Edward L. Green

Graded Artin Algebras

A graded Artin algebra is a graded ring which, neglecting the grading, is an Artin algebra. Although this definition has the merit of brevity, it does not expose some of the basic properties of graded Artin algebras; for instance, that there is a bound on absolute values of degrees of homogeneous elements. Indeed, a different definition is given in Section 1, and the discrepancy is rectified by Theory 1.4. We admit that our chief interest in graded Artin algebras lies in applications to the representation theory of Artin algebras-see [ 5 1. We feel, nonetheless, that graded Artin algebras are interesting objects of study in their own right, and some of our techniques (especially in Sections 5 and 6) are equally applicable to arbitrary graded rings. Thus, this paper, unlike its sequel [ 51, is designed to be read by a general ring theoretical audience. It is, in addition, self-contained from the standpoint of knowledge concerning graded rings required. The major concern of the paper is the existence of gradings on given modules over a graded Artin algebra-we use the term gradable when a gradation exists. Our transcendent result along these lines is that direct summands of finitely generated gradable modules are gradable. It follows immediately (see Section 3) that simple modules and projective modules over a graded Artin algebra are gradable. Also, inasmuch as we do not restrict ourselves to positively graded rings, endomorphism rings of graded finitely generated modules over graded Artin algebras are graded Artin algebras. In a similar vein, duality and horn

GRAPHS WITH RELATIONS, COVERINGS AND GROUP-GRADED ALGEBRAS

The paper studies the interrelationship between coverings of finite directed graphs and gradings of the path algebras associated to the directed graphs. To include gradings of all basic finite-dimensional algebras over an algebraically closed field, a theory of coverings of graphs with relations is introduced. The object of this paper is to relate group gradings on algebras to coverings of a graph which is associated to the algebra. The linking of the theories allows one to relate purely algebraic questions to questions in algebraic topology, group theory or combinatorics. In the representation theory of Artin algebras the association to each algebra of a finite directed graph, called the quiver of the algebra, has been a useful tool. The reason that the quiver of such an algebra is of interest is that there is a natural definition of representations of the quiver so that the category of representations of the quiver satisfying certain relations is equivalent to the category of finitely generated modules over the algebra. §1 gives a slight extension of these concepts to finitely generated algebras. The main emphasis of the paper is to show that the theory of coverings of graphs with relations, introduced by C. Riedtmann (9) and expanded by P. Gabriel (2), and the theory of group-graded algebras are essentially the same. Although the original connection between coverings and gradings was inspired by the similarity of results for Z-graded Artin algebras (3,4) and P. Gabriels announced results (2), the context of this paper is more general and deals with all finitely generated algebras over a field. We associate to each such algebra a finite directed graph which we still call a quiver of the algebra. We show that for each regular covering T of a quiver T0 of an algebra A, with certain prescribed restrictions, we get a G-grading of the algebra A, where G is the automorphism group of the covering Y over ro. Conversely, given a certain type of G-grading of A, where G is a group, we construct a regular covering Y of the quiver T0 of the algebra such that G is isomorphic to the automorphism group of T over ro. Furthermore, if Y is a regular covering of T0 with automorphism group G, we show that the category of representations of Y satisfying a certain set of relations is equivalent to the category of finite-dimensional graded G-modules. We

Synergy in the theories of Gröbner bases and path algebras

A general theory for Grobner basis in path algebras is introduced which extends the known theory for commutative polynomial rings and free associative algebras

On the homology of quotients of path algebras

We extend the results and techniques of [Al] to find a method of constructing projective resolutions for certain simple modules over homomorphic images of path algebras. We provide a number of applications in the case when the image algebra is finite dimensional.

Minimal projective resolutions

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 computational way of obtaining a minimal projective resolution, maps included. This resolution turns out to be a graded resolution if our algebra and module are graded. We apply this resolution to the study of the Ext-algebra of the algebra; namely, we present a new method for computing Yoneda products using the constructions of the resolutions. We also use our resolution to prove a case of the “no loop” conjecture.

Representation theory of graded Artin algebras

Having initiated the study of graded Artin algebras A in [ 91, here we initiate the study of their representation theory. Our point of view is to study graded /i-modules, which we believe to be somewhat more tractable than ungraded ones, in order to obtain information about all /i-modules. This point of view leads to the introduction, in Section 1, of the full subcategory mod,@!) of mod/i consisting of the gradable objects of mod/i; that is, the finitely generated /i-modules which support a gradation. In these terms, one of the major results of the paper asserts that mod,@) has finite representation type if and only if mod/i has finite representation type. Thus, in Section 3 we introduce a number G = G(4) designed to measure the size of mod,(/i). If G = co, we show that /i has infinite representation type. If G < co, we show that there is a graded Artin algebra Q of a certain specified form such that /i has infinite representation type precisely when 0 has infinite representation type. The Artin algebra fl has, in particular, desirable diagrammatic properties; and these we will exploit elsewhere. We speculate that when G is finite, every finitely generated /i-module is gradable. Now, in case n has finite representation type, it is obvious that G is finite. We show, indeed, that for every gradation of a given Artin algebra of finite representation type, every module is gradable. This result, and the others cited, are chiefly consequences of a result proved in Section 4: If a component of the Auslander-Reiten graph of a graded Artin algebra contains a gradable module, then the component

Noncommutative Gröbner Bases, and Projective Resolutions

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

Multiplicative bases, Gröbner bases, and right Gröbner bases

Abstract In this paper, we study conditions on algebras with multiplicative bases so that there is a Grobner basis theory. We introduce right Grobner 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 coefficients in a quotient of a path algebra are studied via right Grobner basis theory.

δ-Koszul Algebras

ABSTRACT 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 K A 1 < ∞. We study those graded algebras A with the property that A 0 , viewed as a graded A -module, has a graded projective resolution, , such that each P i can be generated in a single degree. The paper describes necessary and sufficient conditions for the Ext-algebra of A , , to be finitely generated. We also investigate classes of modules over such algebras and Veronese subrings of the Ext-algebra.

Basic Hopf algebras and quantum groups

Abstract. 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 coalgebra structure are found. In particular, it is shown that only the quivers