Construction of Gorenstein projective modules over tensor rings

TL;DR

This work provides a complete framework to classify complete projective resolutions and Gorenstein projective modules over tensor rings $T_R(M)$ for $N$-nilpotent bimodules $M$. The authors derive explicit necessary and sufficient conditions for induced sequences $\mathrm{Ind}(P^{\bullet})$ to be complete projective resolutions and show that GP$(T_R(M))$ modules correspond to kernels of these maps, with the Ind functor lifting GP from $R$ to $T_R(M)$ under compatibility. This yields a practical construction method for GP$(T_R(M))$ modules from GP$(R)$ and extends to several classical relatives, including trivial ring extensions, Morita context rings, and triangular matrix rings, providing both rederived and new results. The results unify and generalize prior work on Gorenstein homological properties in these tensor-ring settings, offering concrete criteria and explicit resolutions that can be applied to a broad class of rings and bimodules.

Abstract

For a tensor ring $T_R(M)$, we obtain sufficient and necessary conditions to describe all complete projective resolutions and all Gorenstein projective modules. As a consequence, we provide a method for constructing Gorenstein projective modules over $T_R(M)$ from the ones of $R$. Some applications to trivial ring extensions, Morita context rings and triangular matrix rings are given.

Theorems & Definitions (20)

• Theorem A
• Theorem B
• Lemma 2.1
• proof : Proof
• Lemma 2.2
• proof : Proof
• Theorem 2.3
• Corollary 2.4
• Corollary 2.5
• proof : Proof