Crepant resolutions of four-dimensional finite quotients

We give counterexamples to the conjecture of Dais-Henk-Ziegler on projective crepant resolutions of complete-intersection quotient singularities. For finite $G\subset {\mathrm{SL}}_4(\mathbb{Z})$, we determine whether ${\mathbb{C}}^4\mathrm{/}G$ admits a projective crepant resolution, with the representations considered up to conjugacy in ${\mathrm{SL}}_4(\mathbb{C})$. Ten explicit criteria decide all cases. The examples also show that the Molien series and the age polynomial do not determine existence. We also discuss questions of Ito-Reid on maximal cyclic subgroups and the generation of cohomology in degree two.