REPRESENTATION THEORY AND RELATED TOPICS SEMINAR

Luchezar Avramov

ABSTRACT:  It will be shown that the sum of the Loewy length of the homology modules of a finite free complex F over a local ring R is bounded below by an invariant that measures the singularity of R.  When R is the closed fiber of a flat local homomorphism, this result yields unexpected lower bounds on the Loewy length of R-modules of finite projective dimension.  Another application of the theorem is to be the case when R is the group algebra of an elementary abelian group, in which case one recovers results of 
G. Carlsson and of C. Allday and V. Puppe.  The arguments use numerical invariants of objects in general triangulated categories, introduced in the paper and called levels.  One such level models projective dimension; a lower bound for this level implies the New Intersection Theorem for commutative local rings containing fields.  The lower bound on the Loewy length of the homology of F is sharp in general.  Under additional hypothesis on the ring R, stronger estimates are proved for Loewy lengths of modules of finite projective dimension. The talk is based on joint work with R.-O. Buchweitz, S. Iyengar, and C. Miller.