Linear operators acting on a finite-dimensional vector space
are completely classified by their Jordan normal form. On the other hand, it is
very rarely possible to describe all actions of several operators in a similarly
exhaustive way. For this reason algebraists have been looking for ways to
organize representations, or modules, with prescribed properties. A relatively
recent technology associates geometric objects to modules over rings by using
homological algebra and algebraic geometry. Some themes of current research
will be explained through examples.