Q-形导出范畴中的无环与全无环对象
Acyclic and totally acyclic objects in the $Q$-shaped derived category
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中文总结 AI 辅助
本文将Iyengar与Krause的经典结果推广到Q-形导出范畴框架，利用关联Q-形图与链复形的伴随三元函子，刻画了诺特交换环的Gorenstein性质，其主要贡献为该伴随三元函子。
中文摘要 AI 辅助
本文研究基于Q-形图的同调代数，其中Q属于某类小范畴。我们将证明，诺特交换环的Gorenstein性质可通过合适类模的Q-形图的无环性与全无环性相一致的条件来刻画。许多特殊情形可作为推论出现，例如N-复形。这将Iyengar与Krause的经典结果推广到Q-形导出范畴的框架，该框架以Iyama与Minamoto的洞见为基础。我们利用关联Q-形图与链复形的伴随三元函子证明该结果，其中一个函子由Jasso引入，整个三元函子推广了Avramov、Buchweitz、Iyengar以及Nkansah引入的微分模的压缩、余压缩与扩张函子。我们认为该伴随三元函子是本文的主要贡献。
英文摘要
This paper concerns homological algebra based on $Q$-shaped diagrams, where $Q$ belongs to a certain class of small categories. We will show that the Gorenstein property of noetherian commutative rings is characterised by the condition that acyclicity coincides with total acyclicity for $Q$-shaped diagrams of suitable classes of modules. Many special cases occur as corollaries, for instance $N$-complexes. This generalises a classic result by Iyengar and Krause to the programme of $Q$-shaped derived categories, which builds on an insight of Iyama and Minamoto. We prove our result using an adjoint triple of functors relating $Q$-shaped diagrams to chain complexes. One of the functors was introduced by Jasso, and the whole triple generalises the compression, cocompression, and expansion functors for differential modules introduced by Avramov, Buchweitz, and Iyengar and by Nkansah. We consider the adjoint triple to be the main contribution of this paper.
