Algebraic spectral theory and Serre multiplicity formula

TL;DR

This paper develops an algebraic approach to joint spectral theory for Noetherian modules over algebraic extensions, proving key theorems and linking spectral invariants to Serre's multiplicity formula.

Contribution

It introduces an algebraic framework for spectral theory, proving the spectral mapping theorem and connecting the index of tuples to Samuel polynomials via Serre's formula.

Findings

Proved the spectral mapping theorem in algebraic setting

Extended the index function to a numerical Tor-polynomial

Linked Tor-polynomial to Samuel polynomial of local algebra

Abstract

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.