Linearity Defect and Regularity over a Koszul Algebra

Let A = ⊕ i∈N Ai be a Koszul algebra over a filed K = A0, and *modA the category of finitely generated graded left A-modules. The linearity defect ldA(M) of M ∈ *modA is an invariant defined by Herzog and Iyengar. An exterior algebra E is a Koszul algebra which is the Koszul dual of a polynomial ring. Eisenbud et al. showed that ldE(M) < ∞ for all M ∈ *modE. Improving their result, we show that the Koszul dual A of a Koszul commutative algebra A satisfies the following. • Let M ∈ *modA. If { dimK Mi | i ∈ Z } is bounded, then ldA!(M) < ∞. • If A is complete intersection, then ldA!(M) < ∞ for all M ∈ *modA . • If E = ∧ 〈y1, . . . , yn〉 is an exterior algebra, then ldE(M) ≤ c 2 for M ∈ *modE with c := max{ dimK Mi | i ∈ Z }.