Derivations and relation modules for inverse semigroups

We define the derivation module for a homomorphism of inverse semigroups, generalizing a construction for groups due to
Crowell. For a presentation map from a free inverse semigroup, we can then define its relation module as the kernel of
a canonical map from the derivation module to the augmentation module. The constructions are analogues of the first steps
in the Gruenberg resolution obtained from a group presentation. We give a new proof of the characterization
of inverse monoids of cohomological dimension zero, and find a class of examples of inverse semigroups of cohomological
dimension one.