Koszul Modules Research Articles

Reduced resonance schemes and Chen ranks

Abstract 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 ensure the reducedness of the associated projective resonance scheme. We prove an asymptotic formula for the Hilbert series of the associated Koszul module, then discuss applications to vector bundles on algebraic curves and to Chen ranks formulas for finitely generated groups, with special emphasis on Kähler and right-angled Artin groups.

Higher resonance schemes and Koszul modules of simplicial complexes

Each connected graded, graded-commutative algebra A of finite type over a field k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Bbbk $$\\end{document} of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the (higher) Koszul modules of A. In this note, we investigate the geometry of the support loci of these modules, called the resonance schemes of the algebra. When A=k⟨Δ⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A=\\Bbbk \\langle \\Delta \\rangle $$\\end{document} is the exterior Stanley–Reisner algebra associated to a finite simplicial complex Δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta $$\\end{document}, we show that the resonance schemes are reduced. We also compute the Hilbert series of the Koszul modules and give bounds on the regularity and projective dimension of these graded modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group.

Koszul modules with vanishing resonance in algebraic geometry

We discuss various applications of a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace K⊆⋀2V\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K\\subseteq \\bigwedge ^2 V$$\\end{document}, where V is a vector space. Previously Koszul modules of finite length have been used to give a proof of Green’s Conjecture on syzygies of generic canonical curves. We now give applications to effective stabilization of cohomology of thickenings of algebraic varieties, divisors on moduli spaces of curves, enumerative geometry of curves on K3 surfaces and to skew-symmetric degeneracy loci. We also show that the instability of sufficiently positive rank 2 vector bundles on curves is governed by resonance and give a splitting criterion.

Generalized Koszul algebra and Koszul duality

We define generalized Koszul modules and rings and develop a generalized Koszul theory for N-graded rings with the degree zero part noetherian semiperfect. This theory specializes to the classical Koszul theory for graded rings with degree zero part artinian semisimple developed by Beilinson-Ginzburg-Soergel and the ungraded Koszul theory for noetherian semiperfect rings developed by Green and Martinéz-Villa. Let A be a left finite N-graded ring generated in degree 1 with A0 noetherian semiperfect, J be its graded Jacobson radical. By the Koszul dual of A we mean the Yoneda Ext ring Ext_A•(A/J,A/J). If A is a generalized Koszul ring and M is a generalized Koszul module, then it is proved that the Koszul dual of the Koszul dual of A is the associated graded ring GrJA and the Koszul dual of the Koszul dual of M is the associated graded module GrJM. If A is a locally finite algebra, then the following statements are proved to be equivalent: A is generalized Koszul; the Koszul dual Ext_A•(A/J,A/J) of A is (classically) Koszul; GrJA is (classically) Koszul; the opposite ring Aop of A is generalized Koszul. As an application, it is proved that if A is generalized Koszul with finite global dimension then A is generalized AS regular if and only if the Koszul dual of A is self-injective.

Topological invariants of groups and Koszul modules

We provide a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace K⊆∧2V as well as a sharp upper bound for its Hilbert function. This purely algebraic statement has interesting applications to the study of a number of invariants associated to finitely generated groups, such as the Alexander invariants, the Chen ranks, and the degree of growth and virtual nilpotency class. For instance, we explicitly bound the aforementioned invariants in terms of the first Betti number for the maximal metabelian quotients of (1) the Torelli group associated to the moduli space of curves, (2) nilpotent fundamental groups of compact Kähler manifolds, and (3) the Torelli group of a free group.

Bi-graded Koszul modules, K3 carpets, and Green's conjecture

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$0$or at least equal to the Clifford index. Our results confirm a conjecture of Eisenbud and Schreyer regarding the characteristics where the generic statement of Green's conjecture holds. They also recover and extend to positive characteristics the results of Voisin asserting that Green's conjecture holds for generic curves of each gonality.

Koszul modules (and the Ω-growth of modules) over short local algebras

Following the well-established terminology in commutative algebra , any (not necessarily commutative) finite-dimensional local algebra A with radical J will be said to be short provided J 3 = 0 . As in the commutative case , also in general, the asymptotic behavior of the Betti numbers of modules seems to be of interest. As we will see, there are only few possibilities for the growth of the Betti numbers of modules. And there is a class of modules, the Koszul modules, where the Betti numbers can be calculated quite easily. We generalize results which are known for commutative algebras, but some of our results seem to be new also in the commutative case.

Koszul modules and Green’s conjecture

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, we deduce that the general canonical curve of genus g satisfies Green's conjecture in this range. Our results are new in positive characteristic, whereas in characteristic zero they provide a different proof for theorems first obtained in two landmark papers by Voisin. Our strategy involves establishing two key results of independent interest: (1) we describe an explicit, characteristic-independent version of Hermite reciprocity for sl_2-representations; (2) we completely characterize, in arbitrary characteristics, the (non-)vanishing behavior of the syzygies of the tangential variety to a rational normal curve.

Applications of one-point extensions to compute the A∞-(co)module structure of several Ext (resp., Tor) groups

Let A be a nonnegatively graded connected algebra over a noncommutative separable k-algebra K, and let M be a bounded below graded right A-module. If we denote by T the A∞-coalgebra Tor•A(K,K), we know that there exists an A∞-comodule structure on T′=Tor•A(M,K) over T. The structure of the A∞-algebra E=ExtA•(K,K) and the corresponding A∞-module on E′=ExtA•(M,K) are just obtained by taking the bigraded dual. In this article we prove that there is partial description of the A∞-comodule T′ over T and of the structure of A∞-module E′ over E, similar to and also generalizing the partial description of the A∞-algebra structure on E given by Keller's higher-multiplication theorem in [19]. We also provide a criterion to check if a given A∞-comodule structure on T′ is a model by regarding if the associated twisted tensor product is a minimal projective resolution of M, analogous to a theorem of B. Keller explained by the author of this article in [9]. Finally, we give an application of this result by computing the A∞-module structure on E′ for any generalized Koszul algebra A and any generalized Koszul module M.

On the definition of multi-Koszul modules

In [8] we introduced the notion of multi-Koszul algebra: it is an extension of the definition of generalized Koszul algebra given by R. Berger in [1] for homogeneous algebras (see also [7]) that can be applied to any nonnegatively graded connected algebra over a field k. The goal of this article is on the one hand to properly extend the notion of multi-Koszul algebra to the case where the base ring K is a product of copies of a field k, which a priori allows us to treat quiver algebras, and on the other hand to introduce the notion of multi-Koszul module such that it extends the usual definition of generalized Koszul module over a generalized Koszul algebra. We show eventually that multi-Koszul algebras and multi-Koszul modules are strongly linked via the notion of one-point extensions, as in the case of generalized Koszul algebras. Moreover, we describe the complete structure of right A∞-module on ExtA•(M,K) over ExtA•(K,K), where M is a multi-Koszul module over a multi-Koszul algebra A, extending a result in [9]. As a corollary, we obtain that the underlying right module structure of ExtA•(M,K) over ExtA•(K,K) is generated by the component of cohomological degree zero, as in the case of generalized Koszul modules over generalized Koszul algebras.

Koszulity and Koszul modules of dual extension algebras

Let A and B be algebras, and let T be the dual extension algebra of A and B. We provide a different method to prove that T is Koszul if and only if both A and B are Koszul. Furthermore, we prove that an algebra is Koszul if and only if one of its iterated dual extension algebras is Koszul, if and only if all its iterated dual extension algebras are Koszul. Finally, we give a necessary and sufficient condition for a dual extension algebra to have the property that all linearly presented modules are Koszul modules, which provides an effective way to construct algebras with such a property.

Modules with pure resolutions over Koszul algebras revisited

I construct a Koszul algebra A and a finitely generated graded A-module M that together form a counterexample to a recently published claim. M is generated in degree 0 and has a pure resolution, and the graded Jacobson radical of the Yoneda algebra of A does not annihilate the Ext module of M, but nonetheless M is not a Koszul module.

Notes on the linearity defect and applications

The linearity defect, introduced by Herzog and Iyengar, is a numerical measure for the complexity of minimal free resolutions. Employing a characterization of the linearity defect due to Şega, we study the behavior of linearity defect along short exact sequences. We point out two classes of short exact sequences involving Koszul modules, along which linearity defect behaves nicely. We also generalize the notion of Koszul filtrations from the graded case to the local setting. Among the applications, we prove that if $R\to S$ is a surjection of noetherian local rings such that $S$ is a Koszul $R$-module, and $N$ is a finitely generated $S$-module, then the linearity defect of $N$ as an $R$-module is the same as its linearity defect as an $S$-module. In particular, we confirm that specializations of absolutely Koszul algebras are again absolutely Koszul, answering positively a question due to Conca, Iyengar, Nguyen and Römer.

On the Regularity and Koszulness of Modules Over Local Rings

Koszul modules over Noetherian local rings R were introduced by Herzog and Iyengar and they possess good homological properties, for instance their Poincaré series is rational. It is an interesting problem to characterize classes of Koszul modules. Following the idea traced by Avramov, Iyengar, and Sega, we take advantage of the existence of special filtration on R for proving that large classes of R-modules over Koszul rings are Koszul modules. By using this tool we reprove and extend some results obtained by Fitzgerald.

On modules with pure resolutions over d-Koszul algebras

Let A = L i>0 Ai be a d-Koszul algebra and M = L i>0 Mi be any nitely 0-generated graded A-module with a pure resolution. It is proved that M is a discrete Koszul module under some additional conditions. Moreover, we can extend the results to the case of -Koszul algebras.

(p,\\lambda)-Koszul algebras and modules, II

This paper is a continuous work of [14], where the notions of (p,\lambda)-Koszul algebra and (p,\lambda)-Koszul module were first introduced. More precisely, some new criteria for a positively graded algebra to be (p,\lambda)-Koszul are provided. We also generalize (p,\lambda)-Koszul objects to the nongraded case and define the so-called quasi-(p,\lambda)-Koszul objects. Further, the relationships between (quasi-) (p,\lambda)-Koszul modules and minimal Horseshoe Lemma are established.

(p, λ)-Koszul algebras and modules

In this paper, the notions of (p, λ)-Koszul algebra and (p, λ)-Koszul module are introduced. Some criteria theorems for a positively graded algebra A to be (p, λ)-Koszul are given. The notion of weakly (p, λ)-Koszul module is defined as well and let WKλp(A) denote the category of weakly (p, λ)-Koszul modules. We show that M ∈ WKλp(A) if and only if it can be approximated by (p, λ)-Koszul submodules, which is equivalent to that G(M) is a (p, λ)-Koszul module, where G(M) denotes the associated graded module of M. As applications, the relationships of the minimal graded projective resolutions of M, G(M) and (p, λ)-Koszul submodules are established. In particular, for a module M ∈ WKλp(A) we prove that ⊕i≥0 ExtAi(M,A0) ∈ gr0(E(A)), we also get as a consequence that the finitistic dimension conjecture is valid in WKλp(A) under certain conditions.

Minimal free resolution of a finitely generated module over a regular local ring

Numerical invariants of a minimal free resolution of a module M over a regular local ring ( R , n ) can be studied by taking advantage of the rich literature on the graded case. The key is to fix suitable n -stable filtrations M of M and to compare the Betti numbers of M with those of the associated graded module gr M ( M ) . This approach has the advantage that the same module M can be detected by using different filtrations on it. It provides interesting upper bounds for the Betti numbers and we study the modules for which the extremal values are attained. Among others, the Koszul modules have this behavior. As a consequence of the main result, we extend some results by Aramova, Conca, Herzog and Hibi on the rigidity of the resolution of standard graded algebras to the local setting.

Linearity defects of modules over commutative rings

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 applications, it is proved that a local ring is Koszul if and only if it admits a Koszul module that is Cohen–Macaulay of minimal degree. An injective analogue of the linearity defect is introduced and studied. The main results express this new invariant in terms of linearity defects of free resolutions, and relate it to other ring theoretic and homological invariants of the module.

Gorenstein Algebras in Codimension 2 and Koszul Modules

The notion of Koszul module introduced in Grassi (1996) is a generalization of the concept of complete intersection ring to arbitrary finite modules over some base ring A (cf. Def. 1.3). A Gorenstein A-algebra R of codimension 2 is a perfect finite A-algebra such that holds as R-modules, where A is a Cohen–Macaulay local ring with dim A − dim A R = 2. Here we prove (Thm. 1.4) that every Gorenstein algebra R of codimension 2 has a resolution over A (assuming 2 is invertible in A) that is simultaneously Gorenstein-symmetric and of Koszul module type; this is a generalization to algebras of higher rank of the easy fact that, if R is a quotient of A, it has a resolution over A given by a length two Koszul complex.