Linearity Defect and Regularity over a Koszul Algebra

Abstract

Let A be a Koszul algebra, and $modA$ the category of finitely generated graded left A-modules. The "linearity defect" ld_A(M) of $M\in modA$ is an invariant defined by Herzog and Iyengar. An exterior algebra E is a Koszul algebra which is the Koszul dual S^! of a polynomial ring S. Eisenbud et al. showed that $l{d}_E(M)<\infty$ for all $M\in modE$. Improving their result, we show the following (and many other facts): (*) If A is a Koszul complete intersection, then $re{g}_{A^{!}}(M)<\infty$ and $l{d}_{A^{!}}(M)<\infty$ for all $M\in mod{A}^{!}$. (**) There is a uniform bound of $ld(J)$, where J is a graded ideal of E.

13 pages. Several proofs have been simplified, and comments on known results have been revised