DOCSLIB.ORG

NONCOMMUTATIVE RINGS Michael Artin

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
NONCOMMUTATIVE RINGS Michael Artin NONCOMMUTATIVE RINGS Michael Artin class notes, Math 251, Berkeley, fall 1999 I began writing notes some time after the semester began, so the beginning of the course (diamond lemma, Peirce decomposition, density and Wedderburn theory) is not here. Also, the first chapter is sketchy and unreadable. The remaining chapters, though not in good shape, are a fair record of the course except for the last two lectures, which were on graded algebras of GK dimension two and on open problems. I. Morita equivalence 1. Hom 3 2. Bimodules 3 3. Projective modules 4 4. Tensor products 5 5. Functors 5 6. Direct limits 6 7. Adjoint functors 7 8. Morita equivalence 10 II. Localization and Goldie's theorem 1. Terminology 13 2. Ore sets 13 3. Construction of the ring of fractions 15 4. Modules of fractions 18 5. Essential submodules and Goldie rank 18 6. Goldie's theorem 20 III. Central simple algebras and the Brauer group 1. Tensor product algebras 23 2. Central simple algebras 25 3. Skolem-Noether theorem 27 4. Commutative subfields of central simple algebras 28 5. Faithful flatness 29 6. Amitsur complex 29 7. Interlude: analogy with bundles 32 8. Characteristic polynomial for central simple algebras 33 9. Separable splitting fields 35 10. Structure constants 36 11. Smooth maps and idempotents 37 12. Azumaya algebras 40 13. Dimension of a variety 42 14. Background on algebraic curves 42 15. Tsen's theorem 44 1 2 IV. Maximal orders 1. Lattices and orders 46 2. Trace pairing on Azumaya algebras 47 3. Separable algebras 49 4. Maximal orders in separable algebras 51 5. Hensel's lemma 52 6. Maximal orders over complete discrete valuation rings 54 7. Recovering data from the completion 57 8. Addendum to Chapter III 59 V. Irreducible representations 1. Definition 62 2. Examples 62 3. Standard identities 64 4. Central polynomials 66 5. Polynomial identity rings 68 6. Kaplansky's theorem 69 7. Amitsur's theorem 70 8. Posner's theorem 72 9. Intrinsic characterization of Azumaya algebras 72 10. Irreducible representations of the free ring 74 11. The case of two 2 2 matrices X; Y 76 12. Some tensor notation× 77 13. The main theorem of invariant theory 78 14. Procesi's theorem 79 15. The spectrum of the ring of generic matrices 82 VI. Growth of algebras 1. Growth functions 85 2. Warfield’s theorem 87 3. Path algebras 88 4. Bergman's gap theorem 88 5. Theorem of Stephenson and Zhang 92 6. Projective covers 93 7. Hilbert functions of graded algebras of finite global dimension 95 8. Modules with linear growth 98 9. Theorem of Small and Warfield 99 3 I. MORITA EQUIVALENCE This discusses when the module categories over two different rings A and B are equivalent. I.1. Hom When possible, we write homomorphisms f : PB NB of right B-modules as acting on the left: p fp. Writing homomorphisms on the left has−the! advantage of keeping the scalars out of the wa7!y. B-linearity becomes the associative law: f(pb) = (fp)b; and f(p1 + p2) = fp1 + fp2: 0 For the same reason, we try to write homomorphisms g :A M A M of left modules as acting on the right: m mg. −! 7! The set HomB(P; N) of module homomorphisms is an abelian group under addition of functions: [f + g]p = fp + gp. Composition of operators makes HomB(P; N) into a covariant functor of N and a contravariant 0 0 functor of P : If u : N N is a homomorphism, then the map HomB(P; N) HomB(P; N ) −! 0 −! 0 is defined by sending f u f, and similarly, if v : P P , then HomB(P; N) HomB(P; N ) sends f 0 to f 0 v. 7! ◦ −! ◦ Homomorphisms from a module PB to itself are called endomorphisms, and we denote the set HomB(P; P ) by End PB. It is a ring, multiplication being composition of functions. I.2. Bimodules Let A; B be rings. An A; B-bimodule is an abelian group with a structure of right B-module and also a structure of left A-module, such that the operations of A and B commute: (I.2.1) (ap)b = a(pb) for all a A, p P , b B. There are other ways to interpret such a structure: If we consider the 2 2 2 right B-module structure PB, then left multiplication λa by an element a A, the map defined by λ p = ap, is an endomorphism of P . So we obtain a map A End P2 by sending a λ , a B −! B 7! a and this map is a ring homomorphism. Conversely, any ring homomorphism A End PB gives a bimodule structure on P : −! Corollary I.2.2. An A; B-bimodule APB is given by a right module PB together with an arbitrary homomorphism A End P . In particular a right B-module P has a canonical E; B-bimodule −! B structure, where E = End PB. Similarly, P is also determined by a left module P and a homomorphism B End P . A B A −! A 4 Lemma I.2.3. (i) If APB is a bimodule, then HomB(P; N) is a right A-module by the rule [fa]p = f(ap). (ii) If RNB is a bimodule then HomB(P; N) is a left R-module, with the operation defined by [rf]p = r(fp). (iii) Hom (P; ) is a left exact functor on Mod B. More precisely, sequence B − 0 N N N −! 1 −! 2 −! 3 is exact if and only if 0 Hom (P; N ) Hom (P; N ) Hom (P; N ) −! B 1 −! B 2 −! B 3 is exact for all P Mod B. (iv) Hom ( ; N)2is a right exact contravariant functor on Mod B. A sequence B − P P P 0 1 −! 2 −! 3 −! is exact if and only if 0 Hom (P ; N) Hom (P ; N) Hom (P ; N) −! B 1 −! B 2 −! B 3 is exact for all N Mod B. 2 Remark I.2.4: Let A be a commutative ring. By central A-module we mean the A; A-bimodule obtained from a right A-module by decreeing that the left and right actions are the same: am = ma. In commutative algebra, it is customary to move scalars from left to right informally, i.e., to work with this bimodule, calling it a module. This is not a good idea when A isn't commutative, because the associative law for scalar multiplication screws things up: If we declare that am = ma for all a A and m M, then for a; a0 A, (aa0)m = m(aa0). But the associative law requires that (aa20)m = a(a0m2) = a(ma0) = (ma02)a = m(a0a). Thus right multiplication by aa0 and by a0a are forced to be equal, which means that the actions of A on M factor through a commutative quotient ring of A. I.3. Projective modules A right B-module P is called projecive if it satisfies any one of the conditions of the proposition below. For example, a free module B is projective. Proposition I.3.1. The following Lconditions on a right B-module P are equivalent: φ (i) Let N 0 v N be a surjective map of right B-modules, and let P N be any map. There exists 0 −! φ −! a lifting of φ to N 0, a map P N 0 such that φ = v φ0. f−! ◦ (ii) Every surjective map Q P of right B-modules splits, i.e., there is a map Q s P such that −! − f s = idP . (iii)◦ P is a summand of a free module F , i.e., F P P 0 for some module P 0. ≈ ⊕ (iv) The functor HomB(P; ) is exact: If 0 N1 N2 N3 0 is an exact sequence, then − −! −! −! −! 0 Hom (P; N ) Hom (P; N ) Hom (P; N ) 0 −! B 1 −! B 2 −! B 3 −! 5 is also exact. I.4. Tensor products If MA and AP are right and left A-modules, we can form the tensor product M A P . It is defined as the abelian group generated by elements m p, with the relations ⊗ ⊗ (m + m ) p = m p + m p; m (p + p ) = m p + m p ; and ma p = m ap: 1 2 ⊗ 1 ⊗ 2 ⊗ ⊗ 1 2 ⊗ 1 ⊗ 2 ⊗ ⊗ There is no structure of A-module on this tensor product: the actions of A have been used up in the middle. Lemma I.4.1. (i) The tensor product is functorial in M and in P . (ii) If APB is a bimodule, then M A P becomes a right B-module. (iii) The functor P is right exact:⊗ If − ⊗A M M M 0 1 −! 2 −! 3 −! is an exact sequence of right A-modules, then M P M P M P 0 1 ⊗A −! 2 ⊗A −! 3 ⊗A −! is also exact. Similarly, if P P P 0 1 −! 2 −! 3 −! is exact, then so is M P M P M P 0 ⊗A 1 −! ⊗A 2 −! ⊗A 3 −! I.5. Functors We are interested in additive functors Mod A F Mod B, but most considerations carry over to abelian categories. So let's write the functor neutrally−! as F : C −! D Recall that a functor associates to every object X an image F X , and for every pair 2 C 2 D X1; X2 of objects in , a map HomC(X1; X2) HomD(F X1; F X2), these maps on Hom being compatible with compCosition of morphisms in −.! C The adjective \additive" means that the maps on morphisms are homomorphisms of abelian groups, which is a mild restriction.
Load more

Recommended publications
  • AZUMAYA, SEMISIMPLE and IDEAL ALGEBRAS* Introduction. If a Is An
    BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 78, Number 4, July 1972 AZUMAYA, SEMISIMPLE AND IDEAL ALGEBRAS* BY M. L. RANGA RAO Communicated by Dock S. Rim, January 31, 1972 Introduction. If A is an algebra over a commutative ring, then for any ideal I of R the image of the canonical map / ® R A -» A is an ideal of A, denoted by IA and called a scalar ideal. The algebra A is called an ideal algebra provided that / -> IA defines a bijection from the ideals of R to the ideals of A. The contraction map denned by 21 -> 91 n JR, being the inverse mapping, each ideal of A is a scalar ideal and every ideal of R is a contraction of an ideal of A. It is known that any Azumaya algebra is an ideal algebra (cf. [2], [4]). In this paper we initiate the study of ideal algebras and obtain a new characterization of Azumaya algebras. We call an algebra finitely genera­ ted (or projective) if the R-module A is, and show that a finitely generated algebra A is Azumaya iff its enveloping algebra is ideal (1.7). Hattori introduced in [9] the notion of semisimple algebras over a commutative ring as those algebras whose relative global dimension [10] is zero. If R is a Noetherian ring, central, finitely generated semisimple algebras have the property that their maximal ideals are all scalar ideals [6, Theorem 1.6]. It is also known that for R, a Noetherian integrally closed domain, finitely generated, projective, central semisimple algebras are maximal orders in central simple algebras [9, Theorem 4.6].
    [Show full text]
  • Bibliography
    Bibliography [1] Emil Artin. Galois Theory. Dover, second edition, 1964. [2] Michael Artin. Algebra. Prentice Hall, first edition, 1991. [3] M. F. Atiyah and I. G. Macdonald. Introduction to Commutative Algebra. Addison Wesley, third edition, 1969. [4] Nicolas Bourbaki. Alg`ebre, Chapitres 1-3.El´ements de Math´ematiques. Hermann, 1970. [5] Nicolas Bourbaki. Alg`ebre, Chapitre 10.El´ements de Math´ematiques. Masson, 1980. [6] Nicolas Bourbaki. Alg`ebre, Chapitres 4-7.El´ements de Math´ematiques. Masson, 1981. [7] Nicolas Bourbaki. Alg`ebre Commutative, Chapitres 8-9.El´ements de Math´ematiques. Masson, 1983. [8] Nicolas Bourbaki. Elements of Mathematics. Commutative Algebra, Chapters 1-7. Springer–Verlag, 1989. [9] Henri Cartan and Samuel Eilenberg. Homological Algebra. Princeton Math. Series, No. 19. Princeton University Press, 1956. [10] Jean Dieudonn´e. Panorama des mat´ematiques pures. Le choix bourbachique. Gauthiers-Villars, second edition, 1979. [11] David S. Dummit and Richard M. Foote. Abstract Algebra. Wiley, second edition, 1999. [12] Albert Einstein. Zur Elektrodynamik bewegter K¨orper. Annalen der Physik, 17:891–921, 1905. [13] David Eisenbud. Commutative Algebra With A View Toward Algebraic Geometry. GTM No. 150. Springer–Verlag, first edition, 1995. [14] Jean-Pierre Escofier. Galois Theory. GTM No. 204. Springer Verlag, first edition, 2001. [15] Peter Freyd. Abelian Categories. An Introduction to the theory of functors. Harper and Row, first edition, 1964. [16] Sergei I. Gelfand and Yuri I. Manin. Homological Algebra. Springer, first edition, 1999. [17] Sergei I. Gelfand and Yuri I. Manin. Methods of Homological Algebra. Springer, second edition, 2003. [18] Roger Godement. Topologie Alg´ebrique et Th´eorie des Faisceaux.
    [Show full text]
  • Tōhoku Rick Jardine
    INFERENCE / Vol. 1, No. 3 Tōhoku Rick Jardine he publication of Alexander Grothendieck’s learning led to great advances: the axiomatic description paper, “Sur quelques points d’algèbre homo- of homology theory, the theory of adjoint functors, and, of logique” (Some Aspects of Homological Algebra), course, the concepts introduced in Tōhoku.5 Tin the 1957 number of the Tōhoku Mathematical Journal, This great paper has elicited much by way of commen- was a turning point in homological algebra, algebraic tary, but Grothendieck’s motivations in writing it remain topology and algebraic geometry.1 The paper introduced obscure. In a letter to Serre, he wrote that he was making a ideas that are now fundamental; its language has with- systematic review of his thoughts on homological algebra.6 stood the test of time. It is still widely read today for the He did not say why, but the context suggests that he was clarity of its ideas and proofs. Mathematicians refer to it thinking about sheaf cohomology. He may have been think- simply as the Tōhoku paper. ing as he did, because he could. This is how many research One word is almost always enough—Tōhoku. projects in mathematics begin. The radical change in Gro- Grothendieck’s doctoral thesis was, by way of contrast, thendieck’s interests was best explained by Colin McLarty, on functional analysis.2 The thesis contained important who suggested that in 1953 or so, Serre inveigled Gro- results on the tensor products of topological vector spaces, thendieck into working on the Weil conjectures.7 The Weil and introduced mathematicians to the theory of nuclear conjectures were certainly well known within the Paris spaces.
    [Show full text]
  • Artinian Rings Having a Nilpotent Group of Units
    JOURNAL OF ALGEBRA 121, 253-262 (1989) Artinian Rings Having a Nilpotent Group of Units GHIOCEL GROZA Department of Mathematics, Civil Engineering Institute, Lacul Tei 124, Sec. 2, R-72307, Bucharest, Romania Communicated by Walter Feit Received July 1, 1983 Throughout this paper the symbol R will be used to denote an associative ring with a multiplicative identity. R,[R”] denotes the subring of R which is generated over R, by R”, where R, is the prime subring of R and R” is the group of units of R. We use J(R) to denote the Jacobson radical of R. In Section 1 we prove that if R is a finite ring and at most one simple component of the semi-simple ring R/J(R) is a field of order 2, then R” is a nilpotent group iff R is a direct sum of two-sided ideals that are homomorphic images of group algebras of type SP, where S is a particular commutative finite ring, P is a finite p-group, and p is a prime number. In Section 2 we study the artinian rings R (i.e., rings with minimum con- dition) with RO a finite ring, R finitely generated over its center and so that every block of R is as in Lemma 1.1. We note that every finite-dimensional algebra over a field F of characteristic p > 0, so that F is not the finite field of order 2 is of this type. We prove that R is of this type and R” is a nilpotent group iff R is a direct sum of two-sided ideals that are particular homomorphic images of crossed products.
    [Show full text]
  • Exercises and Solutions in Groups Rings and Fields
    EXERCISES AND SOLUTIONS IN GROUPS RINGS AND FIELDS Mahmut Kuzucuo˘glu Middle East Technical University [email protected] Ankara, TURKEY April 18, 2012 ii iii TABLE OF CONTENTS CHAPTERS 0. PREFACE . v 1. SETS, INTEGERS, FUNCTIONS . 1 2. GROUPS . 4 3. RINGS . .55 4. FIELDS . 77 5. INDEX . 100 iv v Preface These notes are prepared in 1991 when we gave the abstract al- gebra course. Our intention was to help the students by giving them some exercises and get them familiar with some solutions. Some of the solutions here are very short and in the form of a hint. I would like to thank B¨ulent B¨uy¨ukbozkırlı for his help during the preparation of these notes. I would like to thank also Prof. Ismail_ S¸. G¨ulo˘glufor checking some of the solutions. Of course the remaining errors belongs to me. If you find any errors, I should be grateful to hear from you. Finally I would like to thank Aynur Bora and G¨uldaneG¨um¨u¸sfor their typing the manuscript in LATEX. Mahmut Kuzucuo˘glu I would like to thank our graduate students Tu˘gbaAslan, B¨u¸sra C¸ınar, Fuat Erdem and Irfan_ Kadık¨oyl¨ufor reading the old version and pointing out some misprints. With their encouragement I have made the changes in the shape, namely I put the answers right after the questions. 20, December 2011 vi M. Kuzucuo˘glu 1. SETS, INTEGERS, FUNCTIONS 1.1. If A is a finite set having n elements, prove that A has exactly 2n distinct subsets.
    [Show full text]
  • Extended Jacobson Density Theorem for Lie Ideals of Rings with Automorphisms
    Publ. Math. Debrecen 58 / 3 (2001), 325–335 Extended Jacobson density theorem for Lie ideals of rings with automorphisms By K. I. BEIDAR (Tainan), M. BRESARˇ (Maribor) and Y. FONG (Tainan) Abstract. We prove a version of the Chevalley–Jacobson density theorem for Lie ideals of rings with automorphisms and present some applications of the obtained results. 1. Introduction In the present paper we continue the project initiated recently in [7] and developed further in [3], [4]; its main idea is to connect the concept of a dense action on modules with the concept of outerness of derivations and automorphisms. In [3] an extended version of Chevalley–Jacobson density theorem has been proved for rings with automorphisms and derivations. In the present paper we consider a Lie ideal of a ring acting on simple modules via multiplication. Our goal is to extend to this context results obtained in [3]. We confine ourselves with the case of automorphisms. We note that Chevalley–Jacobson density theorem has been generalized in various directions [1], [10], [14], [12], [13], [17], [19]–[22] (see also [18, 15.7, 15.8] and [9, Extended Jacobson Density Theorem]). As an application we generalize results of Drazin on primitive rings with pivotal monomial to primitive rings whose noncentral Lie ideal has a pivotal monomial with automorphisms. Here we note that while Mar- tindale’s results on prime rings with generalized polynomial identity were extended to prime rings with generalized polynomial identities involving derivations and automorphisms, the corresponding program for results of Mathematics Subject Classification: 16W20, 16D60, 16N20, 16N60, 16R50.
    [Show full text]
  • Representations of Finite Rings
    Pacific Journal of Mathematics REPRESENTATIONS OF FINITE RINGS ROBERT S. WILSON Vol. 53, No. 2 April 1974 PACIFIC JOURNAL OF MATHEMATICS Vol. 53, No. 2, 1974 REPRESENTATIONS OF FINITE RINGS ROBERT S. WILSON In this paper we extend the concept of the Szele repre- sentation of finite rings from the case where the coefficient ring is a cyclic ring* to the case where it is a Galois ring. We then characterize completely primary and nilpotent finite rings as those rings whose Szele representations satisfy certain conditions. 1* Preliminaries* We first note that any finite ring is a direct sum of rings of prime power order. This follows from noticing that when one decomposes the additive group of a finite ring into its pime power components, the component subgroups are, in fact, ideals. So without loss of generality, up to direct sum formation, one needs only to consider rings of prime power order. For the remainder of this paper p will denote an arbitrary, fixed prime and all rings will be of order pn for some positive integer n. Of the two classes of rings that will be studied in this paper, completely primary finite rings are always of prime power order, so for the completely primary case, there is no loss of generality at all. However, nilpotent finite rings do not need to have prime power order, but we need only classify finite nilpotent rings of prime power order, the general case following from direct sum formation. If B is finite ring (of order pn) then the characteristic of B will be pk for some positive integer k.
    [Show full text]
  • Dynamic Algebras: Examples, Constructions, Applications
    Dynamic Algebras: Examples, Constructions, Applications Vaughan Pratt∗ Dept. of Computer Science, Stanford, CA 94305 Abstract Dynamic algebras combine the classes of Boolean (B ∨ 0 0) and regu- lar (R ∪ ; ∗) algebras into a single finitely axiomatized variety (BR 3) resembling an R-module with “scalar” multiplication 3. The basic result is that ∗ is reflexive transitive closure, contrary to the intuition that this concept should require quantifiers for its definition. Using this result we give several examples of dynamic algebras arising naturally in connection with additive functions, binary relations, state trajectories, languages, and flowcharts. The main result is that free dynamic algebras are residu- ally finite (i.e. factor as a subdirect product of finite dynamic algebras), important because finite separable dynamic algebras are isomorphic to Kripke structures. Applications include a new completeness proof for the Segerberg axiomatization of propositional dynamic logic, and yet another notion of regular algebra. Key words: Dynamic algebra, logic, program verification, regular algebra. This paper originally appeared as Technical Memo #138, Laboratory for Computer Science, MIT, July 1979, and was subsequently revised, shortened, retitled, and published as [Pra80b]. After more than a decade of armtwisting I. N´emeti and H. Andr´eka finally persuaded the author to publish TM#138 itself on the ground that it contained a substantial body of interesting material that for the sake of brevity had been deleted from the STOC-80 version. The principal changes here to TM#138 are the addition of footnotes and the sections at the end on retrospective citations and reflections. ∗This research was supported by the National Science Foundation under NSF grant no.
    [Show full text]
  • Algebra + Homotopy = Operad
    Symplectic, Poisson and Noncommutative Geometry MSRI Publications Volume 62, 2014 Algebra + homotopy = operad BRUNO VALLETTE “If I could only understand the beautiful consequences following from the concise proposition d 2 0.” —Henri Cartan D This survey provides an elementary introduction to operads and to their ap- plications in homotopical algebra. The aim is to explain how the notion of an operad was prompted by the necessity to have an algebraic object which encodes higher homotopies. We try to show how universal this theory is by giving many applications in algebra, geometry, topology, and mathematical physics. (This text is accessible to any student knowing what tensor products, chain complexes, and categories are.) Introduction 229 1. When algebra meets homotopy 230 2. Operads 239 3. Operadic syzygies 253 4. Homotopy transfer theorem 272 Conclusion 283 Acknowledgements 284 References 284 Introduction Galois explained to us that operations acting on the solutions of algebraic equa- tions are mathematical objects as well. The notion of an operad was created in order to have a well defined mathematical object which encodes “operations”. Its name is a portemanteau word, coming from the contraction of the words “operations” and “monad”, because an operad can be defined as a monad encoding operations. The introduction of this notion was prompted in the 60’s, by the necessity of working with higher operations made up of higher homotopies appearing in algebraic topology. Algebra is the study of algebraic structures with respect to isomorphisms. Given two isomorphic vector spaces and one algebra structure on one of them, 229 230 BRUNO VALLETTE one can always define, by means of transfer, an algebra structure on the other space such that these two algebra structures become isomorphic.
    [Show full text]
  • Right Ideals of a Ring and Sublanguages of Science
    RIGHT IDEALS OF A RING AND SUBLANGUAGES OF SCIENCE Javier Arias Navarro Ph.D. In General Linguistics and Spanish Language http://www.javierarias.info/ Abstract Among Zellig Harris’s numerous contributions to linguistics his theory of the sublanguages of science probably ranks among the most underrated. However, not only has this theory led to some exhaustive and meaningful applications in the study of the grammar of immunology language and its changes over time, but it also illustrates the nature of mathematical relations between chunks or subsets of a grammar and the language as a whole. This becomes most clear when dealing with the connection between metalanguage and language, as well as when reflecting on operators. This paper tries to justify the claim that the sublanguages of science stand in a particular algebraic relation to the rest of the language they are embedded in, namely, that of right ideals in a ring. Keywords: Zellig Sabbetai Harris, Information Structure of Language, Sublanguages of Science, Ideal Numbers, Ernst Kummer, Ideals, Richard Dedekind, Ring Theory, Right Ideals, Emmy Noether, Order Theory, Marshall Harvey Stone. §1. Preliminary Word In recent work (Arias 2015)1 a line of research has been outlined in which the basic tenets underpinning the algebraic treatment of language are explored. The claim was there made that the concept of ideal in a ring could account for the structure of so- called sublanguages of science in a very precise way. The present text is based on that work, by exploring in some detail the consequences of such statement. §2. Introduction Zellig Harris (1909-1992) contributions to the field of linguistics were manifold and in many respects of utmost significance.
    [Show full text]
  • Noncommutative Localization in Algebra and Topology
    Noncommutative localization in algebra and topology ICMS Edinburgh 2002 Edited by Andrew Ranicki Electronic version of London Mathematical Society Lecture Note Series 330 Cambridge University Press (2006) Contents Dedication . vii Preface . ix Historical Perspective . x Conference Participants . xi Conference Photo . .xii Conference Timetable . xiii On atness and the Ore condition J. A. Beachy ......................................................1 Localization in general rings, a historical survey P. M. Cohn .......................................................5 Noncommutative localization in homotopy theory W. G. Dwyer . 24 Noncommutative localization in group rings P. A. Linnell . 40 A non-commutative generalisation of Thomason's localisation theorem A. Neeman . 60 Noncommutative localization in topology A. A. Ranicki . 81 v L2-Betti numbers, Isomorphism Conjectures and Noncommutative Lo- calization H. Reich . 103 Invariants of boundary link cobordism II. The Blanch¯eld-Duval form D. Sheiham . 143 Noncommutative localization in noncommutative geometry Z. Skoda· ........................................................220 vi Dedicated to the memory of Desmond Sheiham (13th November 1974 ¡ 25th March 2005) ² Cambridge University (Trinity College), 1993{1997 B.A. Hons. Mathematics 1st Class, 1996 Part III Mathematics, Passed with Distinction, 1997 ² University of Edinburgh, 1997{2001 Ph.D. Invariants of Boundary Link Cobordism, 2001 ² Visiting Assistant Professor, Mathematics Department, University of California at Riverside, 2001{2003 ² Research Instructor, International University Bremen (IUB), 2003{2005 vii Publications: 1. Non-commutative Characteristic Polynomials and Cohn Localization Journal of the London Mathematical Society (2) Vol. 64, 13{28 (2001) http://arXiv.org/abs/math.RA/0104158 2. Invariants of Boundary Link Cobordism Memoirs of the American Mathematical Society, Vol. 165 (2003) http://arXiv.org/abs/math.AT/0110249 3. Whitehead Groups of Localizations and the Endomorphism Class Group Journal of Algebra, Vol.
    [Show full text]
  • The Cardinality of the Center of a Pi Ring
    Canad. Math. Bull. Vol. 41 (1), 1998 pp. 81±85 THE CARDINALITY OF THE CENTER OF A PI RING CHARLES LANSKI ABSTRACT. The main result shows that if R is a semiprime ring satisfying a poly- nomial identity, and if Z(R) is the center of R, then card R Ä 2cardZ(R). Examples show that this bound can be achieved, and that the inequality fails to hold for rings which are not semiprime. The purpose of this note is to compare the cardinality of a ring satisfying a polynomial identity (a PI ring) with the cardinality of its center. Before proceeding, we recall the def- inition of a central identity, a notion crucial for us, and a basic result about polynomial f g≥ f g identities. Let C be a commutative ring with 1, F X Cn x1,...,xn the free algebra f g ≥ 2 f gj over C in noncommuting indeterminateso xi ,andsetG f(x1,...,xn) F X some coef®cient of f is a unit in C .IfRis an algebra over C,thenf(x1,...,xn)2Gis a polyno- 2 ≥ mial identity (PI) for RPif for all ri R, f (r1,...,rn) 0. The standard identity of degree sgõ n is Sn(x1,...,xn) ≥ õ(1) xõ(1) ÐÐÐxõ(n) where õ ranges over the symmetric group on n letters. The Amitsur-Levitzki theorem is an important result about Sn and shows that Mk(C) satis®es Sn exactly for n ½ 2k [5; Lemma 2, p. 18 and Theorem, p. 21]. Call f 2 G a central identity for R if f (r1,...,rn) 2Z(R), the center of R,forallri 2R, but f is not a polynomial identity.
    [Show full text]