Let $K$ be a number field, $\Lambda\subset K$ a lattice (i.e. a free abelian subgroup of rank $[K:\mathbb{Q}]$), and consider $\tilde{\Lambda}:=\mathcal{O}_K\Lambda$, which is the smallest $\mathcal{O}_K$-module containing $\Lambda$. For a problem I'm working on, it looks helpful to be able to understand something about the index $[\tilde{\Lambda}:\Lambda]$, which is finite.
The ring of multipliers $R_\Lambda:=\{x\in K:x\Lambda\subset\Lambda\}$ is an order of $K$, and in the examples I've seen so far, $$ [\mathcal{O}_K:R_\Lambda]=[\tilde{\Lambda}:\Lambda]. $$ I proved the equality is true when $[K:\mathbb{Q}]=2$ (my proof was clunky and unelightening). I don't know whether the equality is true for all lattices, but at this point I would guess not.
The equality is obviously true when $\Lambda$ is the set of $R_\Lambda$-multiples of a single element. I have an argument that the equality is true if $\Lambda$ is a projective $R_\Lambda$-module. So I am interested to know:
Is every lattice in a number field projective over its ring of multipliers?