  We construct a hierarchy of lattice fermions, where the coarser lattice Dirac
operator is the Schur complement of the block UL decomposition of the finer
lattice operator. We show that the construction is an exact gauged
renormalisation group transformation of the lattice action. In particular,
using such a transformation and the QCDLAB tool, it is shown how to implement
the Ginsparg-Wilson strategy for chiral fermions in the presence of a dynamical
gauge field. The scheme allows, for the first time, a full multigrid algorithm
for lattice quarks.
