本科毕业设计（论文）
外文翻译
Some Finite Dimensional Algebras Related To Elliptic Curves
作者：S.Paul Simth
国籍：the United States of America
出处：Representation theory of algebras and related topics (Mexico City,1994),315-348,CMS Conf. Proc.,19,Amer.Math.Soc.,Providence.RI,1996.
ABSTRACT The Koszul dual of a Sklyanin algebra is a finite dimensional graded algebra depending on an elliptic curve, a translation automorphism, and an integer ; it may be defined as . The representation theory and structure of these algebras is studied by using the functor to transfer results from the Sklyanin algebras to the finite dimensional algebras. We show that their representation theory is closely related to the elliptic curve and the automorphism.
Key words and phrases. Sklyanin algebra, finite dimensional algebra, elliptic curve,duality,Yoneda Ext algebra
0.INTRODUCTION
Given an elliptic curve over an algebraically closed field ,a translation automorphism of ,and an integer ,we define in Section 10 a finite dimensional algebra depending on this data.Its Hilbert series is the same as that of the exterior algebra .In particular, is local.It is also of wild representation type,and a Frobenius algebra(in fact symmetric when ).The construction is such that is naturally embedded in ,the projective space of 1-dimensional subspaces of the degree one component of .
These properties of are proved in an indirect fashion;the starting point is that is a Koszul algebra,and its properties are consequences of properties of its Koszul dual .Since this paper is aimed at those whose main interest is finite dimensional algebras,we will treat as the primary object.However, is the object of primary interest to the author.It is a Sklyanin algebra,and has been the object of intense study over the past 6 or 7 years.A survey of what is known about may be found in [24].
Each is a connected graded algebra whose defining relations are homogeneous of degree two;its Koszul dual is,by definition, endowed with the Yoneda product.The contravariant functor sending graded -modules to graded -modules,is the vehicle used for transferring properties from to .
The basic properties of are reviewed in Section 8.The key result,due to Tate and van den Bergh [35],is that is a quantum polynomial ring(Definition 8.5);the terminology suggests that is a non-commutative deformation of a polynomial ring-like the polynomial ring,it is generated in degree one,has Hilbert series ,is right and left Noetherian,is a domain,has global homological dimension ,is Auslander-Gorenstein(Denition4.2),and Cohen-Macaulay(Definition8.4).The Frobenius property for is equivalent to the Gorenstein property for (Proposition 5.10);the Hilbert series for is obtained from that for via the functional equation for Koszul algebras (5-1).
Although the representation theory of is not well-understood(except for and 4),there is,for each ,a particularly important class of graded modules which is understood.These are the linear modules(Definition 9.1).They are analogues of linear subspaces of the projective space .Their properties are discussed in Section 9.The most important point for the present paper is that linear modules have linear resolutions(Definition 1.6),and is a duality between the categories of graded modules over and having linear resolutions(Corollary 6.4).For each effective divisor on of degree ,there is an associated linear module ;the -module is indecomposable,cyclic,graded,and has Hilbert series (hence dimension ).These are the -modules about which we have most information.(For the exterior algebra,the analogous modules are where is the left idea lgenerated by a -dimensional subspace of ,the degree one component of .)That part of the Auslander-Reiten quiver for containing the modules ,,is described.
Our approach to the study of requires some technical background,which the earlier part of the paper provides.Section 1 gives basic terminology and results on the category of graded modules,with particular attention paid to linear resolutions.Section 2 recalls the Yoneda product and gives a result comparing two Yoneda Ext-algebras,one for left,and one for right,modules.Section 3 considers the Frobenius property for connected graded algebras,and defines a lsquo;symmetrizing automorphismrsquo; which measures the failure of a Frobenius algebra to be symmetric- is inner if and only if the algebra is symmetric.The Auslander-Reiten translation is isomorphic to ,where is the pull-back functor along and is the second syzygy functor.Section 4 considers consequences of a version of the Gorenstein property for non-commutative connected graded algebras.Theorem 4.3 shows that if is such an algebra(also left Noetherian and of finite global dimension),then is Frobenius;in particular is Frobenius.Section 5 gives some background on Koszul algebras-we restrict attention to those which are in degree zero.In Section 6 we show that for a Koszul algebra is a duality for modules having a linear resolution.The close relation of this to the Koszul duality results of Beilinson-Ginsburg-Soergel [5] is briefly discussed in Section 7.Sections 8 and 9 discuss the Sklyanin algebras,and in Section 10 we finally get to the finite dimensional algebra .
Sections 11 and 12 attempt to put some of the results for Sklyanin algebras in context.The general features of show up in many other situations.First,there are many quantum polynomial rings.The 3-dimensional ones have been classified by Artin-Schelter [1] and Artin-Tate-van den Bergh [2]:they are classified by geometric data consisting of a scheme and an au
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