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 connection"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.
