  For a finite-dimensional vector space $V$, the common basis complex of $V$ is
the simplicial complex whose vertices are the proper non-zero subspaces of $V$,
and $\sigma$ is a simplex if and only if there exists a basis $B$ of $V$ that
contains a basis of $S$ for all $S\in \sigma$. This complex was introduced by
Rognes in 1992 in connection with stable buildings. In this article, we prove
that the common basis complex is homotopy equivalent to the proper part of the
poset of partial direct sum decompositions of $V$. Moreover, we establish this
result in a more general combinatorial context, including the case of free
groups, matroids, vector spaces with non-degenerate sesquilinear forms, and
free modules over commutative Hermite rings, such as local rings or Dedekind
domains.
