However, we show that McCarthy's theory cannot be both, compatible with localizations and invariant under functors inducing equivalences in the derived category.
This is our motivation for introducing a new theory for which all three properties hold: extension, invariance and localization. Thanks to these properties, the new theory can be computed explicitly for a number of categories of modules and sheaves.
This work goes back to a question by P. Polo. I thank him for the interest he has continued to take in the subject. I am grateful to the referee for his thorough reading of the manuscript. His remarks and questions have been a great help and encouragment in preparing the final version of the article.