AI 中文总结
本文修改最终同调同构定义，引入n-伴随型奇异等价，发展相关理论并应用于同调猜想简化与性质保持，同时建立其与三角函子的对应关系，纳入3-伴随型奇异等价框架。
AI 中文摘要
我们修改了最终同调同构的定义，使其更自然且富有成效。我们引入了n-伴随型奇异等价，可视为不同层级的标准奇异等价，更便于简化各类同调猜想和传递同调性质。我们针对n≥2的n-伴随型奇异等价，以及导出范畴间的标准最终同调同构，发展了相关理论：通过双模的⊗-（余）完备复形给出其特征刻画；通过对偶代数和张量积代数给出两种构造；给出由代数同态和幂等元诱导的例子；并将其应用于有限维数猜想、Gorenstein对称猜想、Auslander-Reiten猜想、Gorenstein投射猜想等同调猜想的简化，以及Gorenstein性、Fg条件、Happel性质、Han性质等同调性质的保持。此外，我们证明标准最终同调同构对应于带典范左伴随的奇点范畴间的完全忠实标准三角函子，而导出范畴间本质满的标准最终同调同构对应于3-伴随型奇异等价，从而将本质满的标准最终同调同构纳入3-伴随型奇异等价的框架中。
英文摘要
We modify the definition of eventually homological isomorphisms so that it becomes more natural and fruitful. We introduce singular equivalences of $n$-adjoint type which can be viewed as standard singular equivalences of different levels and are more convenient for reducing various homological conjectures and transferring homological properties. We develop the theories of singular equivalences of $n$-adjoint type for $n\ge 2$ and standard eventually homological isomorphisms between derived categories by giving their characterizations in terms of $\otimes$-(co)perfect complexes of bimodules, two constructions via opposite algebras and tensor product algebras, examples induced by homomorphisms of algebras and idempotents, and applications to the reduction of homological conjectures such as the finitistic dimension conjecture, Gorenstein symmetry conjecture, Auslander-Reiten conjecture and Gorenstein projective conjecture, as well as to the preservation of homological properties such as Gorensteinness, the Fg condition, Happel's property and Han's property. Moreover, we show that standard eventually homological isomorphisms correspond to fully faithful standard triangle functors between singularity categories with canonical left adjoint, and essentially surjective standard eventually homological isomorphisms between derived categories correspond to singular equivalences of 3-adjoint type, thereby incorporating essentially surjective standard eventually homological isomorphisms into the framework of singular equivalences of 3-adjoint type.
Comments42 pages
