Related papers: Koszul modules with vanishing resonance in algebra…
We describe the progress in the last 10 years related to Koszul modules and syzygies of algebraic varieties. Topics discussed include the general theory of Koszul modules and resonance varieties, applications to Chen ranks of K\"ahler and…
We prove a strong vanishing result for finite length Koszul modules, and use it to derive Green's conjecture for every g-cuspidal rational curve over an algebraically closed field k with char(k) = 0 or char(k) >= (g+2)/2. As a consequence,…
Koszul modules and their associated resonance schemes are objects appearing in a variety of contexts in algebraic geometry, topology, and combinatorics. We present a proof of an effective version of the Chen ranks conjecture describing the…
We discuss recent progress on syzygies of curves, including proofs of Green's and Gonality Conjectures as well as applications of Koszul cycles to the study of the birational geometry of various moduli spaces of curves. We prove a number of…
We introduce Koszul modules associated with (graded) Kac-Moody Lie algebras. We provide a precise criterion for when these modules are of finite length. As an exemplary application we deduce a bound on the dimension of the second graded…
We develop a Koszul-theoretic framework for comparing classical Alexander-type invariants with infinitesimal invariants arising from finite-type commutative differential graded algebra models. The central mechanism is Koszul linearization,…
We investigate Koszul cohomology on irreducible nodal curves. In particular, we prove both Green and Green-Lazarsfeld conjectures for the general k-gonal nodal curve.
We explore a relationship between the classical representation theory of a complex, semisimple Lie algebra \g and the resonance varieties R(V,K)\subset V^* attached to irreducible \g-modules V and submodules K\subset V\wedge V. In the…
We extend the theory of Koszul modules to the bi-graded case, and prove a vanishing theorem that allows us to show that the Canonical Ribbon Conjecture of Bayer and Eisenbud holds over a field of characteristic zero or at least equal to the…
In this paper we continue the study (initiated in a previous article) of linear Koszul duality, a geometric version of the standard duality between modules over symmetric and exterior algebras. We construct this duality in a very general…
Each connected graded, graded-commutative algebra $A$ of finite type over a field $\Bbbk$ of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the…
The resonance varieties are cohomological invariants that are studied in a variety of topological, combinatorial, and geometric contexts. We discuss their scheme structure in a general algebraic setting and introduce various properties that…
Several spectral sequence techniques are used in order to derive information about the structure of finite free resolutions of graded modules. These results cover estimates of the minimal number of generators of defining ideals of…
A key result for syzygies of curves is Voisin's proof of Green's conjecture for the canonical embedding of a general curve of any genus. Her primary tools were the Lazarsfeld Mukai bundle on a K3 surface and a representation of Koszul…
Koszul algebras have arisen in many contexts; algebraic geometry, combinatorics, Lie algebras, non-commutative geometry and topology. The aim of this paper and several sequel papers is to show that for any finite dimensional algebra there…
This paper provides a new class of examples for the Koszul dualities established in~\cite{5}. We study quadratic monomial algebras from the perspective of Koszul duality, with particular emphasis on finitely presented and finitely…
We define generalized Koszul modules and rings and develop a generalized Koszul theory for $\mathbb{N}$-graded rings with the degree zero part noetherian semiperfect. This theory specializes to the classical Koszul theory for graded rings…
Let R be a commutative noetherian local ring with completion R^. We apply differential graded (DG) algebra techniques to study descent of modules and complexes from R^ to R' where R' is either the henselization of R or a pointed \'etale…
We provide a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace K inside the second exterior product of a vector space, as well as a sharp upper bound for its Hilbert function.…
We obtain Koszul-type dualities for categories of graded modules over a graded associative algebra which can be realized as the semidirect product of a bialgebra coinciding with its degree zero part and a graded module algebra for the…