AI 中文总结
研究通过遗传代数导出范畴的覆盖函子探讨高阶簇倾斜对象，覆盖形式可将存在性问题转化，对有限不可分解对象三角范畴给出存在准则与公式，证明有限Frobenius模型中相关自同态代数导出等价，还应用于多个领域。
AI 中文摘要
我们通过遗传代数导出范畴的覆盖函子研究高阶簇倾斜对象。覆盖形式将存在性问题归结为在\(d\)-簇范畴中寻找\(G\)-稳定\(d\)-簇倾斜对象的等变问题。对于具有有限多个不可分解对象的三角范畴，这给出了完整的ADE存在准则和显式计数公式。我们还证明，在有限Frobenius模型中，所有\(d\)-簇倾斜对象的自同态代数是导出等价的。并应用于Cohen-Macaulay有限范畴、有限非交换crepant分解和表示有限自入射代数的刚性维数。
英文摘要
We study higher cluster tilting objects through covering functors from derived categories of hereditary algebras. The covering formalism reduces the existence problem to the equivariant problem of finding \(G\)-stable \(d\)-cluster tilting objects in \(d\)-cluster categories. For triangulated categories with finitely many indecomposable objects this gives a complete ADE existence criterion and explicit counting formulas. We also prove that, in finite Frobenius models, the endomorphism algebras of all \(d\)-cluster tilting objects are derived equivalent. Applications are given to Cohen--Macaulay finite categories, finite noncommutative crepant resolutions, and rigidity dimensions of representation-finite self-injective algebras.
