  We formulate a unimodular N=1, d=4 supergravity theory off shell. We see that
the infinitesimal Grassmann parameters defining the unimodular supergravity
transformations are constrained and show that the conmutator of two
infinitesinal unimodular supergravity transformations closes on transverse
diffeomorphisms, Lorentz transformations and unimodular supergravity
transformations. Along the way, we also show that the linearized theory is a
supersymmetric theory of gravitons and gravitinos. We see that de Sitter and
anti-de Sitter spacetimes are non-supersymmetric vacua of our unimodular
supergravity theory.
