Frobenius Splitting in Commutative Algebra
by Karen E. Smith, Wenliang Zhang
Publisher: arXiv 2014
Number of pages: 53
Frobenius splitting has inspired a vast arsenal of techniques in commutative algebra, algebraic geometry, and representation theory. The purpose of these lectures is to give a gentle introduction to Frobenius splitting, or more broadly 'Frobenius techniques,' for beginners.
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by Pete L. Clark - University of Georgia
Contents: Introduction to Commutative Rings; Introduction to Modules; Ideals; Examples of Rings; Swan's Theorem; Localization; Noetherian Rings; Boolean rings; Affine algebras and the Nullstellensatz; The spectrum; Integral extensions; etc.
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The books cover birational geometry, D-modules, invariant theory, matrix factorizations, noncommutative resolutions, singularity categories, support varieties, tilting theory, etc. These volumes reflect the lively interaction between the subjects.
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An expository account of the theory of twisted commutative algebras, which can be thought of as a theory for handling commutative algebras with large groups of linear symmetries. Examples include the coordinate rings of determinantal varieties, etc.
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These notes give a rapid review of the rudiments of classical commutative algebra. Some of the main results whose proofs are outlined here are: Hilbert basis theorem, primary decomposition of ideals in noetherian rings, Krull intersection theorem.