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By Strömberg G.

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Let R be a commutative noetherian semilocal ring and let A , B be R-algebras which are finitely generated Rmodules. Pu t R = R / J ( R ) , A = A / J ( A ) and B = B / J ( B ) Generalities 656 Then the following properties hold : (i) ( A @ R B ) / J ( A@ R 2 ( A @ R B ) / J _ ( A @ R B) (ii) If R is a field and A is a separable R-algebm, then e) ( A @ R B ) / J ( A@ R B ) g A @R B Proof. Since A / J ( R ) A is artinian, we have J(A)" C_ J ( R ) A for some integer n >_ 1. Let I m ( J ( A )@ R B ) be the canonical image of J ( A ) @ R B in A @ R B.

W, is an F-basis of I n v ( V ) . Proof. It is clear that w1,.. ,w, are F-linearly independent elements of I n v ( V ) . Furthermore, if V = A;v; E I n v ( V ) , X i E F, and vi,Vj lie in the same orbit, then A; = A j . So the lemma is true. 2. Let F be an arbitrary field, let XI,.. ,x m be all irreducible F-characters of G and let K1,.. ,Km be subsets of G which satisfy the following two conditions : Generalities 638 (a) Each xi is constant on each Kj. (b) For any given g E G, there exists j = j ( g ) E (1,.

ZT are F-linearly independent. 5, X I , . . ,xr are F-linearly independent, it follows that each X i = 0, as desired. (ii) Write u = 1u,x with ux E F , 2 E G. Then ug-' = 1u z ( x g - ' ) and so Xi(21g-l) = C uzXi(2g-l) = C uxXi(g-'x) xEG C uxXi(Y-') = ( w ) g , xEG = xY=s as required. (iii) By (ii), we must show that u annihilates V, if and only if x ; ( u g - ' ) = 0 for all g E G. If u annihilates V,, then so does ug-' and hence x ; ( u g - ' ) = 0 for all g E G. Conversely, suppose that x ; ( u g - ' ) = 0 for all g E G.

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A Determination of the Solar Motion and the Stream Motion Based on Radial Velocities and Absolute Ma by Strömberg G.


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