Research Papers

Algebraic spectral theory and the Serre multiplicity formula

A. Dosi

College of Mathematical Sciences, Harbin Engineering University, Nangang District, Harbin 150001, China

Abstract: 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. Based on Serre's multiplicity formula, we deduce that Tor-polynomial is just the Samuel polynomial of the local algebra.

Keywords: Taylor spectrum, Noetherian modules, integral extensions, Samuel polynomial, Serre's multiplicity formula.

Received: 14.10.2024

Language: English