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18 705: Commutative Algebra
3.00 Credits
Massachusetts Institute of Technology
Exactness, direct limits, tensor products, Cayley-Hamilton theorem, integral dependence, localization, Cohen-Seidenberg theory, Noether normalization, Nullstellensatz, chain conditions, primary decomposition, length, Hilbert functions, dimension theory, completion, Dedekind domains.
Prerequisite:
Prereq: 18.702
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18 705 - Commutative Algebra
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18 706: Noncommutative Algebra
3.00 Credits
Massachusetts Institute of Technology
Wedderburn theory, Morita equivalence, localization and Goldie's theorem, central simple algebras and the Brauer group, maximal orders, representations, polynomial identity rings, invariant theory growth of algebras, Gelfand-Kirillov dimension.
Prerequisite:
Prereq: 18.705
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18 706 - Noncommutative Algebra
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18 712: Introduction to Representation Theory
3.00 Credits
Massachusetts Institute of Technology
Algebras, representations, Schur's lemma. Representations of SL(2). Representations of finite groups, Maschke's theorem, characters, applications. Induced representations, Burnside's theorem, Mackey formula, Frobenius reciprocity. Representations of quivers.
Prerequisite:
Prereq: 18.702 or 18.703
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18 712 - Introduction to Representation Theory
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18 721: Introduction to Algebraic Geometry
3.00 Credits
Massachusetts Institute of Technology
Introduction to affine and projective algebraic geometry. Emphasis on basic examples of complex algebraic varieties, including algebraic curves.
Prerequisite:
Prereq: 18.702, 18.901
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18 721 - Introduction to Algebraic Geometry
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18 725: Algebraic Geometry I
3.00 Credits
Massachusetts Institute of Technology
No course description available.
Prerequisite:
Prereq: None. Coreq: 18.705
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18 725 - Algebraic Geometry I
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18 726: Algebraic Geometry II
3.00 Credits
Massachusetts Institute of Technology
Continuation of the introduction to algebraic geometry given in 18.725. More advanced properties of the varieties and morphisms of schemes, as well as sheaf cohomology.
Prerequisite:
Prereq: 18.725
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18 726 - Algebraic Geometry II
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18 727: Topics in Algebraic Geometry
3.00 Credits
Massachusetts Institute of Technology
Topics vary from year to year.
Prerequisite:
Prereq: 18.725
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18 727 - Topics in Algebraic Geometry
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18 735: Topics in Algebra
3.00 Credits
Massachusetts Institute of Technology
Topics vary from year to year.
Prerequisite:
Prereq: 18.705
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18 735 - Topics in Algebra
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18 737: Algebraic Groups
3.00 Credits
Massachusetts Institute of Technology
Structure of linear algebraic groups over an algebraically closed field, with emphasis on reductive groups. Representations of groups over a finite field using methods from etale cohomology. Some results from algebraic geometry are stated without proof.
Prerequisite:
Prereq: 18.705
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18 737 - Algebraic Groups
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18 739: Theory of Invariants
3.00 Credits
Massachusetts Institute of Technology
Hilbert's finiteness theorem for reductive groups. Properties of the orbits and the orbit space. Classical invariant theory. Hilbert-Mumford-Richardson theorem. Rosenlicht's theorem on the existence of invariants. Matsushima criterion. Richardson's theorem on the principal stabilizer. Chevalley-Luna-Richardson theorem. Linear actions with a non-trivial stabilizer. Polar representations. Methods of the orbit classification. Applications to classical problems of linear algebra. Other topics.
Prerequisite:
Prereq: 18.705
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18 739 - Theory of Invariants
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