Department of Mathematics Mathematics 502, section E1 Commutative Algebra 1:00 - 1:50 PM MWF 341 Altgeld Hall Professor Daniel R. Grayson Office: 357 Altgeld Hall Office phone: x3-6209 or 217-333-6209.

The course

Commutative algebra is a beautiful mixture of geometry and algebra. The geometry comes from the examination of solution sets of systems of polynomial equations. For example, consider the following equations: x²+y²=1 yields a circle, x³=y² yields a curve with a cusp, and x³+x²=y² yields a curve with a node. The algebra comes in when we examine the properties of the ring of polynomial functions on such a solution set. Mixing the two areas leads to intriguing algebraic interpretations of geometric notions, such as an interpretation of singularity in terms of resolutions of modules by projective modules.

We'll try to cover a substantial number of the following topics: commutative rings and modules; Gröbner bases; prime ideals; ideal quotient; localization; noetherian rings; primary decomposition; integral extensions and Noether normalization; the Nullstellensatz; dimension; rings of small dimension; flatness, faithful flattness, descent; Hensel's lemma; graded rings; Hilbert polynomial; valuations; regular rings; singularities; unique factorization; homological dimension; depth; completion and Hensel's lemma; smooth and étale extensions; ramification; Cohen-Macaulay modules; and complete intersections.

The text book

The text book is Commutative Algebra with a View Toward Algebraic Geometry, by David Eisenbud. The latest word is that the book will be reprinted by Springer in time for the Fall semester.

The grading

The grade will consist of three parts: homework (25%), three hour exams (50%), and a final exam (25%).

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