On derived projective limits Original Research Article

For any projective system of bounded below (cochain) complexes {Ci*}, there exists two convergent spectral sequences with the same limit, involving the derived projective limits of the cohomologies of the Ci*. Assuming an hypothesis which will be satisfied in the given applications, we show how one can, in each dimension of the complexes Ci*, replace the two previous spectral sequences by a long exact sequence linking the cohomology in this dimension of the projective limit of the Ci*, to the derived projective limits of the cohomologies of the Ci* in the same dimension. This may be used in several directions. We obtain results concerning the cohomology (with values in a module) of an inductive limit of complexes, in terms of the pure injective dimension of the module, or of the cardinality of the base ring, or of its global dimension. This may be applied to the case of the twisted cellular cohomology of a filtering family of subcomplexes of a CW-complex. Similarly, if one considers a derived functor ExtAq(_, M) of an extension functor, and an inductive system {Ci} of modules, one obtains results relative to the behavior of this derived functor with respect to the inductive limit of the Ci, assuming some conditions on the module M, the ring A, or the modules Ci.