Cleft extensions of rings and singularity categories

Panagiotis Kostas
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Abstract

This paper provides a systematic treatment of Gorenstein homological aspects for cleft extensions of rings. In particular, we investigate Goresnteinness, Gorenstein projective modules and singularity categories in the context of cleft extensions of rings. This setting includes triangular matrix rings, trivial extension rings and tensor rings, among others. Under certain conditions, we prove singular equivalences between the algebras in a cleft extension, unifying an abundance of known results. Moreover, we compare the big singularity categories of cleft extensions of rings in the sense of Krause.
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环的劈裂扩展和奇点范畴
本文系统地论述了环的劈裂扩展的戈伦斯坦同调问题。特别是,我们研究了环的劈裂扩展背景下的戈伦斯坦性、戈伦斯坦投影模块和奇异性范畴。这一背景包括三角形矩阵环、琐碎扩展环和张量环等。在特定条件下,我们证明了奇异扩展中的代数方程之间的奇异等价性,统一了大量已知结果。此外,我们还比较了克劳斯意义上的裂环扩展的大奇异性范畴。
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