reference of extension of flat bundle

Question

Does any one know where one can find a reference about the following fact?

Let $X$ be a smooth projective variety over an algebraically closed field $k$. Fix two flat bundles $(L_i,\nabla_i)$ over $X$, (vector bundles with integrable connections) for $i=1,2$. Define a new flat bundle $(M,\nabla_M)$ with $M= L_2^*\otimes L_1$ and $\nabla_M=\nabla_2^*\otimes \nabla_1$.

Then all extension classes $$ 0\to (L_1,\nabla_1)\to (H,\nabla)\to (L_2, \nabla_2)\to 0 $$ are described by the first deRham cohomology of $(M,\nabla_M)$ (hypercohomology of the deRham complex).

Answer 1

You are looking at extensions of $L_2$ by $L_1$ as $\mathcal{D}_X$-modules, which are classified by $\mathrm{Ext}^1_{\mathcal{D}_X}(L_2,L_1)\cong \mathrm{Ext}^1_{\mathcal{D}_X}(\mathcal{O}_X,M)$. Now it is well-known in the theory of $\mathcal{D}$-modules that there is a canonical isomorphism of $R\mathcal{H}om_{\mathcal{D}_X}(\mathcal{O}_X,M)$ with the de Rham complex $\Omega ^._X\otimes M$, so $\mathrm{Ext}^1_{\mathcal{D}_X}(\mathcal{O}_X,M)$ is canonically isomorphic to $H^1_{dR}(M)$.