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abelian algebraically closed groups assume Banach space bundle C*-algebra Cartan Cohen-Macaulay cohomology compact complete intersection construction contained convex sets Corollary corresponding countable curvature defined definition denote diffeomorphism dimension dimensional Dirichlet character divisor e.c. groups element equivalent exact sequence exists fibrations finite follows formula function Gauss map geodesic given group G hence holomorphic homotopy hyper-regular ideals identity implies induced integer isometry isomorphism L-ideal L-projection lattice Lemma Let G Lie algebra linear log H log+ manifold Math maximal meromorphic module monomials neighborhood nonnegative norm normal notation Note operator Poincare duality prime proof of Theorem Proposition prove regular neighborhood Remark representation resp result Riemannian ring satisfies Section semi-simple semigroup simplex simplicial solutions spectral sequence strictly positive structure subgroup of G subset subspace Suppose Theorem theory tion topology trivial unique vector field weak zero