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