Chern classes of multiplicative direct image of vector bundles (Q1174629)

Let \(f:X\to Y\) be a finite étale morphism of nonsingular algebraic varieties. Given a vector bundle \(E\) on \(X\) we have two vector bundles \(f_ *E\) and \(N_ fE\) on \(Y\). \(f_ *E\) is the direct image of \(E\) and \(N_ fE\) is its multiplicative analogue. The fibres of them over a geometric point \(y\) of \(Y\) are given as follows: \[ (f_ *E)_ y=\bigoplus_{x\in f^{-1}(y)}E_ x,\quad (N_ fE)_ y=\bigotimes_{x\in f^{-1}(y)}E_ X. \] The purpose of this paper is to study the Chern polynomial \(c(N_ fE)\). The atuthor gives a formula expressing \(c(N_ fE)\) as a function of \(c(E)\). As in the case of \(c(f_ *E)\), the formula involves operations in Chow rings such as pull- back, push-forward, multiplicative transfer for various morphisms \(g\) between finite etale \(Y\)-schemes. But unlike the case of \(c(f_ *E)\) [see \textit{W. Fulton} and \textit{R. MacPherson}, Ann. Math., II. Ser. 125, 1-92 (1987; Zbl 0628.55010)] it involves certain implicit polynomials of \(c_ i(E)\) which are called monomial symmetric functions in the book by \textit{I. G. Macdonald}: ``Symmetric functions and Hall polynomials'' (1979; Zbl 0487.20007).