Consequences of the transfer principles
Throughout this chapter we assume that V is a locally finite Abelian variety. We will show that if V satisfies the (1,2) transfer principle, then the subvariety V1 = V(S1) defined in Chapter 1 is strongly Abelian. On the other hand, if V is assumed to satisfy the (2,1) transfer principle, then it will be shown that the subvariety V2 = V(S2) is affine. Thus if V is structured, then we will be able to conclude with the help of Theorem 8.9 and Theorem 1.3 that V is the join of an affine subvariety with a strongly Abelian one.
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