AI 中文总结
本文针对有限维代数的余倾斜对，基于其与HRS倾斜心有限表现单对象内射包的对应关系，提出获取HRS倾斜心所有单对象的方法，并给出余倾斜对不可约直和项左、右可突变的充要条件。
AI 中文摘要
设A是有限维代数，模A中挠对的格由余倾斜对控制，余倾斜对是支持τ-倾斜对的无限生成类似物。哈塞箭图中的边（即无挠类的极小包含）对应余倾斜对的不可约突变。与经典τ-倾斜理论的一个重要区别是，余倾斜对的所有不可约直和项并非都可突变，因此识别给定余倾斜对中的可突变不可约直和项至关重要。众所周知，可突变直和项对应HRS倾斜心中有限表现单对象的内射包。基于该对应关系，我们首先提出一种获取HRS倾斜心中所有单对象的方法，随后给出给定余倾斜对的不可约直和项左可突变或右可突变的若干必要与充分条件。
英文摘要
Let $A$ be a finite dimensional algebra. The lattice of torsion pairs in $\rm mod (A)$ is controlled by cosilting pairs, infinitely generated analogues of support $τ^-$-tilting pairs. Then, edges in the Hasse quiver (i.e. minimal inclusions of torsion-free classes) correspond to irreducible mutations of cosilting pairs. An important difference with classical $τ$-tilting theory is that not all indecomposable summands of a cosilting pair are mutable. So, it is very important to identify mutable indecomposable summands in a given cosilting pair. It is well-known that mutable summands correspond to injective envelopes of finitely presented simples in the HRS-tilted heart. Based on this correspondence, we first present a method for obtaining all simples in the HRS-tilted heart, and then give some necessary and sufficient conditions for an indecomposable summand of a given cosilting pair to be left mutable or right mutable.
