Notes on Crystals and Algebraic D-modules

∇ : V → V ⊗ ΩX . Then the flat sections of V determine a local system L on X. For every point x ∈ X, the fiber of the local system Lx can be identified with the fiber Vx. Given a path p : [0, 1] from x = p(0) to y = p(1), there is a map p! : Lx → Ly is given by parallel transport along p, using the connection ∇; moreover the map p! depends only on the homotopy class of the path p. This construction is entirely reversible: the local system L determines the vector bundle V and its connection ∇ up to canonical isomorphism. In other words, the category of vector bundles with flat connection on X is equivalent to the category of local systems of vector spaces on X. Now suppose that X is a smooth algebraic variety over a field k of characteristic zero (fixed through the remainder of this lecture). There is a purely algebraic notion of a vector bundle with flat connection on X: that is, an algebraic vector bundle V on X equipped with a map of sheaves