Linearity Defect and Regularity over a Koszul Algebra
Abstract
Description
Let A be a Koszul algebra, and $mod A$ the category of finitely generated graded left A-modules. The "linearity defect" ld_A(M) of $M \in mod A$ 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 $ld_E(M) < \infty$ for all $M \in mod E$. Improving their result, we show the following (and many other facts):
(*) If A is a Koszul complete intersection, then $reg_{A^!} (M) < \infty$ and $ld_{A^!} (M) < \infty$ 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
13 pages. Several proofs have been simplified, and comments on known results have been revised