No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
Cohen-Macaulay Complexes and Koszul Rings
Journal of The London Mathematical Society
Abstract
Throughout this paper k denotes a fixed commutative ground ring. A Cohen–Macaulay complex is a finite simplicial complex satisfying a certain homological
vanishing condition. These complexes have been the subject of much research; introductions can be found in, for example, Björner,
Garsia and Stanley [6] or Budach, Graw, Meinel and Waack [7]. It is known (see, for example, Cibils [8], Gerstenhaber and Schack [10]) that there is a strong connection between the (co)homology of an arbitrary simplicial complex and that of its incidence
algebra. We show how the Cohen–Macaulay property fits into this picture, establishing the following characterization.
A pure finite simplicial complex is Cohen–Macaulay over k if and only if the incidence algebra over k of its augmented face
poset, graded in the obvious way by chain lengths, is a Koszul ring.
... This is mainly done in Theorem 1.10, where we also recover in a unifying way other well-known equivalent characterizations of these rings. For instance, we show that A is Koszul if and only if the coring T (A) is strongly graded, if and only if the primitive part of T (A) coincides with its homogeneous component of degree one, if and only if T (A), regarded in a canonical way as a bigraded coring, is diagonal (for the last characterization see also [9]). ...
... To check Koszulity of a ring is a difficult task. The case of incidence rings of finite graded posets was approached in [9,Theorem 3.7], where it is proved that such a ring is Koszul if and only if all open intervals of the poset are Cohen-Macaulay. The Koszulity of incidence algebras is investigated in [5,7] as well. ...
... Remark 2.15. In [9,Definition 4.6], the author defines a graded poset Ω as being exactly thin whenever x < y and l(x, y) = 2 imply that the interval (x, y) consists of precisely two elements. Note that what we termed 'planar tilings' are examples of such posets, which we proved to be Koszul. ...
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 \Tor_n^A(R,R). As an application of these characterizations we investigate the Koszulity of the incidence rings for finite graded posets. 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.
... This is mainly done in Theorem 1.10, where we also recover in a unifying way other well-known equivalent characterizations of these rings. For instance, we show that A is Koszul if and only if the coring T (A) is strongly graded, if and only if the primitive part of T (A) coincides with its homogeneous component of degree one, if and only if T (A), regarded in a canonical way as a bigraded coring, is diagonal (for the last characterization see also [9]). ...
... To check Koszulity of a ring is a difficult task. The case of incidence rings of finite graded posets was approached in [9,Theorem 3.7], where it is proved that such a ring is Koszul if and only if all open intervals of the poset are Cohen-Macaulay. The Koszulity of incidence algebras is investigated in [5,7] as well. ...
... Remark 2.15. In [9,Definition 4.6], the author defines a graded poset Ω as being exactly thin whenever x < y and l(x, y) = 2 imply that the interval (x, y) consists of precisely two elements. Note that what we termed 'planar tilings' are examples of such posets, which we proved to be Koszul. ...
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 TornA(R,R). 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.
... We want to apply Koszul theory, which has been proved to be very useful in the representation theory of algebras, to study the Ext groups of representations of finite EI categories. Examples of such applications can be found in [21,23], where Koszul theory has been applied to incidence algebras of posets. In general, this theory applies to graded algebras, and we do not assume that the degree 0 part of the algebra is semisimple, unlike the classical Koszul theory described in [4,9,10,17]. ...
... There do already exist several generalized Koszul theories where the degree 0 part A 0 of a graded algebra A is not required to be semisimple; see [11,15,16,23]. Each Koszul algebra A defined by Woodcock in [23] is supposed to satisfy that A is both a left projective A 0 -module and a right projective A 0 -module. ...
... There do already exist several generalized Koszul theories where the degree 0 part A 0 of a graded algebra A is not required to be semisimple; see [11,15,16,23]. Each Koszul algebra A defined by Woodcock in [23] is supposed to satisfy that A is both a left projective A 0 -module and a right projective A 0 -module. This requirement is too strong for us. ...
Let A be a graded algebra. In this paper we develop a generalized Koszul theory by assuming that is self-injective instead of semisimple and generalize many classical results. The application of this generalized theory to directed categories and finite EI categories is described.
... Therefore, it is reasonable to develop a generalized Koszul theory to study representations and homological properties of the above structures. In [11,18,19,29] several generalized Koszul theories have been described, where the degree 0 part A 0 of a graded algebra A is not required to be semisimple. In [29], A is supposed to be both a left projective A 0-module and a right projective A 0-module. ...
... In [11,18,19,29] several generalized Koszul theories have been described, where the degree 0 part A 0 of a graded algebra A is not required to be semisimple. In [29], A is supposed to be both a left projective A 0-module and a right projective A 0-module. However, in many cases A is indeed a left projective A 0-module, but not a right projective A 0-module. ...
... We start with some preliminary results, most of which are generalized from those described in [5,9,10,21,23]. The reader is also suggested to look at other generalized Koszul theories described in [11,18,19,29]. The following lemmas are proved in [17], where we did not use the condition that A 0 is self-injective (Remark 2.8 in [17]). ...
Let be a graded locally finite
k-algebra such that is an arbitrary finite-dimensional algebra
satisfying a certain splitting condition. In this paper we develop a
generalized Koszul theory preserving many classical results. Moreover, we
define a quotient graded algebra and show that A is a generalized Koszul algebra if and only if
is a classical Koszul algebra and a projective -module. We also
describe an application of this theory to the extension algebras of standard
modules of standardly stratified algebras.
... Say that ∆ is Cohen-Macaulay over k if for every face F of ∆, its link link ∆ (F ) := {G ∈ ∆ : F ∪ G ∈ ∆, F ∩ G = ∅} within ∆ has the property that its reduced homology vanishes below the dimension of the link:H Theorem 1.6. (Polo [23, §1.6], Woodcock [29,Theorem 3.7 ...
... We wish to characterize combinatorially when gr I A for A = k[P ] red is a quadratic ring, generalizing a result of Woodcock [29,Lemma 4.5] for incidence algebras of graded posets P . Woodcock's result uses the well-known notion of galleryconnectedness for pure simplicial complexes. ...
We prove a theorem unifying three results from combinatorial homological and commutative algebra, characterizing the Koszul property for incidence algebras of posets and affine semigroup rings, and characterizing linear resolutions of squarefree monomial ideals. The characterization in the graded setting is via the Cohen-Macaulay property of certain posets or simplicial complexes, and in the more general nongraded setting, via the sequential Cohen-Macaulay property.
... Many interesting algebras with properties similar to a Koszul algebra do not satisfy the assumption that A 0 is semisimple. There do already exist several generalized Koszul theories where the degree 0 part A 0 of a graded algebra A is not required to be semisimple, see [5], [8], and [24]. Each Koszul ring A defined by Woodcock in [24] is supposed to satisfy that A is both a left projective A 0 -module and a right projective A 0 -module. ...
... There do already exist several generalized Koszul theories where the degree 0 part A 0 of a graded algebra A is not required to be semisimple, see [5], [8], and [24]. Each Koszul ring A defined by Woodcock in [24] is supposed to satisfy that A is both a left projective A 0 -module and a right projective A 0 -module. This requirement is too strong. ...
Let R be a commutative algebra. In this paper we show that constant skew PBW extensions of a generalized Koszul algebra R are also generalized Koszul. Let A be a semi-commutative skew PBW extension of R such that A is R-augmented. We show also that if R is augmented Koszul then A is R-augmented Koszul.
... It follows that the Ext-algebra of kB is isomorphic to the incidence algebra of Λ(B). Results proved independently by Polo [93] and Woodcock [127] then yield that every open interval of the semilattice Λ(B) is a Cohen-Macaulay poset. This is particularly important because it has consequences for the Möbius function of Λ(B). ...
... Polo and Woodcock [93,127] independently showed that the incidence algebra I(P ; k) of a graded poset P is a Koszul algebra with respect to the natural grading if and only if each open interval in P is k-Cohen-Macaulay. We thus obtain the following theorem, which will be applied to enumerate cells of connected CW left regular bands. ...
... Many interesting algebras with properties similar to a Koszul algebra do not satisfy the assumption that A 0 is semisimple. There do already exist several generalized Koszul theories where the degree 0 part A 0 of a graded algebra A is not required to be semisimple, see [5], [8], and [24]. Each Koszul ring A defined by Woodcock in [24] is supposed to satisfy that A is both a left projective A 0 -module and a right projective A 0 -module. ...
... There do already exist several generalized Koszul theories where the degree 0 part A 0 of a graded algebra A is not required to be semisimple, see [5], [8], and [24]. Each Koszul ring A defined by Woodcock in [24] is supposed to satisfy that A is both a left projective A 0 -module and a right projective A 0 -module. This requirement is too strong. ...
Let R be a commutative algebra. In this paper, we show that constant skew PBW extensions of a generalized Koszul algebra R are also generalized Koszul. Let A be a semi-commutative skew PBW extension of R such that A is R-augmented. We show also that if R is augmented Koszul then A is R-augmented Koszul.
... It follows that the Ext-algebra of kB is isomorphic to the incidence algebra of Λ(B). Results proved independently by Polo [87] and Woodcock [119] then yield that every open interval of the semilattice Λ(B) is a Cohen-Macaulay poset. This is particularly important because it has consequences for the Möbius function of Λ(B). ...
... Polo and Woodcock [87,119] independently showed that the incidence algebra I(P ; k) of a graded poset P is a Koszul algebra with respect to the natural grading if and only if each open interval in P is k-Cohen-Macaulay. We thus obtain the following theorem, which will be applied to enumerate cells of connected CW left regular bands. ...
In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such structures. In a recent paper, the authors established a close connection between algebraic and combinatorial invariants of a left regular band by showing that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order complexes of posets naturally associated to the left regular band.
The purpose of the present monograph is to further develop and deepen the connection between left regular bands and poset topology. This allows us to compute finite projective resolutions of all simple modules of unital left regular band algebras over fields and much more. In the process, we are led to define the class of CW left regular bands as the class of left regular bands whose associated posets are the face posets of regular CW complexes. Most of the examples that have arisen in the literature belong to this class. A new and important class of examples is a left regular band structure on the face poset of a CAT(0) cube complex. Also, the recently introduced notion of a COM (complex of oriented matroids or conditional oriented matroid) fits nicely into our setting and includes CAT(0) cube complexes and certain more general CAT(0) zonotopal complexes. A fairly complete picture of the representation theory for CW left regular bands is obtained.
... Say that ∆ is Cohen-Macaulay over k if for every face F of ∆, its link link ∆ (F ) := {G ∈ ∆ : F ∪ G ∈ ∆, F ∩ G = ∅} within ∆ has the property that its reduced homology vanishes below the dimension of the link:H Theorem 1.6. (Polo [23, §1.6], Woodcock [29,Theorem 3.7 ...
... We wish to characterize combinatorially when gr I A for A = k[P ] red is a quadratic ring, generalizing a result of Woodcock [29,Lemma 4.5] for incidence algebras of graded posets P . Woodcock's result uses the well-known notion of galleryconnectedness for pure simplicial complexes. ...
We prove a theorem unifying three results from combinatorial homological and commutative algebra, characterizing the Koszul property for incidence algebras of posets and affine semigroup rings, and characterizing linear resolutions of squarefree monomial ideals. The characterization in the graded setting is via the Cohen-Macaulay property of certain posets or simplicial complexes, and in the more general nongraded setting, via the sequential Cohen-Macaulay property. (C) 2010 Elsevier Inc. All rights reserved
... There exist numerous equivalent definitions of a Koszul algebra (see for example [3]). Koszul algebras have been defined in a more general way by some authors (see for example [4], [5], [6], [11], [26]). Other authors have studied some properties of algebras constructed from Koszul algebras (see for example [8] and [20]). ...
Pre-Koszul and Koszul algebras were defined by Priddy. There exist some relations between these algebras and the skew PBW extensions defined. We have established conditions to guarantee that skew PBW extensions over fields it turns out homogeneous pre-Koszul or Koszul algebra. In this paper we complement these results defining graded skew PBW extensions and showing that if R is a finite presented Koszul K-algebra then every graded skew PBW extension of R is Koszul.
... y −(x + y) ⋆ ⋆ 6 z −2y −2x ⋆ ⋆ 7 1 0 0 ⋆ ⋆ ⋆ 8 1 x 0 ⋆ ⋆ ⋆ 9 x 1 0 ⋆ ⋆ ⋆ 10 1 y x ⋆ ⋆ ⋆ where α ∈ K\{0}.3 KoszulitySome authors have defined Koszul algebras in a more general sense than[18] (see for example[4],[14],[27]). Our focus is to study the Koszul property for skew PBW extensions taking into account the definition given in[18]. ...
Koszul and homogeneous Koszul algebras were defined by Priddy in \cite{Priddy1970}. There exist some relations between these algebras and the skew PBW extensions introduced in \cite{LezamaGallego}. In this paper we give conditions to guarantee that skew PBW extensions over fields are Koszul or homogeneous Koszul. We also show that a constant skew PBW extension of a field is a PBW deformation of its homogeneous version.
... For certain classes of posets H * (P ) has nice algebraic structure. For example, if P is a Cohen-Macaulay poset, then its incidence algebra I(P ) is a Koszul algebra [Polo, 1995], [Woodcock, 1998]. Hence, H * (P ) is the Koszul dual algebra of I(P ). ...
This article presents a study of an algebra spanned by the faces of a hyperplane arrangement. The quiver with relations of the algebra is computed and the algebra is shown to be a Koszul algebra. It is shown that the algebra depends only on the intersection lattice of the hyperplane arrangement. A complete system of primitive orthogonal idempotents for the algebra is constructed and other algebraic structure is determined including: a description of the projective indecomposable modules; the Cartan invariants; projective resolutions of the simple modules; the Hochschild homology and cohomology; and the Koszul dual algebra. A new cohomology construction on posets is introduced and it is shown that the face semigroup algebra is isomorphic to the cohomology algebra when this construction is applied to the intersection lattice of the hyperplane arrangement.
... There are several papers containing generalizations of Koszul algebras for various purposes, those most relevant to our setting being [13,19,29]. We will not need these here, however, as the known properties of Koszul algebras over fields are sufficient to obtain our results. ...
Braverman and Gaitsgory gave necessary and sufficient conditions for a nonhomogeneous quadratic algebra to satisfy the Poincare-Birkhoff-Witt property when its homogeneous version is Koszul. We widen their viewpoint and consider a quotient of an algebra that is free over some (not necessarily semisimple) subalgebra. We show that their theorem holds under a weaker hypothesis: We require the homogeneous version of the nonhomogeneous quadratic algebra to be the skew group algebra (semidirect product algebra) of a finite group acting on a Koszul algebra, obtaining conditions for the Poincare-Birkhoff-Witt property over (nonsemisimple) group algebras. We prove our main results by exploiting a double complex adapted from Guccione, Guccione, and Valqui (formed from a Koszul complex and a resolution of the group), giving a practical way to analyze Hochschild cohomology and deformations of skew group algebras in positive characteristic. We apply these conditions to graded Hecke algebras and Drinfeld orbifold algebras (including rational Cherednik algebras and symplectic reflection algebras) in arbitrary characteristic, with special interest in the case when the characteristic of the underlying field divides the order of the acting group.
... The reader is referred to [10] for the definition of a Cohen-Macaulay poset over a field K. What we shall need is that the order complex of a Cohen-Macaulay poset is a chamber complex [10, Proposition 11.7] (note that a poset whose order complex is a chamber complex is called strongly connected in [10]). We shall also need that if P is a poset, then I(P, K) is Koszul if and only if (p, q) is Cohen-Macaulay over K for all p < q in P [52,73]. By the above discussion, if (p, q) is Cohen-Macaulay for all open intervals in P , then ∆(P ) is strongly connected in our sense. ...
The goal of this paper is to use topological methods to compute between an irreducible representation of a finite monoid inflated from its group completion and one inflated from its group of units, or more generally coinduced from a maximal subgroup, via a spectral sequence that collapses on the -page over fields of good characteristic. For von Neumann regular monoids in which Green's - and -relations coincide (e.g., left regular bands), the computation of between arbitrary simple modules reduces to this case, and so our results subsume those of S. Margolis, F. Saliola, and B. Steinberg, Combinatorial topology and the global dimension of algebras arising in combinatorics, J. Eur. Math. Soc. (JEMS), 17, 3037-3080 (2015). Applications include computing between arbitrary simple modules and computing a quiver presentation for the algebra of Hsiao's monoid of ordered G-partitions (connected to the Mantaci-Reutenauer descent algebra for the wreath product ). We show that this algebra is Koszul, compute its Koszul dual and compute minimal projective resolutions of all the simple modules using topology. These generalize the results of S. Margolis, F. V. Saliola, and B. Steinberg. Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry, Mem. Amer. Math. Soc., 274, 1-135, (2021). We also determine the global dimension of the algebra of the monoid of all affine transformations of a vector space over a finite field. We provide a topological characterization of when a monoid homomorphism induces a homological epimorphism of monoid algebras and apply it to semidirect products. Topology is used to construct projective resolutions of modules inflated from the group completion for sufficiently nice monoids.
... 4.6]. By [40] or [46] Proof. We claim that if I ⊆ J are poset ideals and I = I 0 ⊆ I 1 ⊆ · · · ⊆ I n = J a sequence of covering relations (meaning each I p+1 has cardinality one more than I p ), then the product (23) [I n , I n−1 ] · · · [I 1 , I 0 ] ...
We enrich the setting of strongly stable ideals (SSI): We introduce shift modules, a module category encompassing SSIs. The recently introduced duality on SSIs is given an effective conceptual and computational setting. We study SSIs in infinite dimensional polynomial rings, where the duality is most natural. Finally a new type of resolution for SSIs is introduced. This is the projective resolution in the category of shift modules.
... 4.6]. By [37] or [42] Lemma A.1. The largest degree d for which E(P ) d is nonzero is the largest cardinality of an antichain in P . ...
We enrich the setting of strongly stable ideals (SSI): We introduce shift modules, a module category encompassing SSI's. The recently introduced duality on SSI's is given an effective conceptual and computational setting. We study strongly stable ideals in infinite dimensional polynomial rings, where the duality is most natural. Finally a new type of resolution for SSI's is introduced. This is the projective resolution in the category of shift modules.
... The complex B P (M) arises from the Bar resolution of S(1), and the complex K P (M) arises from a minimal resolution (assuming that k is a field). Essentially, the complex K P (M) is a Koszul complex: there is a generalization of the theory of Koszul duality from graded algebras to graded linear categories, such that the category kP is Koszul if P is Cohen-Macaulay [31,12]. In this case, K P (M) is a Koszul complex. ...
We introduce a method for associating a chain complex to a module over a combinatorial category, such that if the complex is exact then the module has a rational Hilbert series. We prove homology--vanishing theorems for these complexes for several combinatorial categories including: the category of finite sets and injections, the opposite of the category of finite sets and surjections, and the category of finite dimensional vector spaces over a finite field and injections. Our main applications are to modules over the opposite of the category of finite sets and surjections, known as modules. We obtain many constraints on the sequence of symmetric group representations underlying a finitely generated module. In particular, we describe its character in terms of functions that we call character exponentials. Our results have new consequences for the character of the homology of the moduli space of stable marked curves, and for the equivariant Kazhdan-Luzstig polynomial of the braid matroid.
... This class of examples was considered initially in [20]. They were also studied even in a more general (nongraded) setting in [18]. ...
... y −(x + y) ⋆ ⋆ 6 z −2y −2x ⋆ ⋆ 7 1 0 0 ⋆ ⋆ ⋆ 8 1 x 0 ⋆ ⋆ ⋆ 9 x 1 0 ⋆ ⋆ ⋆ 10 1 y x ⋆ ⋆ ⋆ where α ∈ K\{0}.3 KoszulitySome authors have defined Koszul algebras in a more general sense than[18] (see for example[4],[14],[27]). Our focus is to study the Koszul property for skew PBW extensions taking into account the definition given in[18]. ...
Koszul and homogeneous Koszul algebras were defined by Priddy in \cite{Priddy1970}. There exist some relations between these algebras and the skew PBW extensions introduced in \cite{LezamaGallego}. In this paper we give conditions to guarantee that skew PBW extensions over fields are Koszul or homogeneous Koszul. We also show that a constant skew PBW extension of a field is a PBW deformation of its homogeneous version.
... Koszul algebras have been defined in a more general way by some authors. These algebras are commonly called "Generalized Koszul algebras" (see for example [3], [6], [12], [34]). In this paper we will consider the classical notion of Kozulity introduced by Priddy. ...
Pre-Koszul and Koszul algebras were defined by Priddy in [15 Priddy, S. (1970). Koszul resolutions. Trans. Am. Math. Soc. 152:39–60.[CrossRef]]. There exist some relations between these algebras and the skew PBW extensions defined in [8 Gallego, C., Lezama, O. (2011). Gröbner bases for ideals of σ-PBW extensions. Comm. Algebra 39(1):50–75.[Taylor & Francis Online], [Web of Science ®]]. In [24 Suárez, H., Reyes, A. Koszulity for skew PBW extensions over fields, submitted.] we gave conditions to guarantee that skew PBW extensions over fields it turns out homogeneous pre-Koszul or Koszul algebra. In this paper we complement these results defining graded skew PBW extensions and showing that if R is a finite presented Koszul 𝕂-algebra then every graded skew PBW extension of R is Koszul.
... where ζ s,t = n i=1 e s,ui ⊗ e ui,t is the element associated to the unique interval [s, t] of length 2 in P. Remark 5.16. In [Wo,Definition 4.6], the author defines a graded poset Ω as being exactly thin whenever x < y and l(x, y) = 2 imply that the interval (x, y) consists of precisely two elements. Note that what we termed 'planar tilings' are examples of such posets, which we proved to be Koszul. ...
In this paper we continue the study of Koszul pairs, focusing on new
applications. First, we prove in a unifying way several equivalent descriptions
of Koszul rings, some of them being well known in the literature. Second, we
use Koszul pairs to show that the dual version of the above mentioned theorem
holds true as well, yielding equivalent characterizations of Koszul corings. As
another application of Koszul pairs we prove that, in the locally finite case,
a graded ring is Koszul if and only if its left (or right) graded dual coring
is Koszul. In particular, for a finite graded poset we get that its incidence
ring is Koszul if and only if its incidence coring is so. In the last part of
the paper, we provide an algorithm to produce new examples of Koszul incidence
(co)rings.
... It has been shown independently by Polo [Pol95] and Woodcock [Woo98] that for a graded finite poset P there is an equivalence between the incidence algebra being Koszul and the order complex ∆P being Cohen-Macaulay. Recently Reiner and Stamate [RS10] have proved that the graded hypothesis is largely unnecessary by showing a similar equivalence between non-graded Koszul and sequentially Cohen-Macaulay. ...
Acyclic categories were introduced by D. Kozlov [Combinatorial algebraic topology. Berlin: Springer (2008; Zbl 1130.55001)] and can be viewed as generalized posets. Similar to posets, one can define their incidence algebras and a related topological complex. We consider the incidence algebra of either a poset or acyclic category as the quotient of a path algebra by the parallel ideal. We show that this ideal has a quadratic Gröbner basis with a lexicographic monomial order if and only if the poset or acyclic category is lexshellable.
... Proposition 1.1.1. [W,Theorem 2.3] Let B be a Koszul algebra. Set M = B 1 and S = B 0 . ...
Let B be a generalized Koszul algebra over a finite dimensional algebra
S. We construct a bimodule Koszul resolution of B when the projective
dimension of equals 2. Using this we prove a Poincar\'{e}-Birkhoff-Witt
(PBW) type theorem for a deformation of a generalized Koszul algebra. When the
projective dimension of is greater than 2, we construct bimodule Koszul
resolutions for generalized smash product algebras obtained from braidings
between finite dimensional algebras and Koszul algebras, and then prove the PBW
type theorem. The results obtained can be applied to standard Koszul
Artin-Schelter Gorenstein algebras in the sense of Minamoto and Mori.
... There are several papers containing generalizations of Koszul algebras for various purposes, those most relevant to our setting being [13,19,29]. We will not need these here, however, as the known properties of Koszul algebras over fields are sufficient to obtain our results. ...
Braverman and Gaitsgory gave necessary and sufficient conditions for a
nonhomogeneous quadratic algebra to satisfy the Poincare-Birkhoff-Witt
property when its homogeneous version is Koszul. We widen their
viewpoint and consider a quotient of an algebra that is free over some
(not necessarily semisimple) subalgebra. We show that their theorem
holds under a weaker hypothesis: We require the homogeneous version of
the nonhomogeneous quadratic algebra to be the skew group algebra
(semidirect product algebra) of a finite group acting on a Koszul
algebra, obtaining conditions for the Poincare-Birkhoff-Witt property
over (nonsemisimple) group algebras. We prove our main results by
exploiting a double complex adapted from Guccione, Guccione, and Valqui
(formed from a Koszul complex and a resolution of the group), giving a
practical way to analyze Hochschild cohomology and deformations of skew
group algebras in positive characteristic. We apply these conditions to
graded Hecke algebras and Drinfeld orbifold algebras (including rational
Cherednik algebras and symplectic reflection algebras) in arbitrary
characteristic, with special interest in the case when the
characteristic of the underlying field divides the order of the acting
group.
... Then the skew group algebra ΛG is still a locally finite graded algebra generated in degrees 0 and 1 with (ΛG) 0 = Λ 0 ⊗ kG. Since (ΛG) 0 in general is not semisimple, the classical Koszul theory described in [4,18] cannot apply, and we have to rely on generalized Koszul theories not requiring the semisimple property of (ΛG) 0 ; for example, [9,12,13,15,24]. In this paper we use the generalized Koszul theory developed in [12,13]. ...
In this paper we study representations of skew group algebras ,
where is a connected, basic, finite-dimensional algebra (or a locally
finite graded algebra) over an algebraically closed field k with
characteristic , and G is an arbitrary finite group each
element of which acts as an algebra automorphism on . We characterize
skew group algebras with finite global dimension or finite representation type,
and classify the representation types of transporter categories for . When is a locally finite graded algebra and the action of G
on preserves grading, we show that is a generalized
Koszul algebra if and only if so is .
... We note that the dimension of X(a, b) is rk Γ (b) − rk Γ (a) − 2. This is consistent with the definition: dim(∆(∅)) = −1. We also take as a (standard) conventionH n (∆(∅)) = 0 for n = −1 andH −1 (∆(∅)) = F (cf. [13]). We are now prepared to restate and then prove Theorem 1.2. ...
We study a finite dimensional quadratic graded algebra R defined from a
finite ranked poset. This algebra has been central to the study of the
splitting algebra of the poset, A, as introduced by Gelfand, Retakh, Serconek
and Wilson . The algebra A is known to be quadratic when the poset satisfies a
combinatorial condition known as uniform, and R is the quadratic dual of an
associated graded algebra of A. We prove that R is Koszul and the poset is
uniform if and only if the poset is Cohen-Macaulay. Koszulity of R implies
Koszulity of A. We also show that when R is Koszul, the cohomology of the order
complex of the poset can be identified with certain cohomology groups defined
internally to the ring R. Finally, we settle in the negative the long-standing
question: Does numerically Koszul imply Koszul for algebras of the form R?
... For certain classes of posets H * (P ) has nice algebraic structure. For example, if P is a Cohen-Macaulay poset, then its incidence algebra I(P ) is a Koszul algebra [Polo, 1995], [Woodcock, 1998]. Hence, H * (P ) is the Koszul dual algebra of I(P ). ...
This article presents a study of an algebra spanned by the faces of a hyperplane arrangement. The quiver with relations of the algebra is computed and the algebra is shown to be a Koszul algebra. It is shown that the algebra depends only on the intersection lattice of the hyperplane arrangement. A complete system of primitive orthogonal idempotents for the algebra is constructed and other algebraic structure is determined including: a description of the projective indecomposable modules; the Cartan invariants; projective resolutions of the simple modules; the Hochschild homology and cohomology; and the Koszul dual algebra. A new cohomology construction on posets is introduced and it is shown that the face semigroup algebra is isomorphic to the cohomology algebra when this construction is applied to the intersection lattice of the hyperplane arrangement.
... Proof. By [9,15], R is Koszul if and only if the order complex ∆(I) is Cohen-Macaulay over k for any open interval I of Σ. Set Σ ′ := Σ \ ∅. Note that ∆(I) = lk ∆(Σ ′ ) F for some F ∈ ∆(Σ ′ ) containing a maximal cell σ ∈ Σ. Set ∆ := st ∆(Σ ′ ) σ. Then ∆(I) = lk ∆ F . ...
Let be a finite regular cell complex with , and regard it as a partially ordered set (poset) by inclusion. Let R be the incidence algebra of the poset over a field k. Corresponding to the Verdier duality for constructible sheaves on , we have a dualizing complex giving a duality functor from to itself. w satisfies the Auslander condition. Our duality is somewhat analogous to the Serre duality for projective schemes ( plays a similar role to that of "irrelevant ideals"). If for exactly one i, then the underlying topological space of is Cohen-Macaulay (in the sense of the Stanley-Reisner ring theory). The converse also holds when is a simplicial complex. R is always a Koszul ring with . The relation between the Koszul duality for R and the Verdier duality is discussed. This result is a variant of a theorem of Vybornov. The Mobius function of the poset is also discussed.
We give a topological description of Ext groups between simple representations of categories via a nerve type construction. We use it to show that the Koszulity of indiscretely based category algebras is equivalent to the locally Cohen-Macaulay property of this nerve. We also provide a class of functors which preserve the Koszulity of category algebras called almost discrete fibrations. Specializing from categories to posets, we show that the equivalence relations of V. Reiner and D. Stamate in arXiv:0904.1683 [math.AC] are exactly almost discrete fibrations and recover their results. As an application, we classify when a shifted dual collection to a full strong exceptional collection of line bundles on a toric variety is strong.
We define a poset of partitions associated to an operad. We prove that the operad is Koszul if and only if the poset is Cohen-Macaulay. In one hand, this characterisation allows us to compute the homology of the poset. This homology is given by the Koszul dual operad. On the other hand, we get new methods for proving that an operad is Koszul.
We show that lattice polytopes cut out by root systems of classical type are normal and Koszul, generalizing a well-known result of Bruns, Gubeladze, and Trung in type A. We prove similar results for Cayley sums of collections of polytopes whose Minkowski sums are cut out by root systems. The proofs are based on a combinatorial characterization of diagonally split toric varieties.
We define generalized Koszul modules and rings and develop a generalized Koszul theory for -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 Martin{\'e}z-Villa. Let A be a left finite -graded ring generated in degree 1 with noetherian semiperfect, J be its graded Jacobson radical and S=A/J. By the Koszul dual of A we mean the Yoneda Ext ring . 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 and the Koszul dual of the Koszul dual of M is . If A is a locally finite algebra, then the following statements are proved to be equivalent: A is generalized Koszul; the Koszul dual of A is (classically) Koszul; is (classically) Koszul; the opposite ring of A is generalized Koszul. It is also 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.
We associate with every pure flag simplicial complex a standard graded Gorenstein -algebra whose homological features are largely dictated by the combinatorics and topology of . As our main result, we prove that the residue field has a k-step linear -resolution if and only if satisfies Serre’s condition over and that is Koszul if and only if is Cohen–Macaulay over . Moreover, we show that has a quadratic Gröbner basis if and only if is shellable. We give two applications: first, we construct quadratic Gorenstein -algebras that are Koszul if and only if the characteristic of is not in any prescribed set of primes. Finally, we prove that whenever is Koszul the coefficients of its -vector alternate in sign, settling in the negative an algebraic generalization of a conjecture by Charney and Davis.
Koszul rings are graded rings which have played an important role in algebraic topology, algebraic geometry, noncommutative algebraic geometry, and in the theory of quantum groups. One aspect of the theory is to compare the module theory for a Koszul ring and its Koszul dual. There are dualities between subcategories of graded modules; the Koszul modules. When Λ is an artin algebra and T is a cotilting Λ-module, the functor HomΛ(, T) induces a duality between certain subcategories of the finitely generated modules over Λ and EndΛ(T). The purpose of this paper is to develop a unified approach to both the Koszul duality and the duality for cotilting modules. This theory specializes to these two cases and also contains interesting new examples. The starting point for the theory is a positively ℤ-graded ring Λ = Λ0 + Λ1 + Λ2+ ⋯ and a (Wakamatsu) cotilting Λ0-module T, satisfying additional assumptions. The theory gives a duality between certain subcategories of the finitely generated graded modules generated in degree zero over Λ on one hand and over the Yoneda algebra ⊕i≥0 ExtΛi(T, T) on the other hand.
There are many structures (algebras, categories, etc) with natural gradings
such that the degree 0 components are not semisimple. Particular examples
include tensor algebras with non-semisimple degree 0 parts, extension algebras
of standard modules of standardly stratified algebras. In this thesis we
develop a generalized Koszul theory for graded algebras (categories) whose
degree 0 parts may be non-semisimple. Under some extra assumption, we show that
this generalized Koszul theory preserves many classical results such as the
Koszul duality. Moreover, it has some close relation to the classical theory.
Applications of this generalized theory to finite EI categories, directed
categories, and extension algebras of standard modules of standardly stratified
algebras are described. We also study the stratification property of standardly
stratified algebras, and classify algebras standardly (resp., properly)
stratified for all linear orders.
We show that lattice polytopes cut out by root systems of classical type are normal and Koszul, generalizing a well-known result of Bruns, Gubeladze, and Trung in type A. We prove similar results for Cayley sums of collections of polytopes whose Minkowski sums are cut out by root systems. The proofs are based on a combinatorial characterization of diagonally split toric varieties.
We define a family of posets of partitions associated to an operad. We prove that the operad is Koszul if and only if the posets are Cohen–Macaulay. On the one hand, this characterization allows us to compute completely the homology of the posets. The homology groups are isomorphic to the Koszul dual cooperad. On the other hand, we get new methods for proving that an operad is Koszul.
Let be a graded locally finite
k-algebra such that is an arbitrary finite-dimensional algebra
satisfying some splitting condition. In this paper we develop a generalized
Koszul theory generalizing many classical results, and use it to classify
algebras stratified for all preorders.
ResearchGate has not been able to resolve any references for this publication.