As an algebraic geometer, I am a firm believer in the unity among the three main brances of mathematics: algebra, analysis, and geometry. (Logic deserves a place here too.) In fact algebra seems to be the main way mathematicians express things in a way amenable to computation and precision. My intention in this oourse is to try to cover some of the most important algebraic concepts and techniques that mathematicians seem to use in real life research. Of course, the choice of topics will be very much influenced by my own peronsal experience, and hence will emphasize commutative algebra at the expense of some important noncommutative topics. I hope in particular that the course will be a good preparation for continuing research in algebraic geometry and number theory. I will continue to emphasize the categorical and functorial point of view that showed its great power in many fields of mathematics. The main text will be Serge Lang's Algebra, but I will also refer to other texts for commutative algebra, especially David Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry.

The course will be graded in a relatively serious manner. Although I don't expect to have a final exam or homework grader, I will assign homeork and will expect students to turn in written assignments for me to look at and to present solutions in class. I also want students to send weekly email questions about topics covered in class, and indeed, this is to be regarded as a course requirement.

Limits

Universal Effective Epimorphisms

Dimension

Regular local rings I

Complexes and cohomology

Mapping cones

Projective dimension (final exam project is to clean this up!)