Formal groups [Lecture notes] by Neil P. Strickland

By Neil P. Strickland

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Define a map θM : k≥2 M → − Z(G; M ) by θ(m) = k mk τk (one can k check that this converges, because δ(x ) = 0 + O(k) for all k). 19 that θM 41 is an isomorphism when IM = 0. 2, we deduce that θR/I k is iso for all k and thus that θR is iso. This implies that Z(G) = R{σp (F ), . . , σpn−1 (F )} ⊕ δ(C(G)) and thus that Ext(G, Ga ) = Z(G)/δ(C(G)) = R{σp , . . , σpn−1 }. 22. Given a formal group scheme H of dimension d over X, we define J = JH = {f ∈ OH | f (0) = 0} and ωH = J/J 2 and tH = HomOX (ωH , OX ).

If OX is torsion-free this implies easily that η itself is additive. In particular, as the Lazard ring is torsion-free we see that η is additive in the case of the universal FGL, and it follows by base change that it is additive for any formal group. 17. Let γ be any p-typical basic curve on G, inverse to a coordinate x. Let η be the canonical additive curve such that η ∗ d0 x = p d0 t. Then there is a unique series of elements uk ∈ OX (for k > 0) such that k γ(uk tp ). p γ(t) = η(t) + k>0 Proof.

As ξk = 0 for k < 3 and ξ3 = 1 it is immediate from the definitions that χ(x0 , x1 , x2 ) = x0 + x1 + x2 + O(2). 11. 8) then χ(a0 , a1 , a2 ) = 0 and thus   x0 x1 x2 1 1  = 0. det  1 ξ(x0 ) ξ(x1 ) ξ(x2 ) Proof. We need to show that φ(a0 )φ(a1 )φ(a2 ) = 1 in Q(C), or equivalently that (1 − a0 /x)(1 − a1 /x)(1 − a2 /x) ∈ U A× . Consider the series h(x) = χ(a0 , a1 , x) ∈ A. As χ(x0 , x1 , x2 ) = x0 + x1 + x2 + O(2), we see that h is a W-series of degree one, and h(a2 ) = 0 so h(x) = v(x)(x − a2 ) for some v ∈ A× .

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