Title: Introduction to Commutative Algebra
Proposed Course Meeting times: Tuesday-Thursdays 10-11:15
Text (tentative): Commutative Algebra by M. Atiyah and I.G. MacDonald
Required course: Algebra core course
Contents: Modules projective, injective, local rings, flatness, power
series rings, spectra of rings and their properties, primary
decomposition theorem, depth, dimension theory, regular local rings,
Auslander-Buchsbaum theorem (characterization of regular local rings).
If time permits Cohen-Macaulay rings and Gorenstein rings or Matlis duality.
Description: Commutative algebra is the foundation stone of modern
algebraic geometry and this course should be viewed as a
preparatory (and even required!) course for the Algebraic Geometry
course which we offer and which runs every alternate year. The
course is designed with this purpose in mind. We will begin with the
notion of localization of rings, notions of free, projective,
injective modules, flat modules and move on to discuss prime and
primary ideals and the primary decomposition theorem. After this we will
introduce depth and dimension and prove the dimension
theorem and introduce and study properties of local rings in some
detail. The course may end with the Auslander-Buchsbaum theorem which
characterizes regular local rings. This is theorem is, roughly speaking,
lays the groundwork for the algebro-geometric
analogue of the notion of smoothness. If time permits we will study
the notion of Cohen-Macaulay and Gorenstein rings and some of their
characterizations.
--
Kirti Joshi,
Department of Mathematics,
University of Arizona,
617 N Santa Rita, bldg #89,
Tucson, AZ 85721,
USA.