Cotorsion pairs in Hopfological algebra

Abstract

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.