On the monomorphism category of large modules
Rasool Hafezi, Javad Asadollahi, Razieh Vahed, Yi Zhang
TL;DR
The paper develops a comprehensive framework linking the monomorphism category of large $R$-modules with contravariant and covariant functor categories on finitely presented modules, using pure-injective modules as a central tool. It constructs, analyzes, and applies two central functors, $\Psi$ and $\Phi$, arising from the morphism category of pure-injectives, and shows that they are full, dense, and objective, yielding quotient equivalences with stable and finitely presented functor categories. A duality $\varrho$ between stable left and right module categories is established, and, in Artinian rings with Morita self-duality, this duality aligns with Auslander's transpose $\mathrm{Tr}$, realized functorially as $\mathrm{Tr}^*$. For Quasi-Frobenius rings of finite representation type, the paper proves that suitable Theta and Im functors are full, dense, and objective, producing equivalences with the stable Auslander algebra $\Gamma$, and thereby linking monomorphism/cokernel structures to $\Gamma$-modules and stable categories. The results collectively provide a unified picture connecting large-module monomorphism categories, functor categories, pure-injective phenomena, and classical dualities, with concrete consequences for representation theory and potential geometric applications via recollements and stable/costable correspondences.
Abstract
Let $R$ be an associative ring with identity. This paper investigates the structure of the monomorphism category of large $R$-modules and establishes connections with the category of contravariant functors defined on finitely presented $R$-modules. Several equivalences and dualities will be presented. Our results highlight the role of pure-injective modules in studying the homological properties of functor categories.