  We construct projections from the space of differential k-forms which belong
to L2 and whose exterior derivative also belongs to L2, to finite dimensional
subspaces of piecewise polynomial differential forms defined on a simplicial
mesh. These projections have the properties that they commute with the exterior
derivative and are bounded independent of the mesh size. Unlike some other
recent work in this direction, the projections are also locally defined in the
sense that they are defined by local operators on overlapping macroelements, in
the spirit of the Clement interpolant.
