I was not able to find much on representation theory (with a geometric perspective) of algebraic groups over local artinian rings. In particular, studying on the classic book of Jantzen, I was asking myself some naive question: Is there a way to define "Frobenius kernels" over some ring of characteristic $p^n$ such that they reduce modulo $p$ to the classical ones? Could it (if it exists) be used to describe representation theory of algebraic groups on local artinian rings? This let me to look at some important result of Greenberg, but I am not so confident: this is my reasoning. Let $R$ be an artinian local ring with algebraically closed residue field $k=\bar{k}$ of characteristic $p>0$ (e.g. $R=W_n(\overline{\mathbb{F}_p})$). In "Schemata over local rings I" (and Schemata over local rings II), the author introduces representable functors (realizations) \begin{equation} \mathcal{F}_R : Sch^{l.f.t.}/R \rightarrow Sch^{l.f.t.}/k \end{equation} (where l.f.t.= "locally of finite type") preserving affine schemes, such that for each $X$ l.f.t over $R$ there is a group isomorphism $\mathcal{F}_RX(k)\simeq X(R)$ (see definitions and theorem in section 4 of "Schemata over local rings I"). Denoting by $\mathcal{G}_R$ its left adjoint, there is a natural map $\lambda_RX:\mathcal{G}_R\mathcal{F}_RX \rightarrow X$ (that as far as I understand by universal property of realizations) corresponding by adjunction to the identity of $\mathcal{F}_RX$.
If $R'$ is another artinian local ring as above, and a ring map $\phi : R \rightarrow R'$ is given, Greenberg defines a connecting morphism $F_{\phi}X: \mathcal{F}_RX \rightarrow \mathcal{F}_{R'}X^{\phi}$, where $X^{\phi}:=X \otimes_{\phi,R} R'$, by mean of a certain commutative diagram (see section 5 for more details).
- When $R=R'$, does the connecting map come from functoriality by a map $X \rightarrow X^{\phi}?$ If not in general, what about some not trivial $\phi$?
Now let us just consider to have $R=R'=W_n(\overline{\mathbb{F}_p})$ and $\phi$ to be the lift of Frobenius over $\overline{\mathbb{F}_p}$ (i.e. the Frobenius on the Witt vectors). Let $X$ be a smooth linear algebraic group over $R$. In this case we have a relative Frobenius $\phi_{X/R}: X \rightarrow X^{\phi}=X^{(1)}$ (and iterated ones $\phi^r_{X/R}: X \rightarrow X^{\phi^r}=X^{(r)}$). Applying the realization functor and the connecting morphism we get $k$-morphisms $f^{(r)}: \mathcal{F}X \rightarrow \mathcal{F}X^{(r)}$. Consider $Z_r:=\ker(f^{(r)})$.
- Does $Z_r$ coincide with the Frobenius kernel defined for example in Jantzen ("Representation of algebraic groups")? If not, is there some relation?
From what I understand it seems to be true once one proves that $F_\phi=\mathcal{F}_R(\phi^r_{X/R})$ and it seems to be true using definition of connecting morphism ad adjunction. Is something missing?
By the way, do you know if there is some nice reference on (more geometric oriented) representation theory over artinian rings where similar questions are investigated? Thanks.
