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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.
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