Lattices in the cohomology of U(3) arithmetic manifolds

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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 alcove algebraic vectors or on principal series types given by the λ-isotypic part of completed cohomology is a local invariant of the Galois representation attached to λ when this Galois representation is residually irreducible locally at places dividing p. As a crucial input, we establish corresponding mod p multiplicity one results. Our main innovation is the combination of integral Hecke theory and the Taylor--Wiles method.