Projective dimension is a lattice invariant

We show that, for a free abelian group $G$ and prime power $p^\nu$, every direct sum decomposition of the group $G/p^\nu G$ lifts to a direct sum decomposition of $G$. This is the key result we use to show that, if $R$ is a commutative von Neumann regular ring, and $\mathcal{E}$ a set of idempotents in $R$, then the projective dimension of the ideal $\mathcal{E} R$ as an $R$-module is the same as the projective dimension of the ideal $\mathcal{EB}$, where $\mathcal{B}$ is the boolean algebra generated by $\mathcal{E} \cup \{1\}$. This answers a thirty year old open question of R. Wiegand. The proof is based on gaussian elimination on an $\omega \times \omega$ matrix, with adaptations enabling one to pass from the integers modulo $p^\nu$ to the integers.