Categories by Aristotle

By Aristotle

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Finally, suppose we start with an additive extension E . Choose splittings r and s in the usual way and define σ(u, v) = r (s (u + v) − s (u) − s (v)). We know that q : E → − G is a homomorphism with q s = 1 so q (s (u + v) − s (u) − s (v)) = 0 and j r = 1 − s q so we see that j σ(u, v) = (1 − s q )(s (u + v) − s (u) − s (v)) = s (u + v) − s (u) − s (v). From this it follows that j σ satisfies the symmetric cocycle conditions and j is a monomorphism so σ satisfies the conditions, so σ ∈ Z(G). We define f : Eσ → − E by f (a, u) = j (a) + s (u).

As fq vq = q we see that q −1 fq vq is idempotent, and therefore the operator −1 fq vq is also idempotent. 6 that these idempotents commute, q = 1−q and thus that is also idempotent. It is clear that if γ is p-typical then (γ) = γ. Conversely, we have fq q = 0 and so fq = 0 for all q = p, so (γ) is always p-typical. This shows that gives a natural retraction Curves(G) → Curvesp (G). 12. Consider the case where G = Ga × X and γ corresponds to a series g(t) = i ci ti . 5 we see that fq γ = 0 iff cjq = 0 for all j, and thus that γ is p-typical iff ck = 0 whenever k is i not a power of p.

For this, we may assume that C = X × A1 . The sections are then the maps of the form σ(a) = (a, u(a)) where u : X → − A1 , in other words u ∈ Nil(OX ). We also have Jσ = {f (x) ∈ OX [[x]] | f (u) = 0}. It is easy to see that this is generated by the W-polynomial x − u. 8 that every ideal in D1+ (C) is generated by a unique W-polynomial of degree one, and these clearly all have the form x − u. The proposition follows. 13. Let C be a formal curve over a scheme X. Given a ring R and a point a ∈ X(R) we have a formal curve Ca = spec(R) ×X C over spec(R), and we define + Div+ n (C)(R) = {(a, D) | a ∈ X(R) and D ∈ Dn (Ca )}.

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