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The normal cohomology functor ExtN is introduced from the category of all normal Hilbert modules over the ball algebra to the category of A(B)-modules. From the calculating of ExtN -groups, we show that every normal C(∂B)-extension of a normal Hilbert module (viewed as a Hilbert module over A(B)) is normal projective and normal injective. It follows that a natural isomorphism is established between Hom of normal Silov modules and that of their quotient modules, which is a new lifting theorem of normal Silov modules. Finally, these results are applied to the discussion of the rigidity and extensions of Hardy submodules over the ball

Normal Hilbert modules over the ball algebra A(B)