Commutative Algebra Seminar

Title

Summands in high syzygies

Abstract

Work on infinite resolutions beyond the cases of complete intersections and Golod rings has tended to focus on the sequence of Betti numbers. Hai Long Dao and I have recently begun to study a question of a different kind, and I will report on this joint work:

Let R = S/I be an Artinian quotient of a regular local ring S, with residue field k. When does it happen that k is a direct summand of a syzygy module in the R-free resolution of k, or indeed in the R-free resolution of every module? We were surprised by what we found experimentally, and were able to prove a little of what we observed.