(projectively coresolved) Gorenstein flat modules over tensor rings
Guoliang Tang, Jiaqun Wei
TL;DR
This paper investigates projectively coresolved Gorenstein flat (PGF) modules over tensor rings $T_R(M)$, where $M$ is an $N$-nilpotent $R$-bimodule. Under the Tor-vanishing condition and finiteness assumptions on $M$ (specifically $ ext{pd}_R M< obreak inite$ and $ ext{fd}_{R^{ ext{op}}} M< obreak inite$, with variants on the left and right), the authors characterize PGF modules over $T_R(M)$ via a natural subcategory $oldsymbol{ ext{ extit{$oldsymbol{ ext{Phi}}$}}}( extsf{PGF}(R))$, identifying PGF$(T_R(M))$ with $oldsymbol{ ext{ extit{$oldsymbol{ ext{Phi}}$}}}( extsf{PGF}(R))$ and showing that a $T_R(M)$-module $(X,u)$ is PGF iff $u$ is monic and $ ext{coker}(u) ext{ is PGF}(R)$. They also relate Gorenstein flat modules over $R$ and $T_R(M)$ via Ind, proving that under finite flat dimensions of $M$, $ ext{Ind}(X)$ is Gorenstein flat over $T_R(M)$ iff $X$ is Gorenstein flat over $R$. These results extend to notable constructions: the trivial extension $R times M$ (when $M$ is $1$-nilpotent) and Morita context rings, yielding explicit criteria for PGF/GF in those settings. The work provides a framework for transferring Gorenstein homological properties across tensor extensions, with potential applications to coherent and non-coherent regimes and to algebraic constructions arising in Morita theory and trivial extensions.
Abstract
Let $T_R(M)$ be a tensor ring, where $R$ is a ring and $M$ is an $N$-nilpotent $R$-bimodule. Under certain conditions, we characterize projectively coresolved Gorenstein flat modules over $T_R(M)$, showing that a $T_R(M)$ module $(X,u)$ is projectively coresolved Gorenstein flat if and only if $u$ is monomorphic and $coker(u)$ is a projectively coresolved Gorenstein flat $R$-module. A class of Gorenstein at modules over $T_R(M)$ are also explicitly described. We discuss applications to trivial ring extensions and Morita context rings.