For an integer N ≥ 2 and three subcategories 𝒲,𝒳,𝒴 of modules, we study the notion of (𝒲,𝒳,𝒴)-Gorenstein N-complexes. We show that under certain hypotheses, an N-complex C is (𝒲,𝒳,𝒴)-Gorenstein if and only if Cn is a (𝒲,𝒳,𝒴)-Gorenstein module for each n ∈ ℤ, HomR(V,C) and HomR(C,U) are N-acyclic for any V ∈𝒴̃ and U ∈𝒳̃, where 𝒳̃ and 𝒴̃ denote the subcategory of 𝒳N-complexes and 𝒴N-complexes, respectively. Also, the stability of (𝒲,𝒳,𝒴)-Gorenstein N-complexes is considered. As applications, not only some known results are encompassed, but also some new results are obtained, for instance, some equivalent characterizations for Ding injective N-complexes and Gorenstein AC-injective N-complexes are given.

中文翻译：

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N-复形类别中的 Gorenstein 对象

对于整数ñ ≥ 2和三个子类别𝒲,𝒳,𝒴模块，我们研究的概念(𝒲,𝒳,𝒴)-戈伦斯坦ñ-复合体。我们表明，在某些假设下，ñ-复杂的C是(𝒲,𝒳,𝒴)——戈伦斯坦当且仅当Cn是一个(𝒲,𝒳,𝒴)- 每个人的 Gorenstein 模块n ∈ ℤ,霍姆R(五,C)和霍姆R(C,ü)是ñ- 无环的任何五 ∈𝒴̃和ü ∈𝒳̃， 在哪里𝒳̃和𝒴̃表示子类𝒳ñ- 复合物和𝒴ñ-复合体，分别。另外，稳定性(𝒲,𝒳,𝒴)-戈伦斯坦ñ-复合体被考虑。作为应用，不仅包含了一些已知的结果，而且还得到了一些新的结果，例如，丁内射的一些等价刻画ñ-复数和 Gorenstein AC-内射ñ-给出了复合物。