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Background:

Let $H$ be a finitely generated commutative Hopf $k$-algebra, where $k$ is a field of non-zero characteristic. For $$ \widehat{H} := \text{Alg}_k\{H; k\}, $$ we recall (see Abe Chapter 4 for example) that the Hopf algebra structure of $H$ induces in a canonical way a group structure on $\widehat{H}$.

Question:

Is it true that the category of finite-dimensional modules over $\widehat{H}$ is equivalent to the category of finite-dimensional co-modules over $H$. If so, what is a good reference for this?

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2 Answers 2

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A more comprehensive reference than Waterhouse is the book Representations of Algebraic Groups by J.C. Jantzen (2nd ed., AMS, 2003). Though he aims after a while at prime characteristic, his foundational material in Part I is much more general and often follows the treatment in the older book (in French) by Demazure-Gabriel. Anyway, the appropriate reference is section I.2, especially subsection 2.8.

Jantzen mostly works here over an arbitrary field (sometimes even over a commutative ring), using the language of group schemes and Hopf algebras. You may or may not need the full generality of what he does, but for instance these foundations are needed for the theory of Frobenius kernels in prime characteristic.

ADDED: Maybe a more direct treatment of the connection between groups and Hopf algebras, along with their modules and comodules, is found at the beginning of Hochschild's textbook Basic Theory of Algebraic Groups and Lie Algebras (GTM 75, Springer, 1981). Of course this predates the development of quantum groups, but the elementary ideas are there. Hochschild's approach eventually works best in characteristic 0, but at the outset he deals just with generalities without scheme language as such.

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  • $\begingroup$ Thanks a lot. The reference you have given deals with group schemes, however I am just interested in group varieties, or algebraic groups. Instead of working with Hom$(H,-)$ (a representable functor from the category of $k$-algebras to sets) I would want to work with Hom$(H,k)$ (as in Sweedler's book). I know that $H$ needs to be finitely generated and reduced, but I not sure how to convert the results using the Hom$(A,-)$-definition to the Hom$(A,k)$-definition. I hope this makes some sense! $\endgroup$
    – Andrea Pena
    Commented Jun 10, 2014 at 19:59
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I think that Waterhouse's "Introduction to Affine Group Schemes" (1979), section 3.2 "Comodules", might be what you're looking for.

I'm not sure why you have the constraint "non-zero characteristic" in your question; this seems irrelevant to me.

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  • $\begingroup$ I put in non-zero just to be safe . . . but it if works for all fields then great! $\endgroup$
    – Andrea Pena
    Commented Jun 10, 2014 at 15:19

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