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.)



Tensor and Hom objects for finite flat group schemes
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.


