  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.
