A Homology Theory for the Semimodules of Radical Submodules

TL;DR

This paper develops a homology theory for radical submodules of modules over a commutative ring, using semimodules over the semiring of radical ideals, and explores its properties and resolutions.

Contribution

It introduces a novel radical homology functor for complexes of modules, extending classical homology concepts to radical submodules via semimodule structures.

Findings

Defines radical homology modules for complexes of modules.

Establishes conditions for long exact sequences in radical homology.

Introduces a projective resolution framework based on semimodules.

Abstract

Let $R$ be a commutative ring with identity, and let $\R(R)$ denote the semiring of radical ideals of $R$. The radical functor $\R$, from the category of $R$-modules $R{-}\boldsymbol{\sf{Mod}}$ to the category of $\R(R)$-semimodules $\R(R){-}\boldsymbol{\sf{Semod}}$, maps any complex $\M=(M_n, f_n)_{n\geq 0}$ of $R$-modules to a complex $\R(\M)=(\R(M_n), \R(f_n))_{n\geq 0}$ of $\R(R)$-semimodules, where $\R(M_n)$ consists of radical submodules of $M_n$, and the $\R(R)$-semimodule homomorphisms $\R(f_n):\R(M_n)\rightarrow \R(M_{n-1})$ are defined by $\R(f_n)(N)=\rad(f_n(N))$. The $n$-th radical homology of the complex $(\R(M_n), \R(f_n))_{n\geq 0}$, denoted $H_n(\R(\M))$, consists of radical submodules $N$ of $M_n$ such that $f_n(N)$ is contained in the radical of the zero submodule of $M_{n-1}$, and two such radical submodules are equivalent under the Bourne relation modulo the image of $\R(f_{n+1})$. $H_n(\R(-))$ is regarded as a covariant functor from the category $\boldsymbol{\sf{Ch}}(R{-}\boldsymbol{\sf{Mod}})$ of chain complexes of $R$-modules to $\R(R){-}\boldsymbol{\sf{Semod}}$, which acts identically on any pair of homotopic maps of complexes of $R$-modules. In particular, if $\M$ and $\M'$ are homotopically equivalent, then $H_n(\R(\M))$ and $H_n(\R(\M'))$ are isomorphic $\R(R)$-semimodules. We provide conditions under which $H_n(\R(-))$ induces a long exact sequence of radical homology modules for any short exact sequence of complexes of $R$-modules, and satisfies the naturality condition for exact homology sequences. Finally, we introduce a projective resolution for an $R$-module $M$ based on $\R(R)$-semimodules and give conditions under which such a projective resolution exists and is unique up to a homotopy.