Commutative diagram

Outside the diagonal, we have the following commutative diagram ... 

To finish the proof we simply remark that the isomorphism of [BBM] 5.2.7.1 induces a commutative diagram... 

Suppose now we have a group scheme G ObC(n)s 0 and an embedding of G into an abelian scheme A over Sq. Let B = A/G. Then the following commutative diagram... 

Remark 9.7. It follows from the proof of the lemma that the schemes S X S and S x S x 5 S satisfy the conditions of 9.1. Further we get various commutative diagrams involving lifts of Frobenii and projection maps. This will allow us to use descent arguments. 

The question whether ir Ms(G) — Mt(Gt0) is an isomorphism for general G ObC(n)s0 is local in the Zariski topology on S, so we may assume that G is embedded into an abelian scheme A over Sq. Put B = A/G. The result follows from the above and the commutative diagram with exact rows... 

Proof. As usual we may assume that G is embedded into an abelian scheme A over So We put B = AjG and look at the commutative diagram... 

Consider the following commutative diagram with exact rows... 

Let G ObC(l)s0 have Fa = 0 and H ObC(l)s0 arbitrary. As we only have to show the assertion Zariski locally on S we may assume that we have an exact sequence 0 — H -+ H — 2 with Hi 6 ObjBT(l)s0. We see from the commutative diagram with exact rows... 

From [IL] Theorem 4.4 we get a truncated Barsotti-Tate group H on S 0 such that i H G. The commutative diagram... 

The commutative diagram expressing this relationship is in Figure 5.1. 

Figure B.l. A commutative diagram for the proof of Proposition B.2. The functions tti and 712 are the natural projection functions. The function i is the inclusion function any element of SU(V) is automatically an element of C(V).

Figure B.2. Commutative diagram for proof that every projective unitary representation of 50(3) comes from a linear representation of 5(7(2).

One implication is trivial. Suppose that f is smooth at all the closed points of X and let xcX be any point. Let y c X be a closed point such that 0 fX is a localization of 0 >y We have a commutative diagram... 

Assume that for an integer n we have a commutative diagram... 

Since (2) is the sheafification of (4), we obtain a commutative diagram with exact rows ... 

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