Commutative Algebra
4 hours of lectures and excersises per week.
Lecturer Anders Buch Content Commutative algebra provides the basic toolbox in a number of fields including number theory, invariant theory, and algebraic geometry. This course will be an introduction to the subject which mostly focuses on applications to algebraic geometry. A follow-up course on algebraic geometry is planned for the Fall semester.
The course will emphasize the viewpoint that a commutative ring corresponds to a geometric space. Furthermore, the geometric properties of this space, like its dimension and singularities, are re ected by the algebraic properties of the ring.
The topics will include the important technique of localizing a ring at aprime ideal, which geometrically corresponds to zooming in on a point of an algebraic variety. Another topic is primary decomposition which generalizes unique factorization of ideals to polynomial rings with several variables, and which in geometry can be used to extract information about the components of a variety. We will also prove the " niteness of integral closure" theorem, which can be used to get rid of the worst singularities on a variety, in fact all of them on an algebraic curve.
Prerequisites Algebra II Literature David Eisenbud:
Commutative Algebra with a view toward algebraic geometry, GTM 150.