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Abstract : The objects of study in this paper are sets of spectral synthesis for the Fourier algebra A(G) of a locally compact group and the Varopoulos algebra V(G) of a compact group with respect to submodules of the dual space. Such sets of synthesis are characterized in terms of certain closed ideals. For a closed set in a closed subgroup H of G, the relations between these ideals in the Fourier algebras of G and H are obtained. The injection theorem for such sets of synthesis is then a consequence. For the Fourier algebra of the quotient modulo a compact subgroup, an inverse projection theorem is proved. For a compact group, a correspondence between submodules of the dual spaces of A(G) and V(G) is set up and this leads to a relation between the corresponding sets of synthesis.