The Homotopy Category of Strongly flat modules
Javad Asadollahi, Somayeh Sadeghi
TL;DR
The paper investigates the homotopy category of $S$-strongly flat modules $K(SF-R)$ and its relationship to the homotopy categories of projectives and flats, building on Neeman's framework for well-generated categories. It establishes adjoint relationships and fully faithful embeddings among $K(Prj-R)$, $K(SF-R)$, and $K(Flat-R)$, and introduces the notion of $S$-almost well generated triangulated categories. Under the Optimistic Conjecture (and verified cases), it proves the existence of right adjoints to quotient maps that yield embeddings of projectives into $S$-strongly flat categories, and it analyzes when $K(Flat-R)$ inherits $S$-almost well generated structure from $R$ being $S$-almost perfect. The work connects cotorsion-pair theory, optimistically flatness concepts, and tensor triangulated structure to shed light on the otherwise mysterious class of $S$-strongly flat modules and their homotopy categories.
Abstract
In this paper, we plan to build upon significant results by Amnon Neeman regarding the homotopy category of flat modules to study ${\mathbb{K}}({S\rm{SF}}\mbox{-}R)$, the homotopy category of $S$-strongly flat modules, where $S$ is a multiplicatively closed subset of a commutative ring $R$. The category ${\mathbb{K}}({S\rm{SF}}\mbox{-}R)$ is an intermediate triangulated category that includes ${\mathbb{K}}({\rm{Prj}\mbox{-}} R)$, the homotopy category of projective $R$-modules, which is always well generated by a result of Neeman, and is included in ${\mathbb{K}}({\rm{Flat}}\mbox{-} R)$, the homotopy category of flat $R$-modules, which is well generated if and only if $R$ is perfect, by a result of Štovíček. We analyze corresponding inclusion functors and the existence of their adjoints. In this way, we provide a new, fully faithful embedding of the homotopy category of projectives to the homotopy category of $S$-strongly flat modules. We introduce the notion of $S$-almost well generated triangulated categories. If $R$ is an $S$-almost perfect ring, ${\mathbb{K}}({\rm{Flat}}\mbox{-} R)$ is $S$-almost well generated. We show that the converse is true under certain conditions on the ring $R$. We hope that this approach provides insights into the largely mysterious class of $S$-strongly flat modules.