I want to see the following thing:


$\ \ $If $X$ is a smooth geometrically connected scheme over a field $k$ of characteristic 0, $U\subseteq X$ is a non-empty open, $(E,\nabla)$ is an integrable connection on $X/k$, then $H\_{DM}^0(X/k,(E,\nabla))\hookrightarrow H\_{DM}^0(U/k,(E,\nabla)|\_{U})$ is an isomorphism.
Here the De Rham cohomology with respect to an integral connection means I take the hypercohomology of the complex 


$$E\xrightarrow{\nabla}E\otimes\Omega^1\_{X/k}\to E\otimes\Omega^2\_{X/k}\to\cdots$$


$\ \ $ I think it comes from the following "base change property of Guass-Manin sheaf", I don't know the precise formulation for that property, but I guess it is the following (0-th version):


If $S$ and $X$ are smooth geometrically connected schemes over a field $k$ of characteristic 0, $X\to S$ is proper smooth $k$-morphism, if we have a commutative diagramme  

$\hspace{100pt}\ Y \xrightarrow{f} X$


$\hspace{100pt}\ |$$a$$\ \ \ \ |b$


$\hspace{100pt}\ T\xrightarrow{g}S$


with the property that $f^\*\Omega\_{X/S}\to \Omega\_{Y/T}$ is an isomorphism, and if $(E,\nabla)$ is an integrable connection on $X/S$, then the canonical map 


$g^\*b\_\*E^{\nabla} \to$


$a\_\*{(f^\*E)^{\nabla}}$is an isomorphism. Here $E^{\nabla}$ is the kernel of the $k$-linear map $\nabla: E\to E\otimes\Omega^1\_{X/S}$. By definition $b\_\*E^{\nabla}$ with the canonical connection (the Gauss-Manin connection) on it is $H\_{DM}^0(X,(E,\nabla))$. The similar notations for $(f^\*E)^{\nabla}$. 


Is that true? Is there a reference? If this was true, then we can take $Y/T$ to be $U/k$, this answers my first question. But I think maybe the complete formulation requires the diagramme to be Cartesian.



Base change for the Gauss-Manin sheaf
Here is a proof of your claim:


We want to show that any flat section of $E|\_U$ extends to a flat section
of $E$. Since the extension is unique if it exists, the question is local
on $X$ so we may assume that the bundle $E = \mathcal{O}\_X^n$. The connection is
then given an $n\times n$ matrix $A$ of $1$-forms on $X$ i.e. for $v$ a
section of $E$,
$$\nabla v = dv + Av $$
where $d$ is the exterior derivative applied componentwise.


Now any flat section $w$ of $E|\_U$ can be viewed as a rational section $v$ of $E$.
Since we are in characteristic zero, if any component $f$ of $w$ has a pole of order
$r$ along some divisor $D$ in $X \backslash U$, then $df$ has a pole of order $n+1$
along $D$. Since none of the entries of $A$ have poles, it follows that $\nabla w$
cannot be $0$. Thus all components of $w$ are regular functions on $X$ so $w$ is
extends to a regular section $v$ of $E$. Since $w$ was a flat section of $E|\_U$,
it follows that $v$ is a flat section of $E$, proving the desired surjectivity.


