  A general framework for investigating topological actions of $Z^d$ on compact
metric spaces was proposed by Boyle and Lind in terms of expansive behavior
along lower-dimensional subspaces of $R^d$. Here we completely describe this
expansive behavior for the class of algebraic $Z^d$-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 $Z^d$-actions, and for
algebraic $Z^d$-actions describe how they are related to each other and to
Krull dimension. For a linear subspace of $R^d$ we define the group of points
homoclinic to zero along the subspace, and prove that this group is constant
within an expansive component.
