REGULARITY AND LINEARITY DEFECT OF MODULES OVER LOCAL RINGS

Given a finitely generated module M over a commutative local ring (or a standard graded k-algebra) (R, m, k), we detect its complexity in terms of numerical invariants coming from suitable m-stable filtrations M on M. We study the Castelnuovo-Mumford regularity of gr(M)(M) and the linearity defect of M, denoted ld(R)(M), through a deep investigation based on the theory of standard bases. If M is a graded R-module, then reg(R)(gr(M)(M)) < infinity implies reg(R)(M) < infinity and the converse holds provided M is of homogenous type. An analogous result can be proved in the local case in terms of the linearity defect. Motivated by a positive answer in the graded case, we present for local rings a partial answer to a question raised by Herzog and Iyengar of whether ld(R)(k) < infinity implies R is Koszul.