根子模半模的同调理论
A Homology Theory for the Semimodules of Radical Submodules
AI总结：
本文提出根子模半模的同调理论，通过根函子将R-模复形映射为半模复形，定义根同调并证明其为同伦不变量，同时给出导出长正合序列的条件及基于半模的投射分解存在唯一性条件。
摘要：
设 $R$ 为有单位元的交换环，$\R(R)$ 表示 $R$ 的根理想的半环。根函子 $\R$ 从 $R$-模范畴 $R{-}\boldsymbol{\sf{Mod}}$ 到 $\R(R)$-半模范畴 $\R(R){-}\boldsymbol{\sf{Semod}}$，将任意 $R$-模复形 $\M=(M_n, f_n)_{n\geq 0}$ 映射为 $\R(R)$-半模复形 $\R(\M)=(\R(M_n), \R(f_n))_{n\geq 0}$，其中 $\R(M_n)$ 由 $M_n$ 的根子模构成，且 $\R(R)$-半模同态 $\R(f_n):\R(M_n)\rightarrow \R(M_{n-1})$ 定义为 $\R(f_n)(N)=\rad(f_n(N))$。复形 $(\R(M_n), \R(f_n))_{n\geq 0}$ 的第 $n$ 个根同调记为 $H_n(\R(\M))$，由 $M_n$ 中满足 $f_n(N)$ 包含在 $M_{n-1}$ 的零子模的根中的根子模 $N$ 构成，且两个这样的根子模在模 $\R(f_{n+1})$ 像的 Bourne 关系下等价。$H_n(\R(-))$ 被视为从 $R$-模链复形范畴 $\boldsymbol{\sf{Ch}}(R{-}\boldsymbol{\sf{Mod}})$ 到 $\R(R){-}\boldsymbol{\sf{Semod}}$ 的协变函子，对 $R$-模复形的任意同伦映射对恒等作用。特别地，若 $\M$ 与 $\M'$ 同伦等价，则 $H_n(\R(\M))$ 与 $H_n(\R(\M'))$ 是同构的 $\R(R)$-半模。我们给出了使 $H_n(\R(-))$ 对任意 $R$-模复形短正合序列导出根同调模长正合序列的条件，并满足正合同调序列的自然性条件。最后，我们引入基于 $\R(R)$-半模的 $R$-模 $M$ 的投射分解，并给出该投射分解存在且在同伦意义下唯一的条件。
英文摘要：
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.
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