Motivated by some properties satisfied by Gorenstein projective and Gorenstein injective modules over an Iwanaga-Gorenstein ring, we present the concept of left and right $n$-cotorsion pairs in an abelian category $\mathcal{C}$. Two classes $\mathcal{A}$ and $\mathcal{B}$ of objects of $\mathcal{C}$ form a left $n$-cotorsion pair $(\mathcal{A,B})$ in $\mathcal{C}$ if the orthogonality relation $\mathsf{Ext}^i_{\mathcal{C}}(\mathcal{A,B}) = 0$ is satisfied for indexes $1 \leq i \leq n$, and if every object of $\mathcal{C}$ has a resolution by objects in $\mathcal{A}$ whose syzygies have $\mathcal{B}$-resolution dimension at most $n-1$. This concept and its dual generalise the notion of complete cotorsion pairs, and has an appealing relation with left and right approximations, especially with those having the so called unique mapping property. The main purpose of this paper is to describe several properties of $n$-cotorsion pairs and to establish a relation with complete cotorsion pairs. We also give some applications in relative homological algebra, that will cover the study of approximations associated to Gorenstein projective, Gorenstein injective and Gorenstein flat modules and chain complexes, as well as $m$-cluster tilting subcategories.

中文翻译：

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n-Cotorsion 对

受到 Iwanaga-Gorenstein 环上 Gorenstein 射影和 Gorenstein 单射模块满足的一些性质的启发，我们在阿贝尔范畴 $\mathcal{C}$ 中提出了左右 $n$-cotorsion 对的概念。$\mathcal{C}$ 对象的两个类 $\mathcal{A}$ 和 $\mathcal{B}$ 在 $\ 中形成左 $n$-cotorsion 对 $(\mathcal{A,B})$ mathcal{C}$ 如果索引 $1 \leq i \leq n$ 满足正交关系 $\mathsf{Ext}^i_{\mathcal{C}}(\mathcal{A,B}) = 0$，并且如果 $\mathcal{C}$ 中的每个对象都被 $\mathcal{A}$ 中的对象解析，其 syzygies 最多具有 $\mathcal{B}$-解析维度 $n-1$。这个概念及其对偶概括了完全 cotorsion 对的概念，并且与左右近似具有吸引人的关系，尤其是那些具有所谓的唯一映射属性的近似。本文的主要目的是描述$n$-cotorsion对的几个性质，并建立与完全cotorsion对的关系。我们还给出了在相对同调代数中的一些应用，这将涵盖与 Gorenstein 射影、Gorenstein 单射和 Gorenstein 平面模块和链复合以及 $m$-cluster 倾斜子类别相关的近似研究。