  We reformulate Fourier-space crystallography in the language of cohomology of
groups. Once the problem is understood as a classification of linear functions
on the lattice, restricted by a particular group relation, and identified by
gauge transformation, the cohomological description becomes natural. We review
Fourier-space crystallography and group cohomology, quote the fact that
cohomology is dual to homology, and exhibit several results, previously
established for special cases or by intricate calculation, that fall
immediately out of the formalism. In particular, we prove that {\it two phase
functions are gauge equivalent if and only if they agree on all their
gauge-invariant integral linear combinations} and show how to find all these
linear combinations systematically.
