Determinant bundle in a family of curves, after A. Beilinson and V. Schechtman

Let pi : X --> S be a smooth projective family of curves over a smooth base S over a field of characteristic 0, together with a bundle E on X. Then A. Beilinson and V. Schechtman define in [1] a beautiful "trace complex" (tr)A(E)(.) on X, the 0(th) relative cohomology of which describes the Atiyah algebra of the determinant bundle of E on S. Their proof reduces the general case to the acyclic one. In particular, one needs a comparison of R pi(*)((tr)A(F)(.)) for F = E and F = E(D), where D is etale over S (see Theorem 2.3.1, reduction ii) in [1]), In this note, we analyze this reduction in more detail and correct a point.