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