A Formula for Gauss-manin Determinants

Let K be a function field over a field k of characteristic 0, and let j : U ⊂ PK be a Zariski open set of the projective line. We consider a flat connection (E,∇) on U . The de Rham cohomology groups H i DR(U/K,∇/K) carry a K/k connection, the Gauß-Manin connection, and taking the alternate tensor of the determinant connections ⊗(detH i DR(U/K,∇/K),Gauß−Manin) (−1)i , one defines the Gauß-Manin determinant connection, denoted by