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112pages, PDF, class notes, Math 251, Berkeley, fall 1999

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The list of lecture notes: 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 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 2x2 matrices 6 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 http://math.mit.edu/~etingof/artinnotes.pdf

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