AI 中文总结
本文研究局部可表示阿贝尔范畴中由容许平衡对诱导的恰当结构，证明相关Gorenstein类构成遗传余挠对，并给出相对纯性下的函子完备性判据及Dedekind域上的显式计算。
AI 中文摘要
设 $(\mathcal{X},\mathcal{Y})$ 是局部可表示阿贝尔范畴 $\mathcal{A}$ 中的一个容许平衡对，并设 $\mathcal{E}$ 为其诱导的恰当结构。我们研究类 $G(\mathcal{X})$，即双边复形中项属于 $\mathcal{X}$ 且同时为右和左 $\mathcal{X}$-无环的循环的类。若 $\mathcal{X}$ 对直和因子和有限直和封闭，且对每个足够大的正则基数 $\kappa$ 是 $\kappa$-Kaplansky 的，我们证明 $(G(\mathcal{X}),G(\mathcal{X})^{\perp *})$ 是相对于 $\mathcal{E}$ 的遗传余挠对。对于由有限呈现模决定的相对纯性，我们给出函子完备性的两个判据，并证明相对周期模生成一个函子完备的遗传余挠对。我们还计算了相对 Gorenstein 对象、其右正交，以及 Dedekind 域上一族恰当结构的显式逼近。
英文摘要
Let $(\mathcal{X},\mathcal{Y})$ be an admissible balanced pair in a locally presentable abelian category $\mathcal{A}$, and let $\mathcal{E}$ be its induced exact structure. We study the class $G(\mathcal{X})$ of cycles of two-sided complexes with terms in $\mathcal{X}$ that are both right and left $\mathcal{X}$-acyclic. If $\mathcal{X}$ is closed under direct summands and finite direct sums and is $κ$-Kaplansky for every sufficiently large regular cardinal $κ$, we prove that $(G(\mathcal{X}),G(\mathcal{X})^{\perp *})$ is a hereditary cotorsion pair relative to $\mathcal E$. For relative purity determined by finitely presented modules, we give two criteria for functorial completeness and show that relative periodic modules generate a functorially complete hereditary cotorsion pair. We also compute the relative Gorenstein objects, their right orthogonal, and explicit approximations for a family of exact structures over Dedekind domains.
Comments17 pages
