  Given a domain $\Omega \subset \mathbb{R}^n$, the de Rham complex of
differential forms arises naturally in the study of problems in
electromagnetism and fluid mechanics defined on $\Omega$, and its
discretization helps build stable numerical methods for such problems. For
constructing such stable methods, one critical requirement is ensuring that the
discrete subcomplex is cohomologically equivalent to the continuous complex.
When $\Omega$ is a hypercube, we thus require that the discrete subcomplex be
exact. Focusing on such $\Omega$, we theoretically analyze the discrete de Rham
complex built from hierarchical B-spline differential forms, i.e., the discrete
differential forms are smooth splines and support adaptive refinements - these
properties are key to enabling accurate and efficient numerical simulations. We
provide locally-verifiable sufficient conditions that ensure that the discrete
spline complex is exact. Numerical tests are presented to support the
theoretical results, and the examples discussed include complexes that satisfy
our prescribed conditions as well as those that violate them.
