Relative multiplicities of graded algebras

Without any finiteness assumption we define a sequence of relative multiplicities for a pair A⊂B of standard graded Noetherian algebras that extends the notion of relative multiplicities of Simis, Ulrich and Vasconcelos and unifies them with the j-multiplicity of ideals introduced by Achilles and Manaresi as well as the j-multiplicity of modules defined by Ulrich and Validashti. Using our relative multiplicities, we give numerical criteria for integrality and birationality of the extension A⊂B.