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Let $k$ be a commutative ring. For every cocommutative bialgebra $A$ over $k$ the symmetric algebra of the underlying $k$-module $S(A)$ carries the structure of a $k$-plethory (Borger, Wieland, 2.5). The corresponding comonad on $\mathrm{CAlg}(k)$ is simply $\mathrm{Hom}_{\mathsf{Mod}(k)}(A,-)$. Are these all plethories in the case $k=\mathbb{Q}$? This was asked by Bergman and Hausknecht in their book (p. 336), after noticing that all the known interesting examples "split" for $\mathbb{Q}$-algebras. Perhaps meanwhile more is known?

I think an equivalent statement would be: Does every continuous endofunctor of $\mathrm{CAlg}(\mathbb{Q})$ factor through the forgetful functor $\mathrm{CAlg}(\mathbb{Q}) \to \mathsf{Mod}(\mathbb{Q})$?

References on plethories (in order of appearance):

  • D. O. Tall, G. C. Wraith, Representable Functors and Operations on Rings, Proc. London Math. Soc. (1970) s3-20(4): 619-643
  • G. M. Bergman, A. O. Hausknecht, Cogroups and co-rings in categories of associative rings, American Mathematical Society, Mathematical Surveys and Monographs # 45, 1996.
  • J. Borger, B. Wieland, Plethystic algebra, arXiv, 2004
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up vote 6 down vote accepted

As far as I know, this question is still open. The only result in this direction I'm aware of is the subject of a paper by Buium called "Arithmetic analogues of derivations" where he gives a complete classification of biring structures on the polynomial algebra in two variables. He doesn't even restrict to $\mathbf{Q}$ coefficients. The answer is that you only get the ones responsible for structures you knew about already: ring endomorphisms, derivations, Frobenius lifts/Witt vectors/$p$-derivations, and lifts of the mod $p$ identity map. (Over $\mathbf{Q}$-algebras, the last two agree with the first.) From what I remember, his argument only uses a little bit of the theory of commutative algebraic groups.

In my heart, I think plethories over $\mathbf{Q}$ and even $\mathbf{Z}$ should be classifiable. Basically you should be able to get anything by performing mod $p$ amplification constructions (as described in my paper with Wieland) to the ones coming from the linear ones. Someone should do this! It would be a classification theorem for the rest of time. I'd like to believe it'd be doable with some meditating on Buium's result and a bit of pushing.

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I should add that Buium did not use the language of birings etc. He had his own point of view and did not cite any of the references you list above (the last of which came out much later than his paper). So he wouldn't have seen himself as giving a partial result to the question you asked, although now he's aware of this. –  JBorger Nov 21 '12 at 11:35
Thank you! I wasn't aware of that work. And since I didn't ask for a classification, just for the current progress, and you are an expert on this, I will accept your answer. –  Martin Brandenburg Nov 21 '12 at 16:04
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