Ringel duality revisited

这是“重新审视林格尔双重性”项目的延伸，它将建立在该项目获得资金的两年内取得的进展（截至2022年3月）。这个项目的基础仍然是描述准遗传代数和模的标准过滤在bocs和a -∞范畴背后。这在我最近与布热津斯基和k<s:1>沙默的合作中得到了补充，翻译成corings的语言。Conde对Ovsienko问题的解决方案以一种比第一个建议中所述的目标更强的形式加强了对bocs本身的描述。特别是，她精确地描述了哪些准遗传代数具有由bocs给出的精确Borel子代数，并获得了关于bocs和精确Borel子代数的非常精确的结构和数值信息。在解决这个问题的同时，她还为林格尔对偶建立了一个精确的功能框架，在这里等同于伯特-巴特勒对偶。在第三年，有三个主要目标要考虑，所有继续和扩展的研究（目标A）环结构和同调结构的Ringel自对偶代数，特别是关于保简对偶。第1部分（目标B）：通过添加第三个对偶，即协模和对偶模对应，扩展并详细推导出Conde的Ringel自对偶框架，使该框架也涵盖分层代数和无限最高权范畴。第2部分（目标B，续）：在最初的建议中，我们推测代数的Ringel自对偶意味着在单模范畴上存在一个简单保持对偶。这将解决第一个建议中提出的许多问题。利用第1部分的框架，结合Frobenius函子的机制，以及关于包芯上的模和控制模之间等价的结果，我们将探讨这个猜想和激发它的问题。第3部分（目标C和D）：关于A-infinity结构的计划工作已经开始，并将扩展到一般的范畴框架。最终的结果将阐明Yoneda扩展代数的a -无穷结构在过渡到“好”商代数或准遗传代数（或更一般的范畴）的“好”子代数下的行为，以及精确Borel子代数与这些操作的相容性。

This is an extension of the project "Ringel duality revisited" and it will build on the progress made during the two years of that project obtaining funding (ending in March 2022). The basis of this project continues to be the description of quasi-hereditary algebras and modules with standard filtrations in terms of bocses and the A-infinity categories behind these. This has been supplemented in my recent joint work with Brzezinski and Külshammer with a translation into the language of corings. The description in terms of bocses itself has been strengthened much by Conde's solution of Ovsienko's problem in a stronger form than stated as objective in the first proposal. In particular, she precisely described which quasi-hereditary algebras have an exact Borel subalgebra, given by a bocs, and she obtained very precise structural and numerical information about bocses and exact Borel subalgebras. While solving this problem, she also put together a precise functorial framework for Ringel duality, which here gets identified with Burt-Butler duality.In the third year, there are three main aims left to be considered, all continuing and extending the investigation (objective A) of ring structure and homological structure of Ringel self-dual algebras, especially with respect to simple-preserving dualities.Part 1 (objective B): Extend and work out in detail Conde's framework for Ringel self-duality by adding a third duality known as co- and contramodules correspondence in such a way that the framework also covers stratified algebras and infinite highest weight categories.Part 2 (objective B, continued): In the original proposal, it has been conjectured that Ringel self-duality of an algebra implies the existence of a simple preserving duality on ist module category. This will solve many problems stated in the first proposal. Using the framework in part 1 together with the machinery of Frobenius functors and also results on equivalences between comodules and contramodules over corings (bocses), this conjecture and the problems motivating it will be approached.Part 3 (objectives C and D): The planned work on A-infinity structure has already started, and is to get extended to the general categorical framework. The final results will clarify the behaviour of A-infinity structures of Yoneda extension algebras under passage to "good" quotient algebras or "good" subalgebras of quasi-hereditary algebras (or more general categories) and on compatibility of exact Borel subalgebras with such operations.