GELFAI\D- KIRI LL OV DIMENS I ON AND POI NCARE S ERIES Martin Lorenz D epartment of Mathematical Sciences Northern Illinois University DeKalb, Illinois 60115 U.S.A.
Agtor: Martin Lorenz Department of Mathematical Sciences Northern IlIionois University DeKalb, Illinois 60115 U. S. A. Conité E-iecutivo: E. Aznar. J. L. Bueso. ( P. Jara Pres id.ente ) . Departamento de Algebra. Facultad de Ciencias. Universidad de Granada. 18071=Granad.a. España. ur L,nrlsune nll -F ^r ( 19BO) : AMS Subject Classification 16A05, 16A55, 16A64, 16A33, 16A03. Reeibido en Octubre de 1.987 UNIVERSIDAD DE GRANADA. CUADERNOS DE ALGEBRA NA 7-1.988. Depósito Lega}: GR. Ltz/7985. ISSN : OZ 13-1 42O. Inrprime: Secretariado de PubI icaciones. Colegio Máximo de Cartr,rja. [Jniversidad de Granada. Granada. Printer in Spain Impreso en España
CONTENTS , Preface Acknowledgements o Notations and Conventions 6 Chapter I: Introduction to Gelfand-Kirillov dimension n $ I Definition and Basic Properties n t $ 2 Some Examples 13 $ 3 Further Properties of GK-dimension l9 Chapte r II: Factors modulo Nilpote nt Ideals 26 26 $ I The Problem 29 $ 2 Strongly Finircly Presented Algebras $ 3 Application Noetherian PI-algebras 34 36 Chapter III: On Graded Algebras a¡¡d Modules and Their Poincard Series $ 1 Rational Power Series 36 42 $ 2 Poincarl Series 46 $ 3 A Non-commutative Hilbert Serre Theorem 49 $ 4 Application to Certain Filrcred Algebras 54 Chapter [V: On Associate d Graded Rings and Modules 54 $ I Generalized Rees R'ngs and Modules 58 $ 2 Examples References
-3- / PR^EFAOE Poincare series. This leads to additional information which, we feel, is interesting in its own right. In contrast with Chapt". I, QL_Ut"._Il concentrat¿s on algebras and ideals rather tha¡r These notes fairly faithfully reflect the conte nts of a series of lectures given by the author at modules, although the original motivation here is a problem about modules (the exactness the University of Granada, Spain, in April 1987. question). the cost of having to introduce further terminology, most of the material At The purpose of these lectures was twofold. Firstly, our aim was to give an introduction to discussed in this chapter can be put in a module theoretic framework, and this is done in [Lo leads to advanced material and some open Gelfand-Kirillov (GK-) dimension which rapidly finitely presented algebra" that 3l . The present treatment hinges on the notion of a "strongly problems. we have tried to collect some wide-spread results about growth Secondly, "folklore" was introduced, in a slightly more restrictive form in [Lo 3] . The main result in this chapter, and power series. An effort has been made to render the present set of notes reasonably self- originally due to Lorenz and Small [Lo - Sm] and Lenagan [L], proves integrality and exactness contained. So, in particular, the basic facts about GK-dimension are presented in full detail in Pl-algebras. This result is fairly quickly derived from a result of GK-dimension for Noetherian Chapter I. On the other ha¡rd, we have included a number of paragraphs surveying, without on strongly finitely presented algebras which possibly admits further applications once the class to the subject but are not pursued proofs, certain a¡eas which a¡e of vital interest further in of strongly finitely presented algebras will be better understood. these notes. Needless to say that we do not intend to give a complete account of the state of presenting some more or less sbandard results Chapter III has a more classical flavor. After the a¡t of GK-dimension. In fact, the overlap with other standa¡d texts on the subject, such as about rational power series, we prove a rationality result of Hilbert^Serre type for Poincare' the book by Krause and Lenagan, is rather small a¡rd essentially consists of the basic [K-L] graded algebras (also taken from [Lo 3] ). series of modules over certain (non-commutative) , although our presentation focusses on modules rather than algebras material ir Qgp'1q¡_! Elaborating on some remarks in 3] , we then apply this result to obtain a notion of [Lo more than it is usually the case. multiplicity and an exactness result, for modules over a certain type of filte..d algebra (Theorem Two general themes have served as a guideline for selecting our material: (III. 4.4)). This material applies in particular to enveloping algebras of finite-dimensional (strictly) positively graded Lie-algebras and to positively graded affine Pl-algebras. and (1) integrality of GK-dimension, The final a!gpt"_I_lY is formally nearly independent of the rest of these notes. GK- the "exa¿tness" question concerning the GK-dimensions of modules in short (2) is not even explicitly mentioned in this chapter, but the motivation for the material dimension 0 - M'-* M - M" -O. exactsequences and Poinca¡J ,..i.. which all presented here comes from the earlier results on GK-dimension In $ 1, we study a use certain finiteness assumptions on (associated) graded rings and modules. of a module M can be interpreted as the In classical situations, the Gl(-dimension, d(M), which is closely related to but different construction of associated graded rings and modules with order of the pole at t :l of a certain rational Poincará series Pu(t) that can be associated and seems to be more perspective in some respects. This from the usual construction In this case, "exactness" of GK-dimension as well as integrality are trivial consequences. M. applied to the I-adic filtration of a ring R , it yields the so-called construction is not new: When in the second half of these notes, is to study GK-dimension via Thus our approach, especially
-4- -5- Rees ring of the ideal 1, and it has also been used by Quillen [Q, proof of Theorem 7], AcknowledFements. This work was supported by the Deutsche Schapira [S, p.57] , and possibly others. Thus some of the results discussed in $ 1 of Chapter [V Forschungsgemeinschaft/Heisenberg Programm (Lo 2Sll2-2), by the Max-Planck-Institut für may very well be known, in some form, but the construction probably deserves to be Mathematik in Bonn, a^nd by the University of Granada, Spain. It is a pleasure to tha¡rk these popula.rized among non-commutative ring theorists. In $ 2, we present some explicit examples institutions. I especially would like to express my thanks to Jose-Luis Bueso, Pascual Jara, a.nd of associated graded rings having bad properties. We also mention the results of some Blas Torrecillas for making my stay at Granada so enlryable, to George Bergmm, Lance Small, computations of growth series for certain groups. and Toby Sta^fford for enlighte ning letters, preprints and conversations concerning the subject of these notes (Toby Stafford has in particular contributed Example (IV.2.3). ) , and to Lisa Bonn, June 1.987 M. Lorenz Thompson for her expert job on the word processor.
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