MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

4 added 1 characters in body

I think the idea is that using the splitting principle everything reduces to the first Chern class of line bundles: Chern classes of a general bundle $E$ are symetric functions of $c_1(L_i)$ where $\bigoplus L_i = E$.

If $Q(z)$ is a power series with constant term 1, you can define $K_n$ by the formula: $$\sum K_n(x_1,\ldots,x_n) = \prod Q(z_j)$$ where $x_i$ is the $i$-th elementary symetric function of the variables $z_j$. Hogomeneity corresponds to the fact that the $z_j$ have degree 1.

I think the statement that every multiplicative sequence comes from such a power series is only true in caracteristic 0. A multiplicative sequence with coefficients in $A$ corresponds to a ring homomorphism from the Lazard ring $\mathbb{L} = \Omega^*(pt)$ to $A$ that is to a formal group law $F(t_1,t_2) \in A[[t_1,t_2]]$. There is a natural action of power series $f(z)$ satistfying $f(0) = 0$ and $f(z) f'(0) = 1$ on the Lazard ring; just change of coordinate of formal group laws $(F^f)(t_1,t_2) := f^{-1}(F(f(t_1),f(t_2))$. Now in caracteristic zero, this action is simply transitive: every law is equivalent to the additive one $(t_1,t_2) \mapsto t_1+t_2$ because we can define the logarithm of a law by formally integrating an invariant differential. This should correspond to the fact that every multiplicative sequence is defined by a power series $Q(z) = z/f(z)$.

3 added 6 characters in body

I think the idea is that using the splitting principle everything reduces to the first Chern class of line bundles: Chern classes of a general bundle $E$ are symetric functions of $c_1(L_i)$ where $\bigoplus L_i = E$.

If $Q(z)$ is a power series with constant term 1, you can define $K_n$ by the formula: $$\sum K_n(x_1,\ldots,x_n) = \prod Q(z_j)$$ where $x_i$ is the $i$-th elementary symetric function of the variables $z_j$. Hogomeneity corresponds to the fact that the $z_j$ have degree 1.

I think the statement that every multiplicative sequence comes from such a power series is only true in caracteristic 0. A multiplicative sequence with coefficients in $A$ corresponds to a ring homomorphism from the Lazard ring $\mathbb{L} = \Omega^*(pt)$ to $A$ that is to a formal group law $F(t_1,t_2) \in A[![t_1,t_2]]$A[[t_1,t_2]]$. Now There is a formal natural action of power series$f(z)$satistfying$f(0) = 0$and$f(z) = 1$operate on formal group laws as the Lazard ring; just change of coordinate : of formal group laws$(F^f)(t_1,t_2) := f^{-1}(F(f(t_1),f(t_2))$hence on the Lazard ring. Now in caracteristic zero, this action is simply transitive: every law is equivalent to the additive one$(t_1,t_2) \mapsto t_1+t_2$(because we can define the logarithm of a law by formally integrating an invariant differential)differential. This should correspond to the fact that every multiplicative sequence is defined by a power series$Q(z) = z/f(z)$. 2 added 902 characters in body; deleted 26 characters in body I think the idea is that using the splitting principle everything reduces to the first Chern class of line bundles: Chern classes of a general bundle$E$are symetric functions of$c_1(L_i)$where$\bigoplus L_i = E$. If$Q(z)$is a power series with constant term 1, you can define$K_n$by the formula: $$\sum K_n(x_1,\ldots,x_n) = \prod Q(z_j)$$ where$x_i$is the$i$-th elementary symetric function of the variables$z_j$. Hogomeneity corresponds to the fact that the$z_j$have degree 1. I think the statement that every multiplicative sequence comes from such a power series is only true in caracteristic 0. A multiplicative sequence with coefficients in$A$corresponds to a ring homomorphism from the Lazard ring$\mathbb{L} = \Omega^*(pt)$to$A$that is to a formal group law$F(t_1,t_2) \in A[![t_1,t_2]]$. Now a formal series$f(z)$satistfying$f(0) = 0$and$f(z) = 1$operate on formal group laws as change of coordinate:$(F^f)(t_1,t_2) := f^{-1}(F(f(t_1),f(t_2))$hence on the Lazard ring. Now in caracteristic zero, this action is simply transitive: every law is equivalent to the additive one$(t_1,t_2) \mapsto t_1+t_2$(because we can define the logarithm of a law by formally integrating an invariant differential). This should correspond to the fact that every multiplicative sequence is defined by a power series$Q(z) = z/f(z)\$.

1