论文标题
霍格斯代数中的cotorsion对
Cotorsion pairs in Hopfological algebra
论文作者
论文摘要
在一份有趣的论文中，Arxiv：Math/0509083 Khovanov提出了同源代数的概括，称为Hopfological代数。从那时起，已经尝试了几次尝试从同源代数到HopFological代数的工具和技术。例如，Qi arxiv：1205.1814在类别中引入了同伴对象的概念$ \ mathbf {c} _ {a {a，h}^{H $ h $ equivariant模块的$ h $ h $ h $ h $ -module algebra $ a $ a $ a dig the Mode flaf felf felfulla nod a dig a dide n of a dide n of a n dide n of flafiand of a $ h $ h}^{h}^{h}^{h}^{h}^{h}^{ $ \ mathbf {c} _ {a，h}^{h} $上的模型结构。 在本文中，我们表明在$ \ mathbf {c} _ {a，h}^{h} $上存在一个Abelian模型结构，其中同伴对象与Qi的同伴对象一致。这是通过在$ \ Mathbf {c} _ {a，h}^{h} $中构造cotorsion对来完成的，从吉莱斯蒂·arxiv（Gillespie arxiv）的意义上讲，它形成了hovey三倍：1512.06001。这可以被视为Enochs，Jenda和Xu以及Avramov，Foxby和Halperin的作品的啤酒节类似物。通过限制紧凑的同伴对象，我们获得了waldhausen类别$ \ mathcal {p} \ mathrm {erf} _ {a，h}^{h}^{h} $的完美对象。通过将这个瓦尔德豪森类别的不变性（例如代数$ k $ - 理论，霍基柴尔德同源性，循环同源性等）等不变，我们获得了这些不变性的Hopfological类似物。
In an intriguing paper arXiv:math/0509083 Khovanov proposed a generalization of homological algebra, called Hopfological algebra. Since then, several attempts have been made to import tools and techiniques from homological algebra to Hopfological algebra. For example, Qi arXiv:1205.1814 introduced the notion of cofibrant objects in the category $\mathbf{C}_{A,H}^{H}$ of $H$-equivariant modules over an $H$-module algebra $A$, which is a counterpart to the category of modules over a dg algebra, although he did not define a model structure on $\mathbf{C}_{A,H}^{H}$. In this paper, we show that there exists an Abelian model structure on $\mathbf{C}_{A,H}^{H}$ in which cofibrant objects agree with Qi's cofibrant objects under a slight modification. This is done by constructing cotorsion pairs in $\mathbf{C}_{A,H}^{H}$ which form a Hovey triple in the sense of Gillespie arXiv:1512.06001. This can be regarded as a Hopfological analogues of the works of Enochs, Jenda, and Xu and of Avramov, Foxby, and Halperin. By restricting to compact cofibrant objects, we obtain a Waldhausen category $\mathcal{P}\mathrm{erf}_{A,H}^{H}$ of perfect objects. By taking invariants of this Waldhausen category, such as algebraic $K$-theory, Hochschild homology, cyclic homology, and so on, we obtain Hopfological analogues of these invariants.
