On the structure of 𝑃(𝑛)_{∗}𝑃((𝑛)) for 𝑝=2

We show that P ( n ) ∗ ( P ( n ) ) P(n)_\ast (P(n)) for p = 2 p=2 with its geometrically induced structure maps is not an Hopf algebroid because neither the augmentation ϵ \epsilon nor the coproduct Δ \Delta are multiplicative. As a consequence the algebra structure of P ( n ) ∗ ( P ( n ) ) P(n)_\ast (P(n)) is slightly different from what was supposed to be the case. We give formulas for ϵ ( x y ) \epsilon (xy) and Δ ( x y ) \Delta (xy) and show that the inversion of the formal group of P ( n ) P(n) is induced by an antimultiplicative involution Ξ : P ( n ) → P ( n ) \Xi :P(n)\rightarrow P(n) . Some consequences for multiplicative and antimultiplicative automorphisms of K ( n ) K(n) for p = 2 p=2 are also discussed.