  The present paper 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 prove the spectral mapping
theorem and analyze 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. Based on Serre's multiplicity
formula, we deduce that Tor-polynomial is just the Samuel polynomial of the
local algebra.
