Syzygies of the residue field over a local ring

The syzygies of the residue field over a commutative local ring are important invariants of the ring. However, their module structure remains largely unexplored. In this talk, I'll discuss our results, focusing on the direct summands of the syzygies of the residue field. The talk consists of two parts.

The first part concerns the simple direct summands of the second syzygy of the residue field. I'll introduce the notion of Burch rings, providing examples and explaining their basic properties. The existence of a simple direct summand of the second syzygy is then characterized in terms of Burch rings.

If time permits, I'll also discuss some applications to the classification of subcategories.

This part is based on my joint work with Hailong Dao and Ryo Takahashi.

The second part focuses on determining all direct summands of the higher syzygies of the residue field over a Golod ring. Our main theorem reveals that the higher syzygies of the residue field decomposes as a direct sum of lower syzygies.

As an application, in the case of codimension 2, we obtain the complete description of all indecomposable direct summands.This part is based on my joint work with Doan Trung Cuong, Hailong Dao, David Eisenbud, Claudia Polini, and Bernd Ulrich.