This is joint work with David Chataur.
  
   There exist a few distinct ways to construct fibrewise localizations, but most of them are very specific to the category of spaces (using the space of self-equivalences of the base, or decomposing the total space as a diagram over the simplices of the base). As one should not expect fibrewise localizations to exist in an arbitrary model category (Hirschhorn shows it fails for pointed spaces!), we impose a few conditions, the major assumption being that the analogue of Mather's cube theorem holds. We construct then a fibrewise localization which we can perform for example in the category of differential graded algebras over an admissible operad. This allows us to achieve further computations of the lower cyclic and Hochschild homology groups of an associative rational algebra.