  We study unfrustrated spin Hamiltonians that consist of commuting tensor
products of Pauli matrices. Assuming translation-invariance, a family of
Hamiltonians that belong to the same phase of matter is described by a map
between modules over the translation-group algebra, so homological methods are
applicable. In any dimension every point-like charge appears as a vertex of a
fractal operator, and can be isolated with energy barrier at most logarithmic
in the separation distance. For a topologically ordered system in three
dimensions, there must exist a point-like nontrivial charge. If the
ground-state degeneracy is upper bounded by a constant independent of the
system size, then the topological charges in three dimensions always appear at
the end points of string operators. A connection between the ground state
degeneracy and the number of points on an algebraic set is discussed. Tools to
handle local Clifford unitary transformations are given.
