自对偶余环、Frobenius余环与Ringel自对偶性的比较
Comparing self-dual corings, Frobenius corings and Ringel self-duality
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中文总结 AI 辅助
本文分类自对偶余环与自对偶代数扩张，将其（自）对偶性与Ringel（自）对偶性比较，揭示差异，给出余环自对偶的等价刻画并解释Ringel自对偶的同调特性。
中文摘要 AI 辅助
对自对偶余环和自对偶代数扩张进行分类，并将这些（自）对偶性与Ringel（自）对偶性进行比较，揭示出根本性差异。对每个余环$\boldsymbol{\textit{C}}$，关联两个代数，即$\boldsymbol{\textit{C}}$的左对偶代数与右对偶代数。研究表明，当且仅当$\boldsymbol{\textit{C}}$是Frobenius余环时，这两个代数以自然方式重合。还给出了$\boldsymbol{\textit{C}}$为自对偶的若干其他等价刻画，涉及某些代数扩张为Frobenius扩张，以及某些遗忘函子或限制函子为Frobenius函子。然而，Ringel自对偶性被证明差异显著，文中给出了同调层面的解释。
英文摘要
Self-dual corings and self-dual algebra extensions are classified and these (self-)dualities are compared with Ringel (self-)duality, revealing fundamental differences. To each coring $\mathcal C$, two algebras are associated, known as the left and the right dual algebra of $\mathcal C$. It is shown that these two algebras coincide in a natural way if and only if $\mathcal C$ is a Frobenius coring. Various other equivalent characterisations of $\mathcal C$ being self-dual are given, in terms of certain algebra extensions being Frobenius extensions and in terms of certain forgetful or restriction functors being Frobenius functors. Ringel self-duality however is shown to be rather different, for which a homological explanation is given.
