The aim of this note is to understand under which conditions invertible modules over a commutative S-algebra in the sense of Elmendorf, Kriz, Mandell \& May give rise to elements in the algebraic Picard group of invertible graded modules over the coefficient ring by taking homotopy groups. If a connective commutative S-algebra R has coherent localizations (R-{*})(m) for every maximal ideal m R-{*}, then for every invertible R- module U, U-{*} = pi({*}) U is an invertible graded R-{*}- module. In some non-connective cases we can carry the result over under the additional assumption that the commutative S-algebra has `residue fields' for all maximal ideals m R-{*} if the global dimension of R-{*} is small or if R is 2-periodic with underlying Noetherian complete local regular ring R-0. We apply these results to finite abelian Galois extensions of Lubin-Tate spectra.