A Leray spectral sequence for noncommutative differential fibrations (Q2836546)

The Leray spectral sequence for the cohomology of a sheaf over the total space of a fibration of smooth manifolds is reformulated in terms of the super-commutative (or graded-commutative) algebras of differential forms. For a possibly non-commutative algebra \(A\), a differential calculus \((\Omega^*A,d)\) is a graded associative differential algebra with \(\Omega^0A=A\) and some additional properties mimicking the algebra of exterior differential forms on a smooth manifold. A noncommutative fibration consists of an algebra map between two algebras \(A\) and \(B\), an extension of this map to differential calculi \((\Omega^*A,d)\) and \((\Omega^*B,d)\), under the requirement that \(\Omega^*A\) is in some noncommutative sense generated by \(\Omega^*B\) and by the associated vertical differential calculus. In this setting, the authors describe a Leray spectral sequence abutting to the cohomology of \((\Omega^*A,d)\). In the last section, the authors discuss a differential calculus on the group algebra on the Heisenberg group of upper unitriangular matrices with integer coefficients. The paper is based on the PhD thesis of the second named author under the supervision of the first.