The present talk is devoted to an algebraic treatment of the joint spectral theory within the  framework of Noetherian modules over an algebra finite extension of an algebraically closed field. We discuss the spectral mapping theorem and analyse the index of tuples in purely algebraic case. The index function over tuples from the coordinate ring of a variety is naturally extended up to a numerical Tor-polynomial which behaves as the Hilbert polynomial and provides a link between the index and dimension of a variety.