Jan-Erik Roos: Bass series of local rings. Computational aspects and conjectures.

In 1963 Hyman Bass published "On the ubiquity of Gorenstein rings". This paper quickly became one of the most cited paper in algebra in the 1980:s. Let (R,m) be a local commutative noetherian ring with maximal ideal m and residue field k=R/m. The ring is a Gorenstein ring if it has a finite injective dimension as a module over itself, i.e. if and only if the vector spaces  Ext^i_R(k,R) are 0 for big i. If R is not Gorenstein the generating series of the dimensions of these vector spaces is called the Bass series of R. These series give a lot of information about the singularity of the local ring (R,m). Recently (feb 2012) Luchezar Avramov has written a paper about the behaviour of these series for local rings of embedding codepth <= 3 (will be defined). I will go further and also stress the computational aspects of the theory that leads to reasonable conjectures.  Most definitions will be given from the beginning.