Resumen de Linearity defect and regularity over a Koszul algebra
Let A=⨁i∈NAi be a Koszul algebra over a field 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 this, we show that the Koszul dual A! of a Koszul commutative algebra A satisfies the following.
Let M∈∗modA!. If {dimKMi∣i∈Z} is bounded, then ldA!(M)<∞.
If A is complete intersection, then regA!(M)<∞ and ldA!(M)<∞ for all M∈∗modA!.
If E=⋀⟨y1,…,yn⟩ is an exterior algebra, then ldE(M)≤cn!2(n−1)! for M∈∗modE with c:=max{dimKMi∣i∈Z}.
