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Under hypotheses required for the Taylor-Wiles method, we prove for forms of$U(3)$ which are compact at infinity that the lattice structure on upper alcovealgebraic vectors or on principal series types given by the $\lambda$-isotypicpart of completed cohomology is a local invariant of the Galois representationattached to $\lambda$ when this Galois representation is residually irreduciblelocally at places dividing $p$. As a crucial input, we establish correspondingmod $p$ multiplicity one results. Our main innovation is the combination ofintegral Hecke theory and the Taylor--Wiles method.

Lattices in the cohomology of $U(3)$ arithmetic manifolds