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Let $H$ be a Hopf algebra and $A$ an $H$-comodule algebra; denote by $M^H_A$ the category of right $(H,A)$-Hopf modules [i.e. $A$-module, $H$-comodules, everything is compatible with everything else]. I would like to know if there is an analogue of Baer's criterion for the category $M^H_A$; in other words,

is it true that if an object $N$ of $M^H_A$ has the lifting property for every sub-object of $A$ (or some other distinguished object of the category), then $N$ is injective in $M^H_A$?

A few partial answers:

  • If $H$ is finite dimensional then $M^H_A \cong M_{A \# H^*}$, where $A \# H^*$ is the smash-product of A and the dual Hopf-algebra $H^*$; thus in this case we can use the usual criterion [with a different base ring!].

  • If $H$ is a group algebra then there is a graded analogue of Baer's criterion, using all possible shifts of $A$.

Both are instances of a more general version of Baer's criterion for Grothendieck categories with a family of generators [which I've seen mentioned yet never found a proof], but I haven't been able to adapt this to the general context.

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    $\begingroup$ Baer's criterion for AB5-categories with a generating family is proven as Théorème 5.2.2 in Grothendieck's Alger lecture notes Introduction au langage fonctoriel (1966) which can be found somewhere on the internet. $\endgroup$
    – Fred Rohrer
    Commented Nov 20, 2014 at 15:41
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    $\begingroup$ ... and also in Kashiwara-Shapiro, Categories and sheaves, Proposition 8.4.7, and in Popescu, Abelian categories, Lemma 3.3.1, and (of course) in the Stacks Project, Tag 079G. $\endgroup$
    – Fred Rohrer
    Commented Nov 20, 2014 at 22:03

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