsilon _ X^{-1}K') \Rightarrow R^{p + q}f_{\tau , *}\epsilon _ X^{-1}K' \]

we reduce this to the case where $K'$ has a single nonzero cohomology sheaf $\mathcal{F}'$ in $\mathcal{A}'_ X$; details omitted. Then Lemma 21.30.3 gives $\epsilon _ Y^{-1}R^ if_{\tau ', *}\mathcal{F}' = R^ if_{\tau , *}\epsilon _ X^{-1}\mathcal{F}'$ for all $i$ and the proof is complete. $\square$

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