Jan-Erik Roos: Homological properties of local rings, far away from regular rings.

Let (Q,m) be a regular local commutative noetherian ring with maximal ideal m. It is well-known that all Q-modules M have a finite free resolution. Let now I be an ideal in Q and assume that I is included in m^2, and let R =Q/I. Let 0 ---> F_c ---> ... ---> F_1 ---> F_0 --->Q/I ---> 0 be a minimal free Q-resolution of R=Q/I. Then c measures how far away the ring R is from being regular and it is called the embedding codepth of R. There are lots of papers about what happens when c <= 3, and there is even a recent paper which handles parts of this case by computer calculations by Christensen-Veliche in Journal of Software for Algebra and Geometry vol 6, 2014 (Local rings of embedding codepth 3: A classification algorithm), and which cites some earlier papers on this subject. I will talk about the strange new phenomena that occur when c > 3, that I will analyze using results by L. Avramov, F. Moore, C. Löfwall, J. Backelin, V. Ufnarovski and others.