Projective spectrum and cyclic cohomology

For a tuple A=(A1,A2,…,An)A=(A1,A2,…,An) of elements in a unital algebra BB over CC, its projective spectrum  P(A)P(A) or p(A)p(A) is the collection of z∈Cnz∈Cn, or respectively z∈Pn−1z∈Pn−1 such that the multi-parameter pencil A(z)=z1A1+z2A2+⋯+znAnA(z)=z1A1+z2A2+⋯+znAn is not invertible in BB. BB-valued 1-form A−1(z)dA(z) contains much topological information about Pc(A):=Cn∖P(A)Pc(A):=Cn∖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 BB does a similar job. In fact, a Chen–Weil type map κ   from the cyclic cohomology of BB to the de Rham cohomology Hd⁎(Pc(A),C) is established. As an example, we prove a closed high-order form of the classical Jacobiʼs formula.