Tensor Products of Flat Cotorsion Modules and Cotorsion Dimension

TL;DR

This paper investigates the behavior of flat cotorsion modules under tensor products over k-algebras, establishing a characterization and providing bounds for cotorsion dimensions of tensor product algebras.

Contribution

It proves that the tensor product of flat cotorsion modules remains flat cotorsion and offers a lower bound for the cotorsion dimension of tensor product algebras.

Findings

Tensor product of flat cotorsion modules is flat cotorsion if and only if each factor is.

Provides a lower bound for the global cotorsion dimension of tensor product algebras.

Establishes conditions under which flat cotorsion modules are preserved under tensor products.

Abstract

This paper studies the tensor product of flat cotorsion modules. Let~$R$~and $S$ be~$k$-algebras. We prove that both~$R$-module\ $M$ and~$S$-module\ $N$ are flat cotorsion modules if and only if~$M\otimes_{k} N$ is a flat cotorsion~$R\otimes_{k} S $-module. Based on this conclusion, we provide a lower bound for the global cotorsion dimension of the tensor product algebra~$R\otimes_{k}S $ under appropriate conditions.