  For a tuple $A=(A_1,\ A_2,\ ...,\ A_n)$ of elements in a unital algebra
${\mathcal B}$ over $\mathbb{C}$, its {\em projective spectrum} $P(A)$ or
$p(A)$ is the collection of $z\in \mathbb{C}^n$, or respectively $z\in
\mathbb{P}^{n-1}$ such that the multi-parameter pencil
$A(z)=z_1A_1+z_2A_2+\cdots +z_nA_n$ is not invertible in ${\mathcal B}$.
${\mathcal B}$-valued $1$-form $A^{-1}(z)dA(z)$ contains much topological
information about $P^c(A):=\mathbb{C}^n\setminus P(A)$. 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 ${\mathcal B}$ does a similar job. In fact, a Chen-Weil type map
$\kappa$ from the cyclic cohomology of ${\mathcal B}$ to the de Rham cohomology
$H^*_d(P^c(A),\ \mathbb{C})$ is established. As an example, we prove a closed
high-order form of the classical Jacobi's formula.

Functional Analysis