4.4. CONTROLLING SUITABLE ODD LOCALS 625Assume (1) fails. Then B := CBH(M+/02(M+)) =/:-1. Observe that we have
the hypotheses of 4.4.13 with T, BH, Bin the roles of "S, B, A", so BT= TB by
4.4.13.2. Hence parts (1) and (2) of Hypothesis 4.4.1 are satisfied, with T in the
role of "T+". Thus by 4.4.5, ·M+ = LB02(M+), where LB:= CM+(B)^00 •
Next since H ~ Ca(Z), Cv 0 (T) =Zn Vo ~ Ca(B), so Vo= ((Zn Vo)M+) =
((ZnV 0 )LE) ~ C 0 (B) by (a). Therefore part (3) of Hypothesis 4.4.1 is also satisfied,
with Vin the role of "VB", so that we may apply Theorem 4.4.3. By (a), Vis not
an FF-module for LB/0 2 (LB), which rules out subcase (i) of case (2) of Theorem
4.4.3. By 4.4.13.1, N H(B) f:_ M, ruling out case (1) of Theorem 4.4.3. Thus subcase
(ii) or (iii) of case (2) of Theorem 4.4.3 must hold, and these are conclusions (2)
and (3) of Theorem 4.4.14. D