Projective spectrum and cyclic cohomology

For a tuple A = (A(1), A(2), ... , A(n)) of elements in a unital algebra B over C, its projective spectrum P (A) or p (A) is the collection of z is an element of C-n, or respectively z is an element of Pn-1 such that the multi-parameter pencil A(z) = z(1)A(1) + z(2)A(2) + ... + z(n)A(n) is not invertible in B. B-valued 1-form A(-1)(z)dA(z) contains much topological information about P-c (A) := C-n \ P (A). In commutative cases, invariant multi-linear functionals are effective tools to extract that information. This paper shows that in non-commutative cases, the cyclic cohomology of B does a similar job. In fact, a Chen-Weil type map kappa from the cyclic cohomology of B to the de Rham cohomology H-d*(P-c(A), C) is established. As an example, we prove a closed high-order form of the classical Jacobi's formula. (C) 2013 Elsevier Inc. All rights reserved.