# Cograde conditions and cotorsion pairs

## TL;DR

This paper investigates conditions under which certain functors preserve monomorphisms and epimorphisms in the context of rings and semidualizing bimodules, linking cograde conditions to cotorsion pairs and finitistic dimensions.

## Contribution

It introduces new cograde conditions that characterize functor preservation and constructs complete cotorsion pairs related to these conditions.

## Key findings

- Double functors preserve monomorphisms and epimorphisms under cograde conditions.
- Constructs of two complete cotorsion pairs based on cograde assumptions.
- Establishes relations between relative finitistic dimensions and projective dimensions of the bimodule.

## Abstract

Let $R$ and $S$ be rings and $_R\omega_S$ a semidualizing bimodule. We study when the double functor $\Tor^S_i(\omega, \Ext^i_{R}(\omega,-))$ preserves epimorphisms and the double functor $\Ext_{R}^i(\omega, \Tor_i^{S}(\omega,-))$ preserves monomorphisms in terms of the (strong) cograde conditions of modules. Under certain cograde condition of modules, we construct two complete cotorsion pairs. In addition, we establish the relation between some relative finitistic dimensions of rings and the right and left projective dimensions of $\omega$.