Commutative algebra 3 (What is a syzygy?)

This lecture introduces the concept of syzygies in commutative algebra, motivated by Hilbert’s theorem on finite generation of invariant rings. The speaker begins by explaining group actions on polynomial rings, emphasizing the need for the inverse in the action on functions. He then presents examples of invariant rings: the orthogonal group acting on R^3 with the invariant x^2+y^2+z^2, the special linear group with determinants as invariants, and the symmetric group with elementary symmetric functions. He proves the fundamental theorem of symmetric functions using lexicographic order. Next, he considers the alternating group, where the discriminant appears as an additional invariant, leading to a first-order syzygy. He then gives a second-order syzygy example with the cyclic group of order three acting on a 2-dimensional space, illustrating the relations among generators and the resulting exact sequence. Finally, he poses three finiteness questions: finite generation of the invariant ring, finite generation of syzygy modules, and finiteness of the syzygy chain, mentioning Hilbert’s results for reductive groups in characteristic zero.