Tor algebra of local rings with decomposable maximal ideal
Saeed Nasseh, Maiko Ono, Yuji Yoshino
TL;DR
Let $(R,\mathfrak{m}_R,k)$ be a Noetherian local ring with a nontrivial decomposition $\mathfrak{m}_R=I\oplus J$. The paper leverages Avramov's machine and the fiber-product viewpoint $R\cong R/I\times_k R/J$ to describe the Tor algebra $A_R$ in terms of the Tor algebras of $R/I$ and $R/J$. The main result gives an explicit description of the positive part of the Tor algebra as a non-unital extension: $A_R^+ \cong \left(\left(\bigwedge k^{t}\otimes_k A^+_{R/I}\right)\times \left(A^+_{R/J} \otimes_k \bigwedge k^{s}\right)\right) \ltimes W$, where $t$ and $s$ are the numbers of generators of $I$ and $J$, and $W=\Sigma^{-1}\left(\dfrac{\bigwedge k^{t}\otimes_k \bigwedge k^{s}}{k\otimes_k \bigwedge k^{s}+\bigwedge k^{t}\otimes_k k}\right)$. This construction expresses $A_R$ in terms of the DG-algebra data from $R/I$ and $R/J$, situates the result within the Koszul and mapping-cone framework, and highlights how the Tate/Avramov machinery translates questions about $R$ into questions about $A_R$. The work also clarifies how the tensor and exterior algebra components interact via a nontrivial extension, and suggests directions for extending Tate resolutions in this fiber-product setting.
Abstract
Let $(R,{\frak{m}}_R)$ be a commutative noetherian local ring. Assuming that ${\frak{m}}_R=$$I\oplus J$ is a direct sum decomposition, where $I$ and $J$ are non-zero ideals of $R$, we describe the structure of the Tor algebra of $R$ in terms of the Tor algebras of the rings $R/I$ and $R/J$.