On direct summands of syzygies of the residue field of a local ring

TL;DR

This paper studies local rings where syzygies of the residue field decompose as direct summands, revealing properties like eventual periodicity of Betti sequences and characterizing Golod rings through syzygy decompositions.

Contribution

It introduces a new class of local rings with syzygy decompositions, proves Betti sequence periodicity, confirms Tachikawa conjecture for these rings, and characterizes Golod rings via syzygy decompositions.

Findings

Betti sequences are eventually periodically non-decreasing in these rings

Tachikawa conjecture holds for Cohen-Macaulay rings with this syzygy property

Recursive syzygy decompositions characterize Golod rings

Abstract

We investigate local rings in which a syzygy of the residue field occurs as a direct summand of another syzygy of the field. This class of local rings includes Golod rings, Burch rings and non-trivial fiber products of local rings. For such rings, we prove that the Betti sequence of any finitely generated module is eventually periodically non-decreasing. As an application, we confirm the Tachikawa conjecture for all Cohen-Macaulay local rings satisfying this syzygy condition. In the second part of the paper, we show that a recursive direct sum decomposition of the syzygy of the residue field characterizes Golod rings, thereby establishing the converse to a recent theorem of Cuong-Dao-Eisenbud-Kobayashi-Polini-Ulrich [10].