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دانلود کتاب Noncommutative Algebra: 18.706 (S23), Full Lecture Notes
دانلود کتاب جبر غیرمتعارف: 18.706 (S23) ، یادداشت های سخنرانی کامل
مشخصات کتاب
Noncommutative Algebra: 18.706 (S23), Full Lecture Notes
ویرایش:
نویسندگان: Roman Bezrukavnikov
سری:
ناشر:
سال نشر: 2023
تعداد صفحات: 83
زبان: English
فرمت فایل : PDF (درصورت درخواست کاربر به PDF، EPUB یا AZW3 تبدیل می شود)
حجم فایل: 59 مگابایت
قیمت کتاب (تومان) : 74,000
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توضیحاتی درمورد کتاب به خارجی
فهرست مطالب
February 7 - Introduction, simple and semisimple modules, skew fields Rings, Modules, Ideals Invariant Basis Number Property Simple modules, Schur Lemma Semisimple modules Semisimple modules, socles, Artinian rings, Wedderburn\'s Theorem More on semisimple modules Socles Isotypic components Classification of semisimple rings Simple rings and Wedderburn\'s Theorem February 14: Isotypic decomposition, Density Theorem, Noetherian and Artinian properties, Jacobson radical k[G]-modules Density Theorem Noetherian and Artinian modules Composition Series Jacobson Radical February 16 - Socle and cosocle filtrations, Jacobson radical, Krull-Schmidt Socle and cosocle filtrations Jacobson radical cont. Local rings and indecomposable modules Krull-Schmidt February 23 - Jacobson radical, primitive and semi-primitive rings Interlude on quiver representations Primitive and semi-primitive rings February 28 - Artinian rings are Noetherian, projective covers The Akizuki-Hopkins-Levitzki Theorem (Artinian rings are Noetherian) Projective covers Preview of Morita theory March 2 - Categories and Morita equivalence Morita equivalence Categories and the Yoneda Lemma Proof of Morita equivalence theorem March 7 - Morita theory continued: (co)centers, functors and bimodules Center and cocenter Morita equivalence via functors and bimodules Serre quotients March 9 - more on Serre quotients, abelian categories More on Serre quotients Adjoint functors and (co)limits Additive categories Abelian categories Compact projective generators and Serre quotients revisited March 14 - Exts and Tors, Resolutions Ext and Tor Projective, injective, and homological dimension Cartan matrices March 16 - Koszul complexes More on the Hattori-Stallings Dennis trace Minimal resolutions and Koszul rings Koszul complexes March 21 - Koszul rings cont., bar complex Finishing up Koszul rings Bar complex and Hochschild (co)homology March 23 - Hochschild (co)homology cont., central simple algebras Deformations and Hochschild cohomology Cobar complex and A! Central simple algebras and Brauer group April 4 - Brauer group and Skolem-Noether Theorems Definition and first properties of Brauer group Torsors and Galois forms Centralizer of a commutative subfield Skolem-Noether Artin-Wedderburn April 6 - Separable splitting fields and cross-product algebras Separable splitting fields Group cohomology Cross-product algebras April 11 - Cohomological description of the Brauer group Cross-product algebras and Galois extensions Maximal commutative subfields and splitting fields Proof of the theorem Applications Index and period April 13 - Brauer groups of central simple algebras, reduced norm and trace Reduced norm and trace C1 fields Second approach to the cohomological description of Brauer group Brauer groups of local fields April 18 - Azumaya algebras Azumaya algebras Cohomological description of the Brauer group over a ring - preliminary discussion Faithfully flat ring homomorphisms and faithfully flat descent Universal splitting Rewriting cochain complex for H (G, E) April 20 - Brauer group of a ring cont., localization Amitsur cohomology Relationship between Brauer group and Amitsur cohomology Final remarks on Brauer group Localization Ore conditions April 25 - Ore localization, Goldie theorem Ore localization as a filtered colimit Ore domains Growth of algebras Semi-prime rings and Goldie\'s theorem April 27 - Goldie rank and Goldie theorem More on essential modules Goldie rank Regular elements in essential ideals May 2 - Goldie Theorem, PI rings Finishing up Goldie Theorem Goldie rings Applications of Goldie\'s Theorem PI rings Amitsur-Levitzki Theorem Amitsur-Levitzki Theorem and the cohomology of gl (n) May 9 - Another proof of Amitsur-Levitski, PI algebras Proof of Amitsur-Levitski Theorem Primitive algebras and Kaplansky\'s theorem Prime PI algebras and Posner theorem Central polynomials Rowen\'s Theorem for semi-primitive algebras Proof of Rowen\'s Theorem (for real) May 11 - Gelfand-Kirillov dimension Growth of algebras and Gelfand-Kirillov dimension Warfield\'s Theorem Bergman gap theorem Ufnarovskii graph Smoktunowicz and Berele theorems GK dimension of a module Poincare series May 16 - Final class: noncommutative geometry Representation varieties Weyl algebra and deformations Coh(X) and D(Coh(X)) Artin-Schelter regular algebras and noncommutative projective geometry
