The .-adic dualizing complex on an excellent surface with rational singularities

In this article, we show that if X is an excellent surface with rational singularities, the constant sheaf .. is a dualizing complex. In coefficient .., we also prove that the obstruction for .. to become a dualizing complex, lies on the divisor class groups at singular points. As applications, we study the perverse sheaves and the weights of .-adic cohomology groups on such surfaces.