Ext and Tor modules play important roles in many areas of algebraic geometry, commutative algebra and representation theory. In the toric world, these modules are endowed with a grading by the character lattice. In this talk, we will examine Ext and Tor of two Weil divisors on a two-dimensional cyclic quotient singularity, i.e. an two-dimensional affine toric variety. We will develop a combinatorial description for these modules, and observe a connection of Ext and Tor as Matlis duals.