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By R. L. Goodstein

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1 Let L / K be a Galois extension of number fields. The second Chinburg invariant (Qo(L/K,2) in our notation) is constructed from a projective Z[G(L/K)]module, X, and the classical fundamental classes of local class field theory (see [I311 Chapter 7). 17 in place of the classical one to construct Galois module structure invariants %-1(LIK, 2) E CC(Z[G(L/K)]) lying in the class-group of Z[G(L/K)]. These invariants measure the Galois module structure of algebraic K-groups in dimensions 2r - 1 and 2r - 2 and are expected to be connected with the values of the L-functions at s = 1 - r (or, equivalently, s = r).

Md) E Ker(X) then ml = vr-lmd,m2 = vr-lml,. . and m d ( l v('-')~) = 0 SO that X is injective. Also we can solve the equation X(ml, . . ,md) = (xl, . . ,xd) by choosing xd, X I , . . ,xd-2 successively but the equations are consistent if and only if - Therefore, if 7r2 is defined to depend only on the coordinates in the second summand and to be given by while a(fi(g% (0, I))) = a(gG1 8 (0, vr-l)) (0, Vr-1) 8 (vr-lui-l St, vr-l) = 92-1 8 a"' = gi-l and these are equal since vr-lui-l = quui-l r qui (modulo t).

Xd) = (FO(xd),xl,.. ,xd-1). Since K-theory is additive there is an induced isomorphism > for each s 0. This isomorphism induces a G(L/K)-action on the right-hand group, induced by the Galois action on L. Since the F'robenius acts on the other factor, KO,in the tensor product we may define a Z[G(L/K)]-homomorphism by the same formula as before. 2). 64 Chapter 3. 6 Suppose that g E G(L/K) satisfies (gl W) = (FjI W) for some 0 5 j 5 d - 1. K ~ (LO) ~ is-given ~ by Then the action of g on (ul, u2, .

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Boolean algebra by R. L. Goodstein


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