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Tensor Products of Flat Cotorsion Modules and Cotorsion Dimension

Yonggang Hu, Linyu Ma, Xintian Wang

TL;DR

This paper investigates when the tensor product of flat cotorsion modules over $k$-algebras $R$ and $S$ remains flat cotorsion over $R\otimes S$. It establishes an if-and-only-if condition: $M$ and $N$ are flat cotorsion iff $M\otimes N$ is flat cotorsion, and it provides a corresponding Ext-based framework to transfer cotorsion properties through tensor products. Building on this, the authors derive additive behavior for cotorsion dimensions in tensor products, yielding precise values in key cases and a general lower bound for the left global cotorsion dimension $l.cot.D(R\otimes S)$ under suitable hypotheses. These results deepen the understanding of cotorsion theory in tensor product algebras and have implications for measuring homological dimensions in composite algebra structures.

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.

Tensor Products of Flat Cotorsion Modules and Cotorsion Dimension

TL;DR

This paper investigates when the tensor product of flat cotorsion modules over -algebras and remains flat cotorsion over . It establishes an if-and-only-if condition: and are flat cotorsion iff is flat cotorsion, and it provides a corresponding Ext-based framework to transfer cotorsion properties through tensor products. Building on this, the authors derive additive behavior for cotorsion dimensions in tensor products, yielding precise values in key cases and a general lower bound for the left global cotorsion dimension under suitable hypotheses. These results deepen the understanding of cotorsion theory in tensor product algebras and have implications for measuring homological dimensions in composite algebra structures.

Abstract

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

Paper Structure

This paper contains 4 sections, 17 theorems, 42 equations.

Table of Contents

  1. introduction
  2. preliminaries
  3. TENSOR PRODUCTS OF COTORSION MODULES
  4. propositions on tensor product of cotorsion dimensions

Key Result

Proposition 2.1

Zhou:3 Let be a short exact sequence in $\mathsf{Mod}\space R$, and be a short exact sequence in $\mathsf{Mod}\space R$. Then there exists a short exact sequence in $\mathsf{Mod}\space R\otimes S$ where and are both homomorphisms of $R\otimes S$-modules; $\mathsf{Ker}\space d_{1}\otimes\partial_{1}=C_{0}\otimes D_{1}+ C_{1}\otimes D_{0}$.

Theorems & Definitions (30)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Remark 2.8
  • Proposition 2.9
  • Proposition 2.10
  • ...and 20 more