Cotorsion pairs in Hopfological algebra

TL;DR

This paper establishes a model structure on a category of H-equivariant modules over an H-module algebra, introducing cotorsion pairs that serve as Hopfological analogues of classical homological algebra tools, enabling the study of invariants like K-theory and Hochschild homology.

Contribution

It constructs cotorsion pairs forming a Hovey triple in the category of H-equivariant modules, creating a foundation for Hopfological homological algebra analogous to classical theories.

Findings

Existence of an Abelian model structure on the category of H-equivariant modules.

Identification of cofibrant objects with Qi's cofibrant objects under a modification.

Development of Hopfological analogues of algebraic K-theory, Hochschild, and cyclic homology.

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.