Linearity defect and regularity over a Koszul algebra - Yanagawa, Kohji (Mathematica Scandinavica, 2009)
Let $A = \bigoplus_{i\in \mathsf{N}}A_i$ be a Koszul algebra over a field $K = A_0$, and $*\operatorname{mod} A$ the category of finitely generated graded left $A$-modules. The linearity defect $ld_A(M)$ of $M \in *\operatorname{mod} A$ 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 $ld_E(M) < ∞$ for all $M \in *\operatorname{mod} E$. Improving this, we show that the Koszul dual $A^!$ of a Koszul commutative algebra $A$ satisfies the following. • Let $M \in *\operatorname{mod} A^!$. If $\{\dim_K M_i \mid i \in {\mathsf Z}\}$ is bounded, then $ld_{A^!}(M) < ∞$. • If $A$ is complete intersection, then $reg_{A^!}(M) < ∞$ and $ld_{A^!}(M) < ∞$ for all $M \in *\operatorname{mod} A^!$. • If $E=\bigwedge \langle y_1, …, y_n\rangle$ is an exterior algebra, then $ld_E(M)≤ c^{n!} 2^{(n-1)!}$ for $M \in *\operatorname{mod} E$ with $c := \max \{\dim_K M_i \mid i \in{\mathsf Z}\}$.