Cancellation of lattices and approximation properties of division algebras
Aiichi Yamasaki · Kyoto journal of mathematics · 1996
tie n t fie ld K .L e t A b e an R-order.In this general setting, it is proved in [3] th a t Roiter-Jacobinski type D ivisibility Theorem holds for A -lattices.A s a consequence, fo r a A -lattice L, the following two cancellation properties are equivalent.(A s w as pointed o u t in [3], putting I' : , ----EnciA L and B : =K T , th ere is an in tim a te c o n n e c tio n b e tw e e n c a n c e lla tio n p ro p e rty and the approxim ation property of the group of Vaserstein E(./ j ) in the idele topology of / 1 3 N x, of which precise definitions will be recalled in §1.H ere w e only indicate, R N : = IIR p, th e d ire c t p ro d u c t of p-adic comple- tio n s o v e r a ll m axim al ideals o f R, XI \ : = M RR \ f o r a n y R-alegbra M, and (C) : =- for any ring C 1. O u r first remark is Proposition 1 (proof in 1 .5 ).The property (c') for L is equivalent with the following property (c") of F (c") 7 É -( / ) CPx.Bx as subsets of 0 . 1 .W e shall consider, fo r any finite dimensional K-algebra B , th e following three approximation properties over R , in the idele topology of .r EÎ".(a) Strong approximation property E (B ) is dense in T E (a') B x -approximation property E ) is contained in the closure of 13'.(a") R'' B'<-approximation property E TO is contained in the closure of /7'W .T here are th e obvious implications (a) (a') (a " ).O ur second (rather