By Richard Pink

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**Extra resources for Finite group schemes**

**Example text**

5. For every morphism u : A1Z × A1Z −→ A1Z there exists a unique morphism v : WZ × WZ −→ WZ such that for all ≥ 0 : Φ ◦v = u◦(Φ ×Φ ). Proof. 3) there exist unique v = (v0 , v1 , . ) with vn ∈ Z[ p1 ][x0 , . . , xn , y0, . . , yn ] satisfying the desired relations. It remains to show that vn ∈ A := Z[x0 , . . , y0 , . ]. Since Φ0 (x) = x0 , this is clear for v0 = u(x0 , y0 ). So fix n ≥ 0 and assume that vi ∈ A for all i ≤ n. For any sequence x = (x0 , x1 , . ) we will abbreviate xp = (xp0 , xp1 , .

For the following we shall again assume that k is a perfect field. 1. Let R be a complete noetherian local ring with perfect residue field k of characteristic p and maximal ideal m. Then there exists a unique section i : k → R with the equivalent properties: (a) i(xy) = i(x)i(y) for all x, y ∈ k, −n n (b) i(x) = limn→∞ s(xp )p for any section s and any x ∈ k. The image i(x) is called the Teichm¨ uller representative of x. Proof. The main point is to show that the limit in (b) is well-defined. First notice that for all n ≥ 1 and x, y ∈ R we have x ≡ y mod mn xp ≡ y p mod mn+1 .

Every commutative finite group scheme of local-local type can be embedded into (Wnm )⊕r for some n, m, and r. Proof. To prove this by induction on |G|, we may consider a short exact sequence 0 → G → G → αp → 0 and assume that there exists an embedding ψ = (ψ1 , . . , ψr ) : G → (Wnm )⊕r . 2, determine an extension m+1 ⊕r of the composite embedding ivψ : G → (Wn+1 ) to a homomorphism m+1 ⊕r G → (Wn+1 ) . The direct sum of this with the composite homomorphism m+1 m+1 ⊕r+1 G αp = W11 → Wn+1 is an embedding G → (Wn+1 ) .