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Is the category of finite flat group schemes equipped with "tensor products" and Hom-objects, encoding bilinear maps? I'm aware that the Cartier dual is $Hom(\mathbb{G}, \mathbb{G}_m)$, and want to know if this is part of a systematic collection of objects. For example, is there a "free ring scheme on $\mathbb{G}$"?

If so, given two affine group schemes whose underlying rings are free over the base, are there explicit descriptions of the tensor product and Hom objects in terms of the multiplication and comultiplication rules on the original rings?

Over a field, is there a description in terms of the Dieudonne correspondence?

(References, if they exist, would be very much appreciated. Thank you.)

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    $\begingroup$ Well, the category of group schemes sits inside the category of fppf sheaves of groups over your base. If you restrict to fppf sheaves of abelian groups, then the latter category has Homs and tensor products (just sheaf Hom and sheaf tensor product). In general, these operations will not preserve the category of group schemes. For instance, if you take G to be the group scheme of roots of unity, then the tensor product of G with itself will not be representable. On the Dieudonne side, this corresponds to the fact that slopes add up to more than 1 under tensor products. $\endgroup$
    – Keerthi Madapusi
    Commented Apr 13, 2011 at 20:08

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The answer to your question is 'no'. If there existed a tensor product for finite flat group scheme, there would exist one for $p$-divisible groups (over rings of integers of $p$-adic fields, for example), and for the corresponding (crystalline) Galois representations. Yet there are certain weights for these representations (that are rational numbers); the weight of an \'etale $p$-divisible group is $0$, the weight of multiplicative one is $1$, and a crystalline representation comes from a $p$-divisible group whenever its weights are between $0$ and $1$. Now, the internal Hom with the multiplicative group turns weight $x$ into $1-x$; this operation preserves the interval $[0,1]$. Yet in general tensor products (that correspond to addition of weights) and innner Homs (that corresoonds to subtraction) do not preserve this interval.

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    $\begingroup$ That's exactly correct. For example, in characteristic $p$, $\mu_p\otimes\mu_p$ is certainly not a finite flat group scheme. It is a flat sheaf, but is not of interest. $\endgroup$
    – mephisto
    Commented Apr 13, 2011 at 20:50
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Sampath and Bondarko have already answered the question. Very interesting in this regard is the important work of Larry Breen, who has computed the ring Ext$^*(G_a, G_a)$ in the category of flat sheaves over a perfect field of positive characteristic; see his paper in Inventiones Math and his paper in IHES.

Also a nice tensor category containing that of Dieudonne modules is the category of modules over the Cartier-Dieudonne-Raynaud ring; see Illusie-Raynaud's paper in IHES.

Hope this helps.

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