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# Base change for the Gauss-Manin connectionsheaf

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.

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", 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.

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) $(E,\nabla)$ 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.

I think it comes from the following "base change property of Guass-Manin connection", 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
$$\xymatrix{Y\ar[r]^{f}\ar[d]^{a}&X\ar[d]^{b}\T\ar[r]^{g}&S}$$ \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}$is g^*b_*E^{\nabla} \to$

$a_*{(f^*E)^{\nabla}}$is an isomorphism .

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.

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