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.
Categories: math.FA