A general framework for investigating topological actions of \mathbb{Z}^d on compact metric spaces was proposed by Boyle and Lind in terms of expansive behavior along lower-dimensional subspaces of \mathbb{R}^d. Here we completely describe this expansive behavior for the class of algebraic \mathbb{Z}^d-actions given by commuting automorphisms of compact abelian groups. The description uses the logarithmic image of an algebraic variety together with a directional version of Noetherian modules over the ring of Laurent polynomials in several commuting variables.

We introduce two notions of rank for topological \mathbb{Z}^d-actions, and for algebraic \mathbb{Z}^d-actions describe how they are related to each other and to Krull dimension. For a linear subspace of \mathbb{R}^d we define the group of points homoclinic to zero along the subspace, and prove that this group is constant within an expansive component.