Ringel duality revisited

这是“重新审视林格尔二元性”项目的延伸，它将建立在该项目获得资金的两年期间（2022年3月结束）所取得的进展的基础上。这个项目的基础仍然是描述准遗传代数和模块的标准过滤方面的bocses和A-无穷大范畴背后。在我最近与布热津斯基和库尔沙默尔的合作作品中，这一点得到了补充，并翻译成了核心语言。孔德对奥夫先科问题的解决，比第一个建议中所述的客观形式更强有力，这大大加强了对bocses本身的描述。特别是，她精确地描述了准遗传代数有一个确切的博雷尔子代数，由一个博茨，她获得了非常精确的结构和数值信息bocses和确切的博雷尔子代数。在解决这一问题的同时，她还为Ringel对偶建立了一个精确的函子框架，在这里将其与Burt-Butler对偶等同起来。在第三年，有三个主要目标需要考虑，所有这些都是继续和扩展研究的（目标A）Ringel自对偶代数的环结构和同调结构，特别是关于保持单对偶的。（目标B）：通过添加第三个对偶（称为余模和逆模对应）来扩展并详细计算孔德的Ringel自对偶框架，使得该框架也涵盖层代数和无限最高权范畴。（目标B，续）：在原建议中，它已被证明，Ringel自对偶的代数意味着存在一个简单的保持对偶的ist模范畴。这将解决第一个提案中提出的许多问题。利用第一部分的框架，结合Frobenius函子的机制以及coring（bocses）上余模和逆模之间的等价性，我们将讨论这个猜想及其产生的问题。第三部分（目标C和D）：关于A-无穷结构的计划工作已经开始，并将扩展到一般范畴框架。最终的结果将澄清的行为A-无穷结构的米田扩展代数下通过“好”商代数或“好”子代数的准遗传代数（或更一般的类别）和兼容性的精确Borel子代数与这些操作。

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.