Projective Spectrum and Cyclic Cohomolgy

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