Example 42.43.6. For every p \geq 1 there is a unique homogeneous polynomial P_ p \in \mathbf{Z}[c_1, c_2, c_3, \ldots ] of degree p such that, for any n \geq p we have
in \mathbf{Z}[x_1, \ldots , x_ n] where s_1, \ldots , s_ p are the elementary symmetric polynomials in x_1, \ldots , x_ n, so
The existence of P_ p comes from the well known fact that the elementary symmetric functions generate the ring of all symmetric functions over the integers. Another way to characterize P_ p \in \mathbf{Z}[c_1, c_2, c_3, \ldots ] is that we have
as formal power series. This is clear by writing 1 + c_1 + c_2 + \ldots = \prod (1 + x_ i) and applying the power series for the logarithm function. Expanding the left hand side we get
In this way we find that
and so on. Since the Chern classes of a finite locally free \mathcal{O}_ X-module \mathcal{E} are the elementary symmetric polynomials in the Chern roots x_ i, we see that
For convenience we set P_0 = r in \mathbf{Z}[r, c_1, c_2, c_3, \ldots ] so that P_0(\mathcal{E}) = r(\mathcal{E}) as a bivariant class (as in Remarks 42.38.10 and 42.43.5).
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