Determinants of finite potent endomorphisms, symbols and reciprocity laws

TL;DR

This paper develops an algebraic framework for infinite determinants of finite potent endomorphisms, generalizing classical concepts and linking algebraic and analytic approaches, with applications to Tate's residues and loop groups.

Contribution

It introduces an algebraic definition of infinite determinants for finite potent endomorphisms, extending Grothendieck's determinant and connecting Tate's residues with loop group pairings.

Findings

Algebraic definition of infinite determinants for finite potent endomorphisms

Equivalence with classical analytic determinant definitions

Connection between Tate's residues and Segal-Wilson pairing

Abstract

The aim of this paper is to offer an algebraic definition of infinite determinants of finite potent endomorphisms using linear algebra techniques. It generalizes Grothendieck's determinant for finite rank endomorphisms and is equivalent to the classic analytic definitions. The theory can be interpreted as a multiplicative analogue to Tate's formalism of abstract residues in terms of traces of finite potent linear operators on infinite-dimensional vector spaces, and allows us to relate Tate's theory to the Segal-Wilson pairing in the context of loop groups.