Algebraic spectral theory and the Serre multiplicity formula

The present paper is devoted to an algebraic treatment of the joint spectral theory within the framework of Noetherian modules over a finitely generated algebra extension of an algebraically closed field. The spectral mapping theorem is presented, together with the analysis of the index of n-tuples in the purely algebraic case. The index function over n-tuples from the coordinate ring of a variety is naturally extended up to a numerical Tor-polynomial. On the basis on Serre’s multiplicity formula, it is deduced that the Tor-polynomial is precisely the Samuel polynomial of the local algebra.