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### Intuitive idea: What is the motivation for relative homological algebra

Basically I was wondering what is the purpous of choosing a smaller class of epis in a category? Does it allow for the existence of "more" "projective-like" objects?
What are some applications? For ...

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### modules and comodules related to BP_*

Let BP be the Brown Peterson spectrum. We know that the category of $BP_* BP$-comodules is an abelian category. Consider the category of $BP_* BP$-comodules, which are finitely presented as ...

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### Changing Left Comodules into Right Comodules via the Antipode, and Comodule Tensor Products

Let $H$ be a Hopf algebra, with invertible antipode, and let $(M,\Delta_M)$ and $(N,\Delta_N)$ be two left $H$-comodules. Now as we all know, we have a left $H$-comodule structure on the tensor ...

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### When is a Surjective Comodule Endomorphism an Automorphism?

Given a Hopf algebra $H$, a left $H$-comodule $V$, and a surjective comodule endomorphism $f: V \to V$. Can somebody give:
(i) a set of neccessary, or sufficient, or both neccessary and sufficient, ...