Koszul Rings Research Articles

On robustness and related properties on toric ideals

A toric ideal is called robust if its universal Gröbner basis is a minimal set of generators, and is called generalized robust if its universal Gröbner basis equals its universal Markov basis (the union of all its minimal sets of binomial generators). Robust and generalized robust toric ideals are both interesting from both a commutative algebra and an algebraic statistics perspective. However, only a few nontrivial examples of such ideals are known. In this work, we study these properties for toric ideals of both graphs and numerical semigroups. For toric ideals of graphs, we characterize combinatorially the graphs giving rise to robust and to generalized robust toric ideals generated by quadratic binomials. As a by-product, we obtain families of Koszul rings. For toric ideals of numerical semigroups, we determine that one of its initial ideals is a complete intersection if and only if the semigroup belongs to the so-called family of free numerical semigroups. Hence, we characterize all complete intersection numerical semigroups which are minimally generated by one of its Gröbner basis and, as a consequence, all the Betti numbers of the toric ideal and its corresponding initial ideal coincide. Moreover, also for numerical semigroups, we prove that the ideal is generalized robust if and only if the semigroup has a unique Betti element and that there are only trivial examples of robust ideals. We finish the paper with some open questions.

Some criteria for detecting large homomorphisms of local rings

We study ideals in a local ring R whose quotient rings induce large homomorphisms of local rings. We characterize such ideals in terms of the maps of Koszul homologies over several classes of local rings, including complete intersections, Koszul rings, and some classes of Golod rings.

Koszul pairs and applications

Let R be a semisimple ring. A pair (A,C) is called almost-Koszul if A is a connected graded R -ring and C is a compatible connected graded R -coring. To an almost-Koszul pair one associates three chain complexes and three cochain complexes such that one of them is exact if and only if the others are so. In this situation (A,C) is said to be Koszul. One proves that a connected R -ring A is Koszul if and only if there is a connected R -coring C such that (A,C) is Koszul. This result allows us to investigate the Hochschild (co)homology of Koszul rings. We apply our method to show that the twisted tensor product of two Koszul rings is Koszul. More examples and applications of Koszul pairs, including a generalization of Fröberg Theorem [12], are discussed in the last part of the paper.

The absolutely Koszul property of Veronese subrings and Segre products

Absolutely Koszul algebras are a class of rings over which any finite graded module has a rational Poincaré series. We provide a criterion to detect non-absolutely Koszul rings. Combining the criterion with Macaulay2 computations, we identify large families of Veronese subrings and Segre products of polynomial rings which are not absolutely Koszul. In particular, we classify completely the absolutely Koszul algebras among Segre products of polynomial rings, when the base field has characteristic 0.

The cup product on Hochschild cohomology via twisting cochains and applications to Koszul rings

Given an acyclic twisting cochain π:C→A, from a curved dg coalgebra C to a dg algebra A, we show that the associated twisted hom complex Homkπ(C,A) has cohomology equal to the Hochschild cohomology of A, as a graded ring. As a corollary we find that the Hochschild cohomology of a Koszul algebra A, along with its cup product, is a subquotient of the tensor product algebra A!⊗A of A with its Koszul dual.

Further Properties and Applications of Koszul Pairs

Koszul pairs were introduced in [arXiv:1011.4243] as an instrument for the study of Koszul rings. In this paper, we continue the enquiry of such pairs, focusing on the description of the second component, as a follow-up of the study in [arXiv:1605.05458]. As such, we introduce Koszul corings and prove several equivalent characterizations for them. As applications, in the case of locally finite $R$-rings, we show that a graded $R$-ring is Koszul if and only if its left (or right) graded dual coring is Koszul. Finally, for finite graded posets, we obtain that the respective incidence ring is Koszul if and only if the incidence coring is so.

On Koszulity of finite graded posets

In this paper, we continue our research on Koszul rings, started in [P. Jara, J. López-Peña and D. Ştefan, Koszul pairs and applications, to appear in J. Noncommut. Geom., http://arxiv.org/pdf/1011.4243.pdf ]. In Theorem 1.9, we prove in a unifying way several equivalent descriptions of Koszul rings, some of which being well known in the literature. Most of them are stated in terms of coring theoretical properties of [Formula: see text]. As an application of these characterizations, we investigate the Koszulity of the incidence rings for finite graded posets, see Theorems 2.8 and 2.9. Based on these results, we describe an algorithm to produce new classes of Koszul posets (i.e. graded posets, whose incidence rings are Koszul). Specific examples of Koszul posets are included.

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.

Modular Koszul duality

AbstractWe prove an analogue of Koszul duality for category$ \mathcal{O} $of a reductive group$G$in positive characteristic$\ell $larger than$1$plus the number of roots of$G$. However, there are no Koszul rings, and we do not prove an analogue of the Kazhdan–Lusztig conjectures in this context. The main technical result is the formality of the dg-algebra of extensions of parity sheaves on the flag variety if the characteristic of the coefficients is at least the number of roots of$G$plus$2$.

Mixed Artin–Tate Motives with Finite Coefficients

The goal of this paper is to give an explicit description of the triangulated categories of Tate and Artin{Tate motives with nite coecients Z=m over a eld K containing a primitive m-root of unity as the derived categories of exact categories of ltered modules over the absolute Galois group of K with certain restrictions on the successive quotients. This description is conditional upon (and its validity is equiv- alent to) certain Koszulity hypotheses about the Milnor K-theory/Galois cohomology of K. This paper also purports to explain what it means for an arbitrary nonnegatively graded ring to be Koszul. Tate motives with integral coecients are discussed in the \Conclusions section. 2010 Mathematics Subject Classification. 12G05, 19D45, 16S37. coecients, Milnor{Bloch{Kato conjecture, Beilinson{Lichtenbaum con- jecture, K(; 1) conjecture, Koszulity conjecture, Koszul algebras, nonat Koszul rings, silly ltrations.

Some Koszul rings from geometry

We give examples of Koszul rings that arise naturally in algebraic geometry. In the first part, we prove a general result on Koszul property associated to an ample line bundle on a projective variety. Specifically, we show how Koszul property of multiples of a base point free ample line bundle depends on its Castelnuovo–Mumford regularity. In the second part, we give examples of Koszul rings that come from adjoint line bundles on minimal irregular surfaces of general type.

The Koszul property in affine semigroup rings

We investigate the Koszul property for quotients of ane semigroup rings by semigroup ideals. Using a combinatorial and topological interpretation for the Koszul property in this context, we recover known results asserting that certain of these rings are Koszul. In the process, we prove a stronger fact, suggesting a more general denition of Koszul rings, al

Koszul Duality Patterns in Representation Theory

The aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of Koszul duality to representation theory. The paper consists of three parts relatively independent of each other. The first part gives a reasonably selfcontained introduction to Koszul rings and Koszul duality. Koszul rings are certain Z \mathbb {Z} -graded rings with particularly nice homological properties which involve a kind of duality. Thus, to a Koszul ring one associates naturally the dual Koszul ring. The second part is devoted to an application to representation theory of semisimple Lie algebras. We show that the block of the Bernstein-Gelfand-Gelfand category O \mathcal {O} that corresponds to any fixed central character is governed by the Koszul ring. Moreover, the dual of that ring governs a certain subcategory of the category O \mathcal {O} again. This generalizes the selfduality theorem conjectured by Beilinson and Ginsburg in 1986 and proved by Soergel in 1990. In the third part we study certain categories of mixed perverse sheaves on a variety stratified by affine linear spaces. We provide a general criterion for such a category to be governed by a Koszul ring. In the flag variety case this reduces to the setup of part two. In the more general case of affine flag manifolds and affine Grassmannians the criterion should yield interesting results about representations of quantum groups and affine Lie algebras.