Rigidification of connective comodules

Abstract

Let $\mathbb{k}$ be a commutative ring with global dimension zero. We show that we can rigidify homotopy coherent comodules in connective modules over the Eilenberg-Mac Lane spectrum of $\mathbb{k}$. That is, the $\infty$-category of homotopy coherent comodules is represented by a model category of strict comodules in non-negative chain complexes over $\mathbb{k}$. These comodules are over a coalgebra that is strictly coassociative and simply connected. The rigidification result allows us to derive the notion of cotensor product of comodules and endows the $\infty$-category of comodules with a symmetric monoidal structure via the two-sided cobar resolution.

arXiv comment15 pages. Final version, to appear in Proceedings of AMS. Some results in the original version are now in arXiv:2108.04835