On Complexes Over a Ring and Restricted Cohomology Groups

V 1MAtHEMATICS: B. ECKMA NNI7 tribution on T, with v(T) = v < 1.Then the difference of potentials fJd;z/ MP -fTdv/MP is unbounded above.In (5) the density p(P) is summable.Hence the boundedness of its potential in the neighborhood of s depends merely on the values of p in a neighborhood of s.Accordingly, with respect to the boundedness of the right-hand member of (5), if ,u(e) were not identically zero, we could discard temporarily as much of the distribution p(P) as We pleased, outside that neighborhood, and assume that fT,p(P)dP < fsd,Ap.But then, by lemma 3, the right-hand member of (5) would be unbounded above.Since this is not the case, we must have ,u(e) identically zero.Hence V(M)-h(M) =-, p)dP J MP for M in T + s.The function p(P), according to its definition, is regular in the neighborhood of T*.We may therefore apply Gauss's theorem to the univalent function V(M) -h(M).Finally then /V-dP = 47r fp(M)dM= JV 2VdM` which is the equation ( 4) to be proved.