On direct summands of syzygies of the residue field of a local ring
Doan Trung Cuong, Toshinori Kobayashi
TL;DR
The paper studies local rings in which a syzygy of the residue field occurs as a direct summand of another, focusing on the syzygy direct summand condition $(\ast)$ and identifying Golod, Burch, and fibre-product rings as key sources. It establishes a Betti-number monotonicity/periodicity phenomenon under $(\ast)$, enabling partial Tachikawa-type conclusions for Cohen–Macaulay rings and confirming Tachikawa conjecture in new CM cases. It then proves a complete Golodness characterization: $R$ is Golod if and only if $\mathrm{syz}_{e+1}^R(k)$ decomposes as $\bigoplus_{j=1}^c \mathrm{syz}_{e-j}^R(k)^{h_j(R)}$ (and the analogous decompositions for all $m\ge 0$), with $e$ the embedding dimension, $c$ the codepth, and $h_j(R)=\dim_k H_j(K_\bullet)$ for the Koszul complex $K_\bullet$. The results connect Ext-vanishing, Betti growth and singularity, providing a framework to recognize Golod rings from syzygy decompositions and tying together several classes of rings through explicit decompositions of higher syzygies.
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].