18.706 | Spring 2023 | Graduate

Noncommutative Algebra

Lecture Notes

Thanks to Serina Hu and Vasily Krylov for note taking.
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18.706 (S23), Full Lecture Notes
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Lecture 01: Introduction, Simple and Semisimple Modules, Skew Fields
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Lecture 02: Semisimple Modules, Socles, Artinian Rings, Wedderburn’s Theorem
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Lecture 03: Isotypic Decomposition, Density Theorem, Noetherian and Artinian Properties, Jacobson Radical
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Lecture 04: Socle and Cosocle Filtrations, Jacobson Radical, Krull-Schmidt
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Lecture 05: Jacobson Radical, Primitive and Semi-Primitive Rings
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Lecture 06: Artinian Rings are Noetherian, Projective Covers
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Lecture 07: Categories and Morita Equivalence
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Lecture 08: Morita Theory Continued: (co)Centers, Functors and Bimodules
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Lecture 09: More on Serre Quotients, Abelian Categories
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Lecture 10: Exts and Tors, Resolutions
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Lecture 11: Koszul Complexes
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Lecture 12: Koszul Rings Continued, Bar Complex
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Lecture 13: Hochschild (co)Homology Continued, Central Simple Algebras
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Lecture 14: Brauer Group and Skolem-Noether Theorems
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Lecture 15: Separable Splitting Fields and Cross-Product Algebras
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Lecture 16: Cohomological Description of the Brauer Group
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Lecture 17: Brauer Groups of Central Simple Algebras, Reduced Norm and Trace
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Lecture 18: Azumaya Algebras
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Lecture 19: Brauer Group of a Ring Continued, Localization
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Lecture 20: Ore Localization, Goldie Theorem
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Lecture 21: Goldie Rank and Goldie Theorem
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Lecture 22: Goldie Theorem, PI Rings
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Lecture 23: Another Proof of Amitsur-Levitski, PI Algebras
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Lecture 24: Gelfand-Kirillov Dimension
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Lecture 25: Final Class: Noncommutative Geometry

Course Info

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As Taught In

Spring 2023

Level

Graduate

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