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

2025/10/28 by Cuong, Doan Trung, Kobayashi, Toshinori
#13H10 #Commutative Algebra (math.AC) #FOS: Mathematics #Primary: 13D02. Secondary: 13D07

paper · doi:10.48550/arxiv.2510.24220

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].

Citations

• Syzygies of the residue field over Golod rings
• k Summands of Syzygies over Rings of Positive Burch Index Via Canonical Resolutions
• Burch index, summands of syzygies and linearity in resolutions
• Persistence of homology over commutative noetherian rings
• Burch ideals and Burch rings
• Homology over trivial extensions of commutative DG algebras
• Direct summands of syzygy modules of the residue class field
• On the growth of the Betti sequence of the canonical module
• Syzygy modules with semidualizing or G-projective summands
• Rings which are almost Gorenstein

Related

• Syzygies of the residue field over Golod rings
• Vector space summands of lower syzygies
• Direct summands of syzygy modules of the residue class field
• k Summands of Syzygies over Rings of Positive Burch Index Via Canonical Resolutions
• Syzygy modules with semidualizing or G-projective summands