  The linearity defect, introduced by Herzog and Iyengar, is a numerical
measure for the complexity of minimal free resolutions. Employing a
characterization of the linearity defect due to \c{S}ega, we study the behavior
of linearity defect along short exact sequences. We point out two classes of
short exact sequences involving Koszul modules, along which linearity defect
behaves nicely. We also generalize the notion of Koszul filtrations from the
graded case to the local setting. Among the applications, we prove that if
$R\to S$ is a surjection of noetherian local rings such that $S$ is a Koszul
$R$-module, and $N$ is a finitely generated $S$-module, then the linearity
defect of $N$ as an $R$-module is the same as its linearity defect as an
$S$-module. In particular, we confirm that specializations of absolutely Koszul
algebras are again absolutely Koszul, answering positively a question due to
Conca, Iyengar, Nguyen and R\"omer.
