Classification of plethories over $\mathbb{Q}$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T07:56:36Z http://mathoverflow.net/feeds/question/113989 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/113989/classification-of-plethories-over-mathbbq Classification of plethories over $\mathbb{Q}$ Martin Brandenburg 2012-11-20T20:50:42Z 2012-11-21T11:26:59Z <p>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 (<a href="http://arxiv.org/pdf/math/0407227.pdf" rel="nofollow">Borger, Wieland</a>, 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 <a href="http://books.google.de/books?id=s6NnkQs3JBMC&amp;lpg=PP1&amp;hl=de&amp;pg=PA336#v=onepage&amp;q&amp;f=false" rel="nofollow">book</a> (p. 336), after noticing that all the known interesting examples "split" for $\mathbb{Q}$-algebras. Perhaps meanwhile more is known?</p> <p>I think an equivalent statement would be: Does every continuous endofunctor of $\mathrm{CAlg}(\mathbb{Q})$ factor through the forgetful functor $\mathrm{CAlg}(\mathbb{Q}) \to \mathsf{Mod}(\mathbb{Q})$?</p> <p>References on plethories (in order of appearance):</p> <ul> <li>D. O. Tall, G. C. Wraith, <em>Representable Functors and Operations on Rings</em>, Proc. London Math. Soc. (1970) s3-20(4): 619-643 </li> <li>G. M. Bergman, A. O. Hausknecht, <em>Cogroups and co-rings in categories of associative rings</em>, American Mathematical Society, Mathematical Surveys and Monographs # 45, 1996.</li> <li>J. Borger, B. Wieland, <em>Plethystic algebra</em>, <a href="http://arxiv.org/abs/math/0407227v1" rel="nofollow">arXiv</a>, 2004</li> </ul> http://mathoverflow.net/questions/113989/classification-of-plethories-over-mathbbq/114048#114048 Answer by James Borger for Classification of plethories over $\mathbb{Q}$ James Borger 2012-11-21T11:26:59Z 2012-11-21T11:26:59Z <p>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. </p> <p>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.</p>