Math 831: Abstract Algebra II

Math 831, Abstract Algebra II

TuTh 2:00 - 3:15pm, Snow Hall 408

This is an introductory course on commutative algebra. Because there is too much material to cover at this level in a one semester course, I think it is best that we take a particular fundamental result and make it our goal to cover enough basic material in order to prove the result by the end of the semester. The theorem I have in mind is that every regular local ring is a unique factorization domain. Regular local rings come from algebraic geometry, in that they are the local rings associated to nonsingular points on an algebraic variety. In the geometric context, it was already known in the early 1940s that such rings are unique factorization domains. In the late 1940s, Krull defined the notion of a regular local ring in purely algebraic terms and conjectured that these rings are unique factorization domains. This was eventually shown to be the case in the late 1950s by Auslander and Buchsbaum using homological techniques introduced by Serre. In order to reach our goal, we will cover a number of standard introductory topics in commutative algebra. The specific topics covered will depend upon the background of students enrolled in the class, but we will certainly cover primary decomposition, UFDs, and dimension theory in local rings, plus the relevant tools from homological algebra. If time permits, we will also cover the rudiments of multiplicity theory--another important topic in commutative algebra motivated by algebraic geometry.

None required.

Professor D. Katz

Snow Hall 501

TuTh 10:30 - 11:30am and by appointment, either in-office or via Zoom. Please feel free to schedule a meeting or send an email whenever you have questions about the course or questions about problems on a homework assignment. You are also welcome to drop by any time I happen to be in my office.

dlk53 AT ku dot edu

All important course information and relevant resources can also be found on Canvas.

After each class, a synopsis of what was covered will be posted on Canvas and here: Daily Update

Daily problems will be posted to Canvas and our course web page. These problems are entirely optional, but students will gain a better understanding of the course material if they try working as many of the problems as possible. Each math class has a language of its own, and like any language, fluency can only be obtained through regular practice. Daily homework can be found here.

On the day of the final exam, each student will give a presentation of a published paper of their choosing. Students will also submit a short paper summarizing the main points of the paper they present.

Friday May 15, 1:30-4:00pm.