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 $\mathbb{M}$ on $M$. We study the Castelnuovo-Mumford regularity of $gr_{\mathbb{M}}(M)$ 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_{\mathbb{M}}(M)) <\infty implies \reg_R(M)<\infty 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)<\infty implies $R$ is Koszul.Comment: 15 pages, to appear in Journal of Commutative Algebr