Let $X$ be a sooth algebraic variety over $\mathbb{C}$.
Let us assume that there exists the commutative diagram
$\require{AMScd}$
\begin{CD}
U @>{i}>> \hat{X}\\
@| @VV{\phi}V\\
U @>{j}>> X
\end{CD}


where $\phi : \hat{X} \rightarrow X$ is the blowing-up, and assume that the inclusion maps $i : U\hookrightarrow \hat{X}$, $j : U\hookrightarrow X$ are affine.


Let $M$ be a $D\_U$-module on $U$.
Then we have the minimal extension functor $j\_{!\*}$ (resp. $i\_{!\*}$) from the (derived) category of $D\_U$-modules to the (derived) category of $D\_X$-modules (resp. $D\_{\hat{X}}$-modules),
and direct image functor $\phi\_+(\bullet) := R\phi\_\*(D\_{X \leftarrow \hat{X}}\otimes ^L\_{D\_{\hat{X}}}\bullet)$ (there are various notations in the literature for the direct image , e.g. $\int\_{\phi}$).


My question is


*is it true that
$$ j\_{!\*}(M) \cong \phi\_+(i\_{!\*}(M)) $$
as $D\_X$-modules?*


Or, are there any reference for them?



The commutativity of minimal extension and direct image by blowing-down
No. Here's an example. Let $X=\Bbb C^2$, $\hat{X}$ the blow-up of $X$ at the origin, $U=(\Bbb C^\*)^2$, and $M=\mathcal{O}\_U$. Then $j\_{!\*}M=\mathcal{O}\_X$ and $i\_{!\*}M=\mathcal{O}\_{\hat{X}}$.



> 
> **Claim:** $\phi\_+\mathcal{O}\_{\hat{X}}\ncong \mathcal{O}\_X$.
> *Proof.* Let $E=\phi^{-1}(0)$, and let $i\_E\colon E\hookrightarrow \hat{X}$ and $i\_0\colon \{0\} \hookrightarrow X$ be inclusion. It suffices to show that $i\_0^+\mathcal{O}\_X\ncong i\_0^+\phi\_+\mathcal{O}\_{\hat{X}}$.[1] First, we have
> $$i\_0^+\mathcal{O}\_X = Li\_0^\*\mathcal{O}\_X[-2] = \Bbb{C}[-2].$$
> Next, denoting by $a\_E$ the map from $E$ to a point, we have
> $$i\_0^+\phi\_+\mathcal{O}\_{\hat{X}} = (a\_E)\_+ i\_E^+\mathcal{O}\_{\hat{X}} = (a\_E)\_+ \mathcal{O}\_E[-1],$$
> where the first equality is via base change. But $(a\_E)\_+\mathcal{O}\_E[-1]$ is (up to shift) the de Rham cohomology of $E\cong \Bbb P^1$. In particular, it has cohomology in more than one degree. This finishes the proof of the claim.
> 
> 
> 


[1] Given a morphism $f\colon X\to Y$ of smooth varieties, I denote by $f^+$ the (shifted) $D$-module inverse image functor. I.e.
$$f^+:=Lf^\*[\dim X - \dim Y].$$


