First a duality theory is developed for arbitrary finite-dimensional algebras Λ and Λ′. It provides a characterization of the contravariant equivalences which link resolving subcategories of \(\cal P^{< \infty}\)(Λ-mod), the category of finitely generated left Λ-modules of finite projective dimension, to resolving subcategories of \(\cal P^{< \infty}\)(mod-Λ′). The pertinent theorem expands Miyashita’s work on tilting. As a consequence, we find: If there are resolving subcategories of \(\cal P^{< \infty}\)(Λ-mod) and \(\cal P^{< \infty}\)(mod-Λ′), respectively, which are dual via functors satisfying a strict exactness condition, then Λ and Λ′ are derived equivalent.

The core of the paper addresses the tilting theory of truncated path algebras, i.e., of path algebras modulo ideals generated by all paths of a given fixed length in the underlying quiver. (These algebras provide a natural environment for the study of finite-dimensional representations of quivers with oriented cycles in that, for growing Loewy length, they reflect the combinatorics of the quiver in undiluted form.)

If Λ is a truncated path algebra, the category \(\cal P^{< \infty}\)(Λ-mod) is known to be contravariantly finite in Λ-mod, whence Λ has a strong tilting module. It is shown here that all algebras Δ obtained from A via iterated strong tilting retain these assets, their strong tilting modules being explicitly available from the quiver and Loewy length of Λ. The iteration process becomes periodic with period 2 after the initial tilting step. While structurally the algebras Δ that arise from an iteration of strong tilting have little in common with the original truncated algebra A, we decode their homological properties by combining the mentioned dualities with an algebraic-combinatorial approach to their \(\cal P^{< \infty}\)-categories. This analysis permits us to recognize the Δ-modules of finite projective dimension in terms of their intrinsic structure.

首先，为任意有限维代数Λ和Λ'发展了对偶理论。它提供了逆变等价的表征，这些等价将解析\(\cal P^{< \infty}\) (Λ-mod) 的子类别，有限生成的有限投影维度的左 Λ-模的类别，解析子类别\(\cal P^{< \infty}\) (mod-Λ')。相关定理扩展了 Miyashita 在倾斜方面的工作。因此，我们发现： 如果存在\(\cal P^{< \infty}\) (Λ-mod) 和\(\cal P^{< \infty}\) (mod-Λ′ )，分别是满足严格精确性条件的对偶通孔函子，则 Λ 和 Λ' 等价导出。

论文的核心解决了截断路径代数的倾斜理论，即路径代数模理想的理论，由基础箭袋中给定固定长度的所有路径生成。（这些代数为研究具有定向循环的箭袋的有限维表示提供了一个自然环境，因为随着洛伊长度的增加，它们以未稀释的形式反映了箭袋的组合。）

如果 Λ 是截断路径代数，则已知范畴\(\cal P^{< \infty}\) (Λ-mod) 在 Λ-mod 中是逆变有限的，因此 Λ 具有强倾斜模。这里表明，通过迭代强倾斜从 A 获得的所有代数 Δ 都保留了这些资产，它们的强倾斜模块可以从 Λ 的箭袋和洛伊长度明确获得。在初始倾斜步骤之后，迭代过程以周期 2 为周期。虽然结构上由强倾斜迭代产生的代数 Δ 与原始截断代数 A 几乎没有共同之处，但我们通过将上述对偶性与代数组合方法结合来​​解码它们的同调性质\(\cal P^{< \infty}\)- 类别。这种分析使我们能够根据它们的内在结构识别有限投影维度的 Δ 模。