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Let $G$ be an affine groups scheme over $\mathbb Z$. As such it has an associated Hopf algebra, $A=\mathbb Z[G]$ such that $G(R)$ is naturally identified with the set $\hom_{Rng}(A,R)$ of ring homomorphisms, where the group operations (multiplication, inverse, unit) are given on this set from the co-operations of the algebra $A$.

Fix $m\in\mathbb{N}$ and $p$ a prime number, and let $W_m$ be the functor of $p$-typical Witt vectors of length $m+1$. The functor $W_m$ is represenatble as well, with representing algebra $\mathbb{Z}[x_0,\ldots,x_m]$.

I am interested in the structure of the group scheme $R\mapsto G(W_m(R))$. Greenberg's results imply that this functor is a an affine group scheme as well, and hence representable. Is there any known construction for the associated Hopf-algebra of $G\circ W_m$?


Some obvious facts-

In the case where the prime $p$ is invertible in $R$, the Witt ring $W_m(R)$ is isomorphic to the product ring $\prod_{i=0}^m R$. Since the group scheme $R\mapsto G(\prod_{i=0}^{m} R)=\prod_{i=0}^{m}G( R)$ is represented by the Hopf algebra $A^{\otimes m+1}=\underbrace{A\otimes\cdots\otimes A}_{m+1\text{-fold}}$, I somehow expect there to exist a map between the representing algebra of $G\circ W_m$ and $A^{\otimes m+1}$, which becomes an isomorphism under localization by $p$.

On the other hand, in the complementary case, the group $G(W_m(R))$ is usually nothing like $G(R)\times G(R)$. For example, in the case $G= GL_n$, $R=\mathbb{F}_p$ and $m=2$, we have an exact sequence $$1\to M_n(R)\to GL_n(W_2(R))\to GL_n(R)\to 1.$$ In particular, $|GL_n(W_2(R))|\ne |GL_n(R)\times GL_n(R)|$, and one cannot expect any sort of bijection to exist between the two.


I'm new to the subject, and my foundations on algebraic groups or algebraic geometry are not incredibly solid, so I apologize if anything I wrote above does not make complete sense, or is obviously false.

I would very much appreciate any clue or reference to the construction of the Hopf algebra of $G\circ W_m$, or any other interesting facts regarding the structure of $G\circ W_m$.

Thank you very much! Shai

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    $\begingroup$ In the notations of James Borger, Ben Wieland, Plethystic algebra: The coordinate Hopf algebra of $G \circ W_m$ will be the biring $S_m \odot A$, where $S_m$ is the coordinate Hopf algebra of $W_m$ (that is, $\mathbb{Z}\left[x_0,\ldots,x_m\right]$ with an appropriate coalgebra structure). This is just abstract nonsense. The question is, of course, how much we can say about this biring. $\endgroup$ – darij grinberg Dec 31 '17 at 12:24
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    $\begingroup$ For a perfect field $k$ of characteristic $p>0$ (so $W_2(k)/(p)=k$) and smooth affine $W_2(k)$-group $G$, $G\circ W_2$ is a smooth affine $k$-group and $\ker(G\circ W_2\to G_0:= G\bmod p)=({\rm{Lie}}(G_0))^{(p)}$ respecting $G_0$-actions (so your exact sequence for $G={\rm{GL}}_n$ has a Frobenius twist on the left, invisible for $k=\mathbf{F}_p$). This is discussed in A.6 of the book Pseudo-reductive Groups (the assumption there that $k=\overline{k}$ is not needed for this), where a harder fact is also shown: for non-commutative connected reductive $G$, $G\circ W_2$ has no Levi $k$-subgroup! $\endgroup$ – nfdc23 Dec 31 '17 at 13:04
  • $\begingroup$ Can I convince you that $\mathbb{Z}[G]$ is really not good notation for the coordinate algebra of $G$? It clashes badly with the notation for group algebras (consider the case where $G$ itself is a finite constant group scheme). I would suggest $\mathcal{O}_G$ instead, or $\Gamma(G, \mathcal{O}_G)$ to be fully pedantic. $\endgroup$ – Qiaochu Yuan Dec 31 '17 at 20:11
  • $\begingroup$ @QiaochuYuan: The notation $A[G]$ for the coordinate ring of a group scheme $G$ over a ring $A$ is very standard (especially widespread for work with linear algebraic groups over fields), so one relies on context to avoid any confusion with group algebras in the case of finite constant groups (which is rarely ever a possible source of confusion for work with group schemes in practice anyway). $\endgroup$ – nfdc23 Dec 31 '17 at 21:57
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    $\begingroup$ I agree with @QiaochuYuan, for what it's worth. I have used the square-bracket notation for coordinate rings myself and got disappointed by it; it clashes with the notation for a subalgebra generated by a given subset (not just with the notation for a group algebra). And it looks like it should be covariant in $G $. $\endgroup$ – darij grinberg Jan 2 '18 at 12:59
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The Hopf algebra of $G\circ W_m$ does have an explicit construction. It's appeared in a few of my papers and zillions of Buium's papers, and no doubt many others which aren't coming to mind right now. Some people call $G\circ W_m$ the `order $m$ arithmetic jet space of $G$' and denote it $J^mG$. The reason is that the usual jet space is defined in the same way but where you replace $W_m$ by the functor $R\mapsto R[t]/(t^{m+1})$. But one key difference is that any explicit description of the function algebra of the arithmetic jet space has to be fundamentally nonlinear because the addition law on Witt vectors is nonlinear. So you won't be able to write it down by combining multilinear constructions in a nice way.

So let $A_m$ denote the function algebra $\mathscr{O}(J^mG)$. You want to describe $A_m$ explicitly. It can be done in a few different ways, which roughly correspond to how you think about the Witt vectors. This is similar to how you can construct tensor products in different ways. In fact this is not an accident, as both are left adoints---in other words, constructions of objects which are free in a certain sense.

Let me start with the most concrete first. So fix a presentation $\mathbb{Z}[x_i]/(f_j)$ of the function algebra $A=\mathscr{O}(G)$.

  1. There's the way with the usual Witt components/coordinates. $A_m$ is (canonically) $\mathbb{Z}[x_i^{[k]}]/I^{[m]}$, where the notation is the following: $x_i^{[k]}$ is an indeterminate, $k$ runs from $0$ to $m$, and $i$ runs over the indexing set of the original generators; $I^{[m]}$ is the ideal generated by all elements $f_j^{[k]}$, as $j$ runs over the indexing set for the relations (and $k$ from $0$ to $m$) and where $f_j^{[k]}$ comes in a certain purely syntactic way from the $k$-th Witt component of the expansion of $f_j$. It's easier (for me) to explain it with examples: if $f=x_1+x_2$, then $f^{[0]}$ will be $x_1^{[0]}+x_2^{[0]}$, and $f^{[1]}$ will be $x_1^{[1]}+x_2^{[1]}+\sum_{0<a<p}\frac{1}{p}\binom{p}{a}(x_1^{[0]})^a(x_2^{[0]})^{p-a}$. If $f=x_1 x_2$, then you get the Witt polynomials for multiplication. If $f=-x_1$, you get the Witt polynomials for negation. In fact this determines everything for general $f$ because any polynomial is built out of iterated addition, multiplication, and negation.

  2. There's the way with the Buium-Joyal $\delta$-components. Then similarly $A_m$ is $\mathbb{Z}[x_i^{(k)}]/I^{(m)}$, but now $I^{(m)}$ is generated by the `arithmetic derivatives' $f_j^{(k)}$, where $f_j^{(k)}$ is what you get by applying the procedure above for $k=1$ but done $k$ times. So instead of looking at the higher Witt components, you iterate what you do for the first component.

  3. Then there's the purely formal, hands-off way, mentioned by Darij Grinberg in the comments. Then $A_m$ is $\mathbb{Z}[f\circ g]/I_m$. Here $f$ runs over all elements of the function algebra $\mathscr{O}(W_m)\approx \mathbb{Z}[\theta_0,\dots,\theta_m]$, and $f\circ g$ is a formal symbol, and $I_m$ is the ideal generated by all relations of the form $(f_1+f_2)\circ g = f_1\circ g + f_2\circ g$, similarly for multiplication, $f\circ (g_1+g_2)=P^+_f(g_1,g_2)$, where $P^+_f$ the what you get by applying `$f$'s Leibniz rule for addition' to the sum $g_1+g_2$. What does this mean? You see how $f$ pulls back under the addition map $W_m\times W_m \to W_m$. This will give you a tensor $\sum_\alpha f^1_\alpha \otimes f^2_\alpha$, which can then be coupled with the pair $(g_1,g_2)$ to give you $\sum_\alpha (f^1_\alpha \circ g_1)\cdot (f^2_\alpha\circ g_2)$. This is $P^+_f(g_1,g_2)$. You do the same thing for multiplication and scalar multiplication.

Approach 2 is given in pretty much every one of Buium's papers in the past twenty years, although he usually $p$-adically completes everything. See p 315 of his Duke paper "Differential characters of abelian varieties over p-adic fields". Approach 3, as Darij Grinberg pointed out, is given in my paper with Wieland. It's also essentially in the papers of Tall and Wraith from the 70s, Joyal's papers in the Canadian Comptes Rendus in the 80s, and the book by Bergman and Hausknecht. Approaches 1 and 2 are discussed together in my paper "Basic Geometry of Witt Vectors, I: the affine case", section 3.4.

Not sure how clear that is, and hopefully there are no typos. I'm happy to clarify it if needed.

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  • $\begingroup$ Thank you very much for you detailed response! I will read thoroughly into the references you suggested and see which one is best suited for what I'm trying to do. Just one small clarification: I might be missing out on something in Approach 3, but I can't find the point where the group structure enters into the picture here.. (the ideal $I_m$ is different here, and seems similar to that occurring in one of the constructions for the tensor product; could it be that it should have somehow included the defining relations of the group's function-algebra as well?) $\endgroup$ – kneidell Jan 2 '18 at 14:48
  • $\begingroup$ Good point! What I wrote applies if $G$ is any affine scheme. It doesn't have to be a group scheme. When it is, the Hopf algebra structure on $A_m$ is also pretty formal. In the notation of Darij Grinberg's comment, there is a canonical isomorphism $S_m\odot(A\otimes A)=(S_m\odot A)\otimes (S_m\odot A)$. You can combine this with the map $S_m\odot A \to S_m\odot(A\otimes A)$ coming from functoriality to get the coproduct on $A_m$. $\endgroup$ – JBorger Jan 5 '18 at 18:46
  • $\begingroup$ But to reiterate what Grinberg said, this is all very formal and applies to any functor like $W_m$. If you actually want to prove something non-formal about $G\circ W_m$, then you have to really use the definition of the Witt vectors somewhere. Since their defining universal property is about lifting the Frobenius map, you kind of have to use that. In other words, you kind of have to use the map $G\circ W_m \to G^{m+1}$, as you suspected. $\endgroup$ – JBorger Jan 5 '18 at 18:58
  • $\begingroup$ Also, it is true that the canonical map $A^{\otimes(m+1)}\to A_m$ becomes an isomorphism after $p$ is inverted, as you guessed. $\endgroup$ – JBorger Jan 5 '18 at 19:02

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