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Generalized Koszul properties of commutative local rings  [PDF]
Justin Hoffmeier,Liana M. ?ega
Mathematics , 2014,
Abstract: We study several properties of commutative local rings that generalize the notion of Koszul algebra. The properties are expressed in terms of the Ext algebra of the ring, or in terms of homological properties of powers of the maximal ideal of the ring. We analyze relationships between these properties and we identify large classes of rings that satisfy them. In particular, we prove that the Ext algebra of a compressed Gorenstein local ring of even socle degree is generated in degrees $1$ and $2$.
Strongly Koszul edge rings  [PDF]
Takayuki Hibi,Kazunori Matsuda,Hidefumi Ohsugi
Mathematics , 2013,
Abstract: We classify the finite connected simple graphs whose edge rings are strongly Koszul. From the classification, it follows that if the edge ring is strongly Koszul, then its toric ideal possesses a quadratic Gr\"obner basis.
Symmetry in the vanishing of Ext over Gorenstein rings  [PDF]
Craig Huneke,David Jorgensen
Mathematics , 2014,
Abstract: We investigate symmetry in the vanishing of Ext for finitely generated modules over local Gorenstein rings. In particular, we define a class of local Gorenstein rings, which we call AB rings, and show that for finitely generated modules $M$ and $N$ over an AB ring $R$, $Ext^i_R(M,N)=0$ for all $i >> 0$ if and only if $Ext^i_R(N,M)=0$ for all $i >> 0$.
On Ext-indices of ring extensions  [PDF]
Saeed Nasseh,Yuji Yoshino
Mathematics , 2007,
Abstract: In this paper we are concerned with the finiteness property of Ext-indices of several ring extensions. In this direction, we introduce some conjectures and discuss the relationship of them. Also we give affirmative answers to these conjectures in some special cases. Furthermore, we prove that the trivial extension of an Artinian local ring by its residue class field is always of finite Ext-index and we show that the Auslander-Reiten conjecture is true for this type of rings.
The telescope conjecture for hereditary rings via Ext-orthogonal pairs  [PDF]
Henning Krause,Jan Stovicek
Mathematics , 2008, DOI: 10.1016/j.aim.2010.04.027
Abstract: For the module category of a hereditary ring, the Ext-orthogonal pairs of subcategories are studied. For each Ext-orthogonal pair that is generated by a single module, a 5-term exact sequence is constructed. The pairs of finite type are characterized and two consequences for the class of hereditary rings are established: homological epimorphisms and universal localizations coincide, and the telescope conjecture for the derived category holds true. However, we present examples showing that neither of these two statements is true in general for rings of global dimension 2.
The Ext algebra and a new generalisation of D-Koszul algebras  [PDF]
Joanne Leader,Nicole Snashall
Mathematics , 2015,
Abstract: We generalise Koszul and D-Koszul algebras by introducing a class of graded algebras called (D,A)-stacked algebras. We give a characterisation of (D,A)-stacked algebras and show that their Ext algebra is finitely generated as an algebra in degrees 0, 1, 2 and 3. In the monomial case, we give an explicit description of the Ext algebra by quiver and relations, and show that the ideal of relations has a quadratic Gr\"obner basis; this enables us to give a regrading of the Ext algebra under which the regraded Ext algebra is a Koszul algebra.
Some Koszul Rings from Geometry  [PDF]
Krishna Hanumanthu
Mathematics , 2009,
Abstract: 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 irregular surfaces of general type.
On the Koszul property of toric face rings  [PDF]
Dang Hop Nguyen
Mathematics , 2011,
Abstract: Toric face rings is a generalization of the concepts of affine monoid rings and Stanley-Reisner rings. We consider several properties which imply Koszulness for toric face rings over a field $k$. Generalizing works of Laudal, Sletsj\o{}e and Herzog et al., graded Betti numbers of $k$ over the toric face rings are computed, and a characterization of Koszul toric face rings is provided. We investigate a conjecture suggested by R\"{o}mer about the sufficient condition for the Koszul property. The conjecture is inspired by Fr\"{o}berg's theorem on the Koszulness of quadratic squarefree monomial ideals. Finally, it is proved that initially Koszul toric face rings are affine monoid rings.
Koszul pairs. Applications  [PDF]
Pascual Jara Martínez,Javier López Pe?a,Drago? ?tefan
Mathematics , 2010,
Abstract: 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 Fr\"oberg Theorem, are discussed in the last part of the paper.
Multigraded regularity and the Koszul property  [PDF]
Milena Hering
Mathematics , 2007,
Abstract: We give a criterion for the section ring of an ample line bundle to be Koszul in terms of multigraded regularity. We discuss an application to polytopal semigroup rings.
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