We combine the theory of Pommaret bases with a
(slight generalisation of a) recent construction
by Skoldberg based on discrete Morse theory. This
combination allows us the explicit determination
of a (generally non-minimal) free resolution for a
graded polynomial module with the computation of
only one Pommaret basis. If only the Betti numbers
are needed, one can considerably simplify the
computations by determining only the constant
part of the dierential. For thespecial case of
a quasi-stable monomial ideal, we show that the
induced resolution is a mapping cone resolution.
We present an implementation within the CoCoALib
and test it with some common benchmark ideals.