  We introduce the notion of joint torsion for several commuting operators
satisfying a Fredholm condition. This new secondary invariant takes values in
the group of invertibles of a field. It is constructed by comparing
determinants associated with different filtrations of a Koszul complex. Our
notion of joint torsion generalizes the Carey-Pincus joint torsion of a pair of
commuting Fredholm operators. As an example, under more restrictive
invertibility assumptions, we show that the joint torsion recovers the
multiplicative Lefschetz numbers. Furthermore, in the case of Toeplitz
operators over the polydisc we provide a link between the joint torsion and the
Cauchy integral formula. We will also consider the algebraic properties of the
joint torsion. They include a cocycle property, a symmetry property, a
triviality property and a multiplicativity property. The proof of these results
relies on a quite general comparison theorem for vertical and horizontal
torsion isomorphisms associated with certain diagrams of chain complexes.
