It is well known that if I have a differentialable manifold (holomorphic maniford) $M$, then I have a functor from the categroy of vector bundles on $M$ with flat connections to the categroy of local systems on $M$, given by $$(V,\nabla)\mapsto V^{\nabla}$$ and this functor is an equivalence of categories.
Now if I change the setting a little bit. I assume $X$ is a smooth scheme over a field $k$ of characteristic 0. Do I still have a functor like the above? Or in other words, is it true that
for any vector bundle $(V,\nabla)$ on $X$ the sheaf of horizontal sections $V^{\nabla}$ (i.e. the kernel of $\nabla$ as $k$-vector spaces) is still a locally constant sheaf of $k$-vector spaces?
I think it is true, and it should be written somewhere, could you tell me the reference for that?
Best Answer
Obviously you can still define the functor. But it won't have nice properties because the Zariski open sets are just too big for the concept of locally constant sheaf to apply in an interesting way and the solutions of algebraic differential equations are not algebraic functions in general.
Just take a trivial vector bundle $O_X^2$ on $X = \mathbb{G}_m$ with connection $$ \nabla \begin{pmatrix} f_1 \cr f_2\end{pmatrix} = d\begin{pmatrix} f_1 \cr f_2\end{pmatrix} - \begin{pmatrix} 0 & 0 \cr 1 & 0 \end{pmatrix} \begin{pmatrix} f_1 \cr f_2\end{pmatrix} \frac{dz}{z} $$ Holomorphic horizontal sections are linear combinations $A \begin{pmatrix} 0 \cr 1 \end{pmatrix} + B \begin{pmatrix} 1 \cr log(z) \end{pmatrix}$. You get a rank 2 local system.
But if you look at algebraic horizontal sections you will only get the constant sheaf $A \begin{pmatrix} 0 \cr 1 \end{pmatrix}$.
PS: For the same reason (Zariski open sets are too big), one needs to use the hypercohomology of the algebraic de Rham complex to define a reasonnable algebraic de Rham cohomology. Also for the same reason (not enough algebraic solutions) only the holomorphic solution complex or the holomorphic de Rham complex of a D-module are relevant to the Riemann-Hilbert correspondance.