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Additional info for A crash course on Kleinian groups; lectures given at a special session at the January 1974 meeting of the American Mathematical Society at San Francisco

Example text

3, D(S) is a division ring. 13, EndAs(VS ) is a finitely generated S-algebra and J(S)EndASCVs) £ J(EndAsCVs)), so D(S) is a finite-dimensional S-algebra, S = SjJ(S).

Choose n so vp " = 1, and set T = {c E kG I cP' E S, some integer i > O}. 1 (ii), (uv - uy" == (uvy" - up" = uP"v P" - up" = up" - up" = 0 (mod S), 298 Group Representation Theory so uv - U E T. S s; T, so every element of kG is congruent modulo T to one of Xl' ... I(iii) we conclude m Step 4 m ~ ~ dim kG - dim T = N. N, so (2) holds. Proof With T as in Step 3, it is enough to show that Xl' ... , Xm are linearly independent modulo T. If PI' . . , am E k satisfy L~ I aix i E T, we shall show all a i = 0.

A = EB t Ae i , so X s is an indecomposable Aet-module. In fact, A K = EB t (AeJK' so if Y is an irreducib'e (Aet)rmodule then Y is an irreducible AK-module and Y;::;. )K, some X s E Bt. Also, the Vi E B t are a complete set of principal indecomposables of Ae t , so the {X,} and {L;} index rows and columns of the decomposition matrix of Bt. The composition series of 1's as A-module is its composition series as Ai't-module, so these several remarks show that (3) holds. 3. L7'= L7'= §49 Finite Extensions of Complete Local Domains Definitions Let R be a complete commutative Noetherian integral domain, such that RjJ(R) is Artinian and J(R) = (n) is a principal ideal.

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