In this article, we show that if $X$ is an excellent surface with rational singularities, the constant sheaf $\mathbb{ Q}_{ \ell}$ is a dualizing complex. In coefficient $\mathbb{ Z}_{ \ell}$, we also prove that the obstruction for $\mathbb{ Z}_{ \ell}$ to become a dualizing complex lying on the divisor class groups at singular points. As applications, we study the perverse sheaves and the weights of $\ell$-adic cohomology groups on such surfaces.
Categories: math.AG