  We present an explicit expression of the cohomology complex of a
constructible sheaf of abelian groups on the small \'etale site of an
irreducible curve over an algebraically closed field, when the torsion of the
sheaf is invertible in the field. This expression only involves finite groups,
and is functorial in both the curve and the sheaf. In particular, we explain
how to compute the Galois action on this complex. We also present an algorithm
which computes it and study its complexity.
