  We use covariant phase space methods to study the metric and tetrad
formulations of General Relativity in a manifold with boundary and compare the
results obtained in both approaches. Proving their equivalence has been a
long-lasting problem that we solve here by using the cohomological approach
provided by the relative bicomplex framework. This allows us to prove the
equivalence without having to make any \emph{ad hoc} choices. We also compute
several relevant charges in both schemes and show that, as expected, they are
equivalent.
