  We construct an Eliahou-Kervaire-like minimal free resolution of the
alternative polarization $b-pol(I)$ of a Borel fixed ideal $I$. It yields new
descriptions of the minimal free resolutions of $I$ itself and $I^sq$, where
$(-)^sq$ is the squarefree operation in the shifting theory. These resolutions
are cellular, and the (common) supporting cell complex is given by discrete
Morse theory. If $I$ is generated in one degree, our description is equivalent
to that of Nagel and Reiner.
