Graded comodule categories with enough projectives

TL;DR

This paper proves that connective graded comodules over certain Hopf algebroids have enough projectives, enabling new algebraic topology applications, and shows these categories are not equivalent to module categories over rings.

Contribution

It establishes the existence of enough projectives in categories of connective graded comodules over specific Hopf algebroids, a property not previously known.

Findings

Categories of connective graded comodules over certain Hopf algebroids have enough projectives.

Applications to algebraic topology include stable co-operations in complex bordism and homology theories.

Categories of connective graded comodules are not equivalent to categories of graded modules over rings.

Abstract

It is well-known that the category of comodules over a flat Hopf algebroid is abelian but typically fails to have enough projectives, and more generally, the category of graded comodules over a graded flat Hopf algebroid is abelian but typically fails to have enough projectives. In this short paper we prove that the category of connective graded comodules over a connective, graded, flat, finite-type Hopf algebroid has enough projectives. Applications to algebraic topology are given: the Hopf algebroids of stable co-operations in complex bordism, Brown-Peterson homology, and classical mod $p$ homology all have the property that their categories of connective graded comodules have enough projectives. We also prove that categories of connective graded comodules over appropriate Hopf algebras fail to be equivalent to categories of graded connective modules over a ring.