Hello,

I would like to work out some examples of deformation of isogenies via Grothendieck-Messing theory. Let's take an easy example: Let A be an abelian variety over $k=\overline{\mathbb{F}_p}$ and $f:A\rightarrow A$ be the multiplication by $p$ map. Let $M=\mathbb{D}(A)$ be the Dieudonné module of $A$. It is a free $W(k)$-module of rank $2 dim(A)$. Finally let $R=\frac{k[t]}{(t^2)}$.

Then lifting $A$ to $R$ boils down to lifting the Hodge filtration $$0 \rightarrow \frac{FM}{pM} \rightarrow \frac{M}{pM} \rightarrow \frac{VM}{pM} \rightarrow 0 $$ to a filtration over $R$: $$0 \rightarrow P \rightarrow \frac{M}{pM}\otimes R \rightarrow P' \rightarrow 0 $$

Here I have used the fact that what is written sometimes $\mathbb{D}(A)_k$ is actually $\frac{M}{pM}$, and $\mathbb{D}(A)_R=\mathbb{D}(A)_k \otimes R$ (am I right ?). Now, let's move on to deforming the isogeny $f$. Lifting $f$ to $R$ is equivalent to lifting two such Hodge filtrations in a compatible way with $f$. Here is what I don't understand:

The map $f_k$ induced by $f$ on $\mathbb{D}(A)_k=\frac{M}{pM}$ is obviously the zero map. But how can I compute the map induced on $\mathbb{D}(A)_R$ ? Is it simply $f_k\otimes R=0$ ? I doubt it.

Thank you!

Jean-Stefan