I am trying to understand "de Cataldo, Migliorini. The perverse filtration and the Lefschetz hyperplane theorem. *Annals of Mathematics*, 171(2010), 2089-2113." My question is about one detail in the paper. Here is my reproduction of the situation. 


Let $\Sigma$ be a stratification of $\mathbb{P}^N$ adapted to a bounded complex $K$ with $\Sigma$-constructible cohomology. Let $\Sigma$ be also adapted to the embedding $Y \subseteq \mathbb{P}^N$ of an affine variety $Y$. Let $\Lambda \subset \mathbb{P}^N$ be a hyperplane, $H = \Lambda \cap Y$ and $\bar{H} = \Lambda \cap \bar{Y}$. Set $\bar{U} = \bar{Y} \setminus \bar{H}$ and $U = Y \setminus H$. Consider the cartesian diagram


$$\begin{array}[c]{ccccc}
H&{\xrightarrow{i}}&Y&{\xleftarrow{j}}&U\\
\downarrow\scriptstyle{J}&&\downarrow\scriptstyle{J} && \downarrow\scriptstyle{J}\\
\bar{H}&{\xrightarrow{i}}&\bar{Y}&{\xleftarrow{j}}&\bar{U}
\end{array}$$


Now the two authors stated (page 2101, following the same diagram) "By the octahedron axiom, the map $J\_!j\_\*j^\*K \to j\_\*J\_!j^\*K$ is an isomorphism if and only if the natural base change map $i^\*J\_\*K \to J\_\*i^\*K$ is an isomorphism."


My question is: what is exactly used about the octahedron axiom for the statement?


Thanks!



Base change and the octahedron axiom
I'm not sure what's going on, but here's a guess:


Firstly, the base change map goes the other way: $i^\*J\_\*\mathcal{F}\rightarrow J\_\*i^\*\mathcal{F}$. Say this is an isomorphism for the Verdier dual of $K$ (they end up saying that the condition they impose makes the base change morphism an isomorphism independent of $K$, so this is ok).


Then taking the Verdier dual of this isomorphism we get an isomorphism $J\_!i^!K\rightarrow i^!J\_!K$. Next note that the similarly defined base change map $J\_!j^\*K\rightarrow j^\*J\_!K$ gives an isomorphism, since $j$ is an open embedding and $j^\*=j^!$. 


The distinguished triangles to plug into the octahedron axiom are then


$J\_!K\rightarrow J\_!j\_\*j^\*K\rightarrow J\_!i\_\*i^!K[1]\rightarrow$


and


$J\_!K\rightarrow j\_\*j^\*J\_!K\rightarrow i\_\*i^!J\_!K[1]\rightarrow$


and 


$J\_!j\_\*j^\*K\xrightarrow{f}j\_\*j^\*J\_!K\rightarrow cone(f)\rightarrow$


By the axiom, there is a distinguished triangle


$J\_!i\_\*i^!K[1]\xrightarrow{g} i\_\*i^!J\_!K[1]\rightarrow cone(f)\rightarrow$


and so if $g$ is an isomorphism, $cone(f)=0$ and $f$ is an isomorphism.


