We prove regularity bounds for Koszul cycles holding for every ideal of dimension ≤ 1 in a polynomial ring; see Theorem 3.5. In Theorem 4.7 we generalize the “c + 1” lower bound for the Green–Lazarsfeld index of Veronese rings proved in (Bruns et al., arXiv:0902.2431) to the multihomogeneous setting. For the Koszul complex of the c-th power of the maximal ideal in a Koszul ring we prove that th...

This article concerns linear parts of minimal resolutions of finitely generated modules over commutative local, or graded rings. The focus is on the linearity defect of a module, which marks the point after which the linear part of its minimal resolution is acyclic. The results established track the change in this invariant under some standard operations in commutative algebra. As one of the ap...

We define generalized Koszul modules and rings develop a theory for N-graded with the degree zero part noetherian semiperfect. This specializes to classical graded artinian semisimple developed by Beilinson-Ginzburg-Soergel ungraded semiperfect Green Martinéz-Villa. Let A be left finite ring generated in 1 A0 semiperfect, J its Jacobson radical. By dual of we mean Yoneda Ext Ext_A•(A/J,A/J). If...

We show that there exists a natural non-degenerate pairing of the homomorphism space between two neighbor standard modules over a quasi-hereditary algebra with the first extension space between the corresponding costandard modules and vise versa. Investigation of this phenomenon leads to a family of pairings involving standard, costandard and tilting modules. In the graded case, under some ”Kos...

We show that there exists a natural non-degenerate pairing of the homomorphism space between two neighbor standard modules over a quasi-hereditary algebra with the first extension space between the corresponding costandard modules and vise versa. Investigation of this phenomenon leads to a family of pairings involving standard, costandard and tilting modules. In the graded case, under some ”Kos...

Koszul algebras, introduced by Priddy in [P], are positively graded Kalgebras R whose residue field K has a linear free resolution as an R-module. Here linear means that the non-zero entries of the matrices describing the maps in the R-free resolution of K have degree 1. For example, the polynomial ring S = K[x1, . . . ,xn] over a field K (i.e the symmetric algebra SymK(V ) of a n-dimensional K...

In this paper we devote to generalizing some results of componentwise linear modules over a polynomial ring to the ones over a Koszul algebra. Among other things, we show that the i-linear strand of the minimal free resolution of a componentwise linear module is the minimal free resolution of some module which is described explicitly for any i ∈ Z. In addition we present some theorems about whe...

In this paper we show that if Λ = ∐ i≥0 Λ i is a Koszul algebra with Λ 0 isomorphic to a product of copies of a field, then the minimal projective resolution of Λ 0 as a right Λ-module provides all the information necessary to construct both a minimal projective resolution of Λ 0 as a left Λ-module and a minimal projective resolution of Λ as a right module over the enveloping algebra of Λ. The ...

We present new examples of deformations of smash product algebras that arise from Hopf algebra actions on pairs of module algebras. These examples involve module algebras that are Koszul, in which case a PBW theorem we established previously applies. Our construction generalizes several ‘double’ constructions appearing in the literature, including Weyl algebras and some types of Cherednik algeb...