Edward L. Green

• PhD
• Professor Emeritus at Virginia Tech

Let $KQ$ be a path algebra, where $Q$ is a finite quiver and $K$ is field. We study $KQ/C$ where $C$ is the two-sided ideal in $KQ$ generate by all differences of parallel paths in $Q$. We show that $KQ/C$ is always finite dimensional and its global dimension is finite. Furthermore, we prove the $KQ/C$ is Morita equivalent to an incidence algebra....

In this paper we introduce new affine algebraic varieties whose points correspond to associative algebras. We show that the algebras within a variety share many important homological properties. In particular, any two algebras in the same variety have the same dimension. The cases of finite dimensional algebras as well as that of graded algebras ar...

In this paper we develop new reduction techniques for testing the finiteness of the finitistic dimension of a finite dimensional algebra over a field. Viewing the latter algebra as a quotient of a path algebra, we propose two operations on the quiver of the algebra, namely arrow removal and vertex removal. The former gives rise to cleft extensions...

We return to the fusion rules for the Drinfeld double of the duals of the generalised Taft algebras that we studied in Erdmann et al. (J. Pure Appl. Algebra 204, 413-454 2006). We first correct some proofs and statements in Erdmann et al. (J. Pure Appl. Algebra 204, 413-454 2006) that were incorrect, using stable homomorphisms. We then complete thi...

In this paper we develop new reduction techniques for testing the finiteness of the finitistic dimension of a finite dimensional algebra over a field. Viewing the latter algebra as a quotient of a path algebra, we propose two operations on the quiver of the algebra, namely arrow removal and vertex removal. The former gives rise to cleft extensions...

Previous investigations in gene expression changes in blood after radiation exposure have highlighted its potential to provide biomarkers of exposure. Here, FDXR transcriptional changes in blood were investigated in humans undergoing a range of external radiation exposure procedures covering several orders of magnitude (cardiac fluoroscopy, diagnos...

Previous investigations in gene expression changes in blood after radiation exposure have highlighted its potential to provide biomarkers of exposure. Here, FDXR transcriptional changes in blood were investigated in humans undergoing a range of external radiation exposure procedures covering several orders of magnitude (cardiac fluoroscopy, diagnos...

We survey results on multiserial algebras, special multiserial algebras and Brauer configuration algebras. A structural property of modules over a special multiserial algebra is presented. Almost gentle algebras are introduced and we describe some results related to this class of algebras. We also report on the structure of radical cubed zero symme...

In this paper we introduce an easily verifiable sufficient condition to determine whether an algebra is quasi-hereditary. In the case of monomial algebras, we give conditions that are both necessary and sufficient to show whether an algebra is quasi-hereditary.

Establishing whether an algebra is quasi-hereditary or not is, in general, a difficult problem. In this paper we introduce a sufficient criterion to determine whether a general finite dimensional algebra is quasi-hereditary by showing that the question can be reduced to showing that a closely associated monomial algebra is quasi-hereditary. For mon...

In this paper we introduce new affine algebraic varieties whose points correspond to associative algebras. We show that the algebras within a variety share many important homological properties. In particular, any two algebras in the same variety have the same dimension. The case of finite dimensional algebras as well as that of graded algebras ari...

Let $\cQ$ be a quiver and $K$ a field. We study the interrelationship of homological properties of algebras associated to convex subquivers of $\cQ$ and quotients of the path algebra $K\cQ$. We introduce the homological heart of $\cQ$ which is a particularly nice convex subquiver of $\cQ$. For any algebra of the form $K\cQ/I$, the algebra associate...

Let $\cQ$ be a quiver and $K$ a field. We study the interrelationship of homological properties of algebras associated to convex subquivers of $\cQ$ and quotients of the path algebra $K\cQ$. We introduce the homological heart of $\cQ$ which is a particularly nice convex subquiver of $\cQ$. For any algebra of the form $K\cQ/I$, the algebra associate...

We survey results on multiserial algebras, special multiserial algebras and Brauer configuration algebras. A structural property of modules over a special multiserial algebra is presented. Almost gentle algebras are introduced and we describe some results related to this class of algebras. We also report on the structure of radical cubed zero symme...

Koszul algebras with quadratic Groebner bases, called strong Koszul algebras, are studied. We introduce affine algebraic varieties whose points are in one-to-one correspondence with certain strong Koszul algebras and we investigate the connection between the varieties and the algebras.

We return to the fusion rules for the Drinfeld double of the duals of the generalised Taft algebras that we studied in [Erdmann et al., J. Pure Appl. Algebra 2006]. We first correct some proofs and statements in [Erdmann et al., J. Pure Appl. Algebra 2006] that were incorrect, using stable homomorphisms. We then complete this with new results on fu...

In this paper we define almost gentle algebras. They are monomial special multiserial algebras generalizing gentle algebras. We show that the trivial extension of an almost gentle algebra by its minimal injective co-generator is a symmetric special multiserial algebra and hence a Brauer configuration algebra. Conversely, we show that admissible cut...

In this paper we define almost gentle algebras. They are monomial special multiserial algebras generalizing gentle algebras. We show that the trivial extension of an almost gentle algebra by its minimal injective co-generator is a symmetric special multiserial algebra and hence a Brauer configuration algebra. Conversely, we show that any almost gen...

In this paper we give a new definition of symmetric special multiserial algebras in terms of defining cycles. As a consequence, we show that every special multiserial algebra is a quotient of a symmetric special multiserial algebra.

In this paper we study multiserial and special multiserial algebras. These algebras are a natural generalization of biserial and special biserial algebras to algebras of wild representation type. We define a module to be multiserial if its radical is the sum of uniserial modules whose pairwise intersection is either 0 or a simple module. We show th...

In this paper we introduce a generalization of a Brauer graph algebra which we call a Brauer configuration algebra. As with Brauer graphs and Brauer graph algebras, to each Brauer configuration, there is an associated Brauer configuration algebra. We show that Brauer configuration algebras are finite dimensional symmetric algebras. After studying a...

Let R be a Koszul algebra over a field k and M be a linear R-module. We study a graded subalgebra ΔM of the Ext-algebra ExtR⁎(M,M) called the diagonal subalgebra and its properties. Applications to the Hochschild cohomology ring of R and to periodicity of linear modules are given. Viewing R as a linear module over its enveloping algebra, we also sh...

If $f$ is an idempotent in a ring $\Lambda$, then we find sufficient \linebreak conditions which imply that the cohomology rings $\oplus_{n\ge 0}Ext^n_{\Lambda}(\Lambda/{\br},\Lambda/{\br})$ and \linebreak $\oplus_{n\ge 0}Ext^n_{f\Lambda f}(f\Lambda f/f{\br} f,f\Lambda f/f{\br} f)$ are eventually isomorphic. This result allows us to compare finite...

The paper begins with a detailed study of the category of modules over two different rings using a coproduct construction. If C is a commutative ring and R and S are rings together with ring homomorphisms from C to R and C to S, then we show that the category of C-modules that are also left R-modules and right S-modules is equivalent to the categor...

We study Morita rings $\Lambda_{(\phi,\psi)}=\bigl({smallmatrix} A &_AN_B_BM_A & B {smallmatrix}\bigr)$ in the context of Artin algebras from various perspectives. First we study covariant finite, contravariant finite, and functorially finite subcategories of the module category of a Morita ring when the bimodule homomorphisms $\phi$ and $\psi$ are...

In this paper we study finite generation of the Ext algebra of a Brauer graph algebra by determining the degrees of the generators. As a consequence we characterize the Brauer graph algebras that are Koszul and those that are K_2.

In this article, we study subrings of the Ext-algebra of a graded module over a graded ring R. We show these subrings can be defined by equivalence relations on exact sequences over the ring. In particular, the shriek ring, R , and the even part of an Ext-algebra of a d-Koszul algebra can be defined by equivalence relations on exact sequences.

We develop a theory of group actions and coverings on Brauer graphs that parallels the theory of group actions and coverings of algebras. In particular, we show that any Brauer graph can be covered by a tower of coverings of Brauer graphs such that the topmost covering has multiplicity function identically one, no loops, and no multiple edges. Furt...

In this paper we study Auslander-Reiten sequences of modules with finite complexity over selfinjective artin algebras. In particular, we show that for all eventually Ω-perfect modules of finite complexity, the number of indecomposable non projective summands of the middle term of such sequences is bounded by 4. We also describe situations in which...

Let G{{\mathcal G}} be a group, Λ a G{{\mathcal G}}-graded Artin algebra and gr(Λ) denote the category of finitely generated G{{\mathcal G}}-graded Λ-modules. This paper provides a framework that allows an extension of tilting theory to Db(gr(L)){{\mathcal D}}^b(\rm gr(\Lambda)) and to study connections between the tilting theories of Db(L){{\mathc...

Let R be a connected selfinjective Artin algebra, and M an indecomposable nonprojective R-module with bounded Betti numbers lying in a regular component of the Auslander-Reiten quiver of R. We prove that the Auslander-Reiten sequence ending at M has at most two indecomposable summands in the middle term. Furthermore we show that the component of th...

The relationship between an algebra and its associated monomial algebra is investigated when at least one of the algebras is d-Koszul. It is shown that an algebra which has a reduced Gröbnerbasis that is composed of homogeneous elements of degree d is d-Koszul if and only if its associated monomial algebra is d-Koszul. The class of 2-d-determined a...

Suppose that R is a group graded K-algebra, where K is a commutative ring and R is graded by a group G. The G-grading of R leads to a G-grading of certain Ext-algebras of R. On the other hand, with the G-grading of R, one associates a ‘covering’ algebra S. This paper begins by studying the relationship between Ext-algebras of the covering S and the...

Let R = R(0) circle plus R(1) circle plus R(2) circle plus ... be a graded algebra over a field K such that R(0) is a finite product of copies of K and each R(i) is finite dimensional over K. Set J = R(1) circle plus R(2) circle plus ... and S = circle plus(n >= 0) Ext(R)(n) (R/J, R/J). We study the properties of the categories of graded R-modules...

A major result in Algebraic Geometry is the theorem of Bernstein–Gelfand–Gelfand that states the existence of an equivalence of triangulated categories: grΛ ≅ 𝒟(Coh ℙ), where grΛ denotes the stable category of finitely generated graded modules over the n + 1 exterior algebra and 𝒟(Coh ℙ) is the derived category of bounded complexes of coherent shea...

Let AA be a finite-dimensional hereditary algebra of finite or tame representation type over a finite field, and let MM be a rigid AA-module. Then the element [M][M] in the Ringel–Hall algebra H( A )\mathcal{H}{\left( A \right)} is an iterated skew commutator of the isoclasses of simple AA-modules. This gives a new characterization of the rigidness...

Without Abstract

The main result of this article is the establishment of a new connection between combinatorics and noncommutative algebra. This is done by linking a certain class of directed graphs, called full graphs, to quotients of path algebras that are Koszul algebras.

This paper continues the study of n-full graphs and their connection to certain Koszul algebras started in Green and Hartman (to appear). We provide constructive methods for creating new full graphs from old and study the associated Koszul algebras and the projective resolution of simple modules over such algebras.

In this paper we study the finite generation of Ext-algebras of a class of algebras called δ-resolution determined algebras. We characterize the δ-resolution determined algebras which are monomial algebras. If Λ is a graded algebra such that the associated monomial algebra is δ-resolution determined, we classify when the Ext-algebra of Λ is finitel...

This paper studies the Hochschild cohomology of finite-dimensional monomial algebras. If Λ = KQ/I with I an admissible monomial ideal, then we give sufficient conditions for the existence of an embedding of K[x1,…, xr]/ 〈xaxb for a ≠ b〉 into the Hochschild cohomology ring HH*(Λ). We also introduce stacked algebras, a new class of monomial algebras...

We provide an algorithmic method for constructing projective resolutions of modules over quotients of path algebras. This algorithm is modified to construct minimal projective resolutions of linear modules over Koszul algebras.

Let Λ = kQ/I be a Koszul algebra over a field k, where Q is a finite quiver. An algorithmic method for finding a minimal projective resolution 𝔽 of the graded simple modules over Λ is given in [E. L. Green and Ø. Solberg, An algorithmic approach to resolutions, J. Symbolic Comput., 42 (2007), 1012–1033]. This resolution is shown to have a ‘comultip...

In this paper we show that if L = \mathop \coprod i \geqq 0 L i \Lambda = \mathop \coprod \limits_{i \geqq 0} \Lambda _i is a Koszul algebra with Λ0 isomorphic to a product of copies of a field, then the minimal projective resolution of Λ0 as a right Λ-module provides all the information necessary to construct both a minimal projective resolution...

Let A = A 0 ⊕ A 1 ⊕ A 2 ⊕ ⋯ be a graded K-algebra such that A 0 is a finite product of copies of the field K, A is generated in degrees 0 and 1, and dim KA 1 < ∞. We study those graded algebras A with the properly that A 0, viewed as a graded A-module, has a graded projective resolution, ⋯ → P t → ⋯ P 1 → P 0 → A 0 → 0, such that each P i can be ge...

We study the subcategory of modules with linear presentations and the subcategory of modules with linear resolutions,i.e.,Koszul modules,over a Koszul algebra. In particular,we investigate conditions when these categories coincide.

We investigate the Drinfeld doubles D(Λn,d) of a certain family of Hopf algebras. We determine their simple modules and their indecomposable projective modules, and we obtain a presentation by quiver and relations of these Drinfeld doubles, from which we deduce properties of their representations, including the Auslander–Reiten quivers of the D(Λn,...

In this paper we study d-Koszul algebras which were introduced by Berger. We show that when d⩾3, these are classified by the Ext-algebra being generated in degrees 0, 1, and 2. We show the Ext-algebra, after regrading, is a Koszul algebra and present the structure of the Ext-algebra.

Dieter Happel asked the following question: If the $n$-th Hochschild cohomology group of a finite dimensional algebra $\Gamma$ over a field vanishes for all sufficiently large $n$, is the global dimension of $\Gamma$ finite? We give a negative answer to this question.

For a finite dimensional monomial algebra Λ over a field K we show that the Hochschild cohomology ring of Λ modulo the ideal generated by homogeneous nilpotent elements is a commutative finitely generated K-algebra of Krull dimension at most one. This was conjectured to be true for any finite dimensional algebra over a field in [13].

We give a general construction which shows that a large class of quantum complete intersections can be realized as the basic algebras of non‐principal blocks of finite groups. We investigate the Ext rings of these algebras. We describe how to construct a finite p′‐covering for one of these quantum complete intersections, which supports a Hopf algeb...

Given a nite alphabet X and an ordering on the letters, the map sends each monomial on X to the word that is the ordered product of the letter powers in the monomial. Motivated by a question on Grobner bases, we characterize ideals I in the free commutative monoid (in terms of a generating set) such that the ideal h (I)i generated by (I) in the fr...

In this paper we construct explicitly the first terms in the minimal projective bimodule resolution of a finite-dimensional algebra from the minimal right resolution of each of the simple modules. This result is used to give vanishing results for HH2 of a finite-dimensional algebra, and in particular shows that HH2 = 0 for all Möbius algebras, with...

This paper describes the Hochschild cohomology ring of a selfinjective algebra Λ \Lambda of finite representation type over an algebraically closed field K K , showing that the quotient HH ∗ ⁡ ( Λ ) / N \operatorname {HH}^*(\Lambda )/\mathcal {N} of the Hochschild cohomology ring by the ideal N {\mathcal N} generated by all homogeneous nilpotent el...

Given a finite alphabet X and an ordering on the letters, the map \sigma sends each monomial on X to the word that is the ordered product of the letter powers in the monomial. Motivated by a question on Groebner bases, we characterize ideals I in the free commutative monoid (in terms of a generating set) such that the ideal <\sigma(I)> generated by...

Algebras for which every module is a Koszul module are classified. A necessary condition for the subcategory of graded modules with linear presentations to be equal to the subcategory of Koszul modules is given. This condition is also a sufficient condition when the algebra is radical cube zero. Finally, these subcategories are studied when the alg...

This paper studies the ring structure of the Hochschild cohomology ring of an algebra. The first main result gives a ring homomorphism from the Hochschild cohomology ring of an algebra A to the Ext-algebra of an A-module. Then, for a one point extension B of a finite dimensional algebra A, we relate the ring structures of the Hochschild cohomology...

This paper describes the Hochschild cohomology ring of a selfinjective algebra Λ of finite representation type over an algebraically closed field K, showing that the quotient HH^*(Λ)/ N of the Hochschild cohomology ring by the ideal N generated by all homogeneous nilpotent elements is isomorphic to either K or K[x], and is thus finitely g...

In this paper we study N-koszul algebras which were introduced by R. Berger. We show that when n 3, these are classified by the Ext-algebra being generated in degrees 0, 1, and 2. We give a description of the Ext-algebra using the analogous of the Koszul complex and we also show that it is, is a Koszul algebra, after regrading. This notions can be...

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 m...

Given a finite-dimensional algebra, we present sufficient conditions on the projective presentation of the algebra modulo its radical for a tilted algebra to be a Koszul algebra and for the endomorphism ring of a tilting module to be a quasi-Koszul algebra. One condition we impose is that the algebra has global dimension no greater than 2. One of t...

We present a new deterministic algorithm for constructing homomorphism spaces and endomorphism rings of finite dimensional modules. The modules are given via vertex projective presentations over path algebras and finite dimensional quotients of path algebras. We use the theory of right Grobner bases to encode modules and to construct appropriate sy...

In this paper, we present an algorithmic method for computing a projective resolution of a module over an algebra over a field. If the algebra is finite dimensional, and the module is finitely generated, we have a computa-tional way of obtaining a minimal projective resolution, maps included. This resolution turns out to be a graded resolution if o...

Submitted for publication to Journal of Symbolic Computation. October 22, 1999 A Hopf algebra is an algebra that also has a compatible coalgebraic structure and an antipode. The Hopf project at Virginia Tech is developing in GAP the Hopf system to study and compute with Hopf algebras. The Drinfel'd double is an important and complex construction th...

When an algebra is graded by a group, any additive character of the group induces a diagonalizable derivation of the ring. This construction is studied in detail for the case of a path algebra modulo relations and its fundamental group. We describe an injection of the character group into the first cohomology group following Assem-de la Peña. Rathe...

When an algebra is graded by a group, any additive character of the group induces a diagonalizable derivation of the ring. This construction is studied in detail for the case of a path algebra modulo relations and its fundamental group. We describe an injection of the character group into the rst cohomology group following Assem-de la Pena. Rather...

We present a new deterministic algorithm for constructing endomorphism rings of a finite dimensional module M, given via a vertex projective presentation, over finite dimensional quotients of path algebras. We use the theory of right Gröbner basis to encode M and to construct appropriate systems of equations for finding the endomorphism ring of M....

In this paper, we study conditions on algebras with multiplicative bases so that there is a Gröbner basis theory. We introduce right Gröbner bases for a class of modules. We give an elimination theory and intersection theory for right submodules of projective modules in path algebras. Solutions to homogeneous systems of linear equations with coeffi...

eatures of the theory of Grobner basis is the selection of a well-ordered basis. Recall that ! is a well-order on B if ! is a total order on B and every nonempty subset of B has a minimal element. The standard axioms of set theory imply that every set can be well-ordered. Let ! be a well-order on B. We recall a basic properties of !. Proposition 1....

We introduce the notions of self-dual (graded) Hopf algebras and of structurally simple (graded) Hopf algebras. We prove that the self-dual Hopf algebras are structurally simple and provide a construction of self-dual Hopf algebras. Finally, we classify the self-dual quotients of the form TB (M)/I, where TB (M) is a path algebra with a graded Hopf...

These notes consist of five sections. The aim of these notes is to provide a summary of the theory of noncommutative Gröbner bases and how to apply this theory in representation theory; most notably, in constructing projective resolutions.

In this paper we find necessary and sufficient conditions for an algebra to be a monomial algebra. These are conditions on finite abelian group gradings of the algebra and the first Hochschild cohomology group of the associated covering algebra of the grading. In particular, for a certain class of algebras, we show that an algebra A is a monomial a...

This paper investigates the structure of basic finite dimensional Hopf algebras H over an algebraically closed field k. The algebra H is basic provided H modulo its Jacobson radical is a product of the field k. In this case H is isomorphic to a path algebra given by a finite quiver with relations. Necessary conditions on the quiver and on the coalg...

This paper develops new techniques for studying modules over a ring from knowledge of homomorphic images of the modules. The concept of disjoint ideals is introduced in this regard. Applications are given to the construction of almost split sequences over Artinian algebras.

We extend some of the basic results about Koszul algebras to the nonlocal case. In doing so, we provide new elementary proofs of some of the fundamental results in this area. We also investigate nongraded noetherian semiperfect algebras; defining and studying quasi-Koszul algebras in this setting. The Yoneda algebra of an Auslander algebra is studi...

Suppose thatGis a finite group andkis a field of characteristicp0. In this paper we describe a scheme for computing the Ext algebra ofkG, i.e. the algebra Ext*kg(T,T) whereTis the sum of irreduciblekG-modules.

Without Abstract

This book contains seven lectures delivered at The Maurice Auslander Memorial Conference at Brandeis University in March 1995. The variety of topics covered at the conference reflects the breadth of Maurice Auslander's contribution to mathematics, which includes commutative algebra and algebraic geometry, homological algebra and representation theo...

Graded Artin algebras whose category of graded modules is locally of finite representation type are introduced. The representation theory of such algebras is studied. In the hereditary case and in the stably equivalent to hereditary case, such algebras are classified.

General properties of finite dimensional Hopf algebras are investi-gated. In particular, we generalize some of the work on almost split sequences of Auslander and Carlson on group rings to (not necessarily cocommutative) Hopf algebras with involutive antipode. We also give a new proof of a theorem of Larson, [2], which says that for a finite dimens...

Given a group-graded free associative algebra, we show that in many cases the path algebra associated to the covering coming from the grading has a Hopf algebra structure. Our structure on the path algebra is that of a quantum group far most of these constructions. Adding more restrictions, we create some finite-dimensional quotients which inherit...

Gröbner bases and minimal graded projective resolutions of the trivial module over the straightening closed algebras generated by minors of matrices, or, more generally, over the straightening closed subalgebras of supersymmetric letterplace algebras, are studied. Such algebras provide a new class of Koszul algebras.

A local theory of graded modules is introduced in this paper. In the case when the graded ring is a quotient ring of a path algebra, it is shown how to retrieve the original algebra from knowledge of the local theory. Almost split sequences are investigated from a local point of view.

In this paper we study the algebra structure of the cohomology ring of a monomial algebra.

In this paper we consider a specific example A of a finite dimensional algebra over an algebraically closed field K. This algebra is local and basic. That is, the algebra A is given by a quiver with only one vertex I and loops with some relations. We prove that the cohomology ring Ext* (S, S) of the only simple A-module S is not finitely generated....