Spectral Sequences for Noncommutive Serre Fibrations

We prove in this paper a noncommutative version of Leray Theorem and then Serre Spectral Theorem for noncommutative Serre fibrations. We prove that for NC Serre fibration there are converging spectral sequences with E2 terms as E2 p,q = HPp(B,A;HPq(A)) =⇒ HPp+q(B) and E 2 p,q = HPp(B,A;Kq(A)) =⇒ Kp+q(B). Introduction The ideas of using spectral sequences to operator algebras was started in [C]. In that work the author constructed for an arbitrary algebras some CCRor CTcomposition series and tried to define the structure of the algebras through Busby invariants of extensions. This reduced to the ideas of using spectral sequences. However, the technique of spectral sequence meets some difficulties, for example convergence problem. It is well-known that for every short exact sequence of ideal and algebras 0 −−−→ A −−−→ B −−−→ C −−−→ 0 where A is some ideal in B and C ∼= B/A is the quotient algebra, there is a natural spectral sequence converging to the cyclic periodic homology with E2-term E p,q = HPp(B,A; HPq(A)) =⇒ HPn(B), with n = p+ q The work was supported in part by the National Project of Research in Fundamental Sciences of Vietnam.