MATH 7320: Seminar in Algebra: Differential graded techniques in commutative algebra (Spring 2011)

Instructor: W. Frank Moore

For a ring R, the presence of an algebra structure on a complex of R-modules allows one to describe the differentials more succinctly, and in some cases provides one with finite descriptions of infinite complexes. As such, questions regarding the free resolutions can in some cases be answered using techniques from differential graded (DG) algebra.

In this course, we will be covering some of the basic material on DG techniques and applications to commutative algebra. I will start from scratch assuming only a modest background in commutative and homological algebra and will cover the introductory material as well as give enough background on DG algebras so that the students participating can decide which topics to focus on. I will also give overviews of some of the commutative algebra topics necessary to cover the examples that we pick.

Although I will focus primarily on applications to commutative algebra, some of the applications have an analogue in rational homotopy theory which will also be mentioned at various points in the course.

The background material will include:

  1. Basic facts about DG algebras and DG modules
  2. The Koszul complex and other important examples of DG algebras
  3. Poincaré series of modules, upper bounds
  4. The hierarchy of commutative local rings

A subset of the topics we can cover depending on interest:

  1. Deviations of a local ring
  2. Golod rings, trivial Massey operations, and the Golod resolution
  3. Growth of Betti numbers (several topics here actually)
  4. The homotopy Lie algebra of a local ring
  5. Poincaré series for commutative algebras with monomial relations
  6. The Koszul homology algebra of a Gorenstein ring
  7. Homological characterization of complete intersections
  8. Cohomology operators on complete intersections
  9. The Koszul complex of Stanley-Reisner rings via cohomology of moment-angle complexes