Linearity defect and regularity over a Koszul algebra

Authors

  • Kohji Yanagawa

DOI:

https://doi.org/10.7146/math.scand.a-15095

Abstract

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 $\mathrm{ld}_A(M)$ 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 $\mathrm{ld}_E(M) < \infty$ 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 $\mathrm{ld}_{A^!}(M) < \infty$.
  • If $A$ is complete intersection, then $\mathrm{reg}_{A^!}(M) < \infty$ and $\mathrm{ld}_{A^!}(M) < \infty$ for all $M \in *\operatorname{mod} A^!$.
  • If $E=\bigwedge \langle y_1, \ldots, y_n\rangle$ is an exterior algebra, then $\mathrm{ld}_E(M)\leq c^{n!} 2^{(n-1)!}$ for $M \in *\operatorname{mod} E$ with $c := \max \{\dim_K M_i \mid i \in{\mathsf Z}\}$.

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Published

2009-06-01

How to Cite

Yanagawa, K. (2009). Linearity defect and regularity over a Koszul algebra. MATHEMATICA SCANDINAVICA, 104(2), 205–220. https://doi.org/10.7146/math.scand.a-15095

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Articles