I am learning Cartier theory of commutative formal groups by the book of Zink. It is a powerful tool but I don't understand it's motivation. The Cartier module of a formal group $G$ over a $\mathbb{Z}_{(p)}$-algebra is $\mathcal{H}om(\widehat{W},G)$ which are $p$-type curves. I think it is a generalization of the tangent space and hence contains enough information of the group. But why the formal Witt vectors(or $\Lambda$ for global case) work? It seems we can compute formal groups through Witt vectors: assume $m_1,\cdots,m_n$ is a $V$-base of $M=M_G$, and $Fm_i=\sum_{i,j}V^n[c_{n,i,j}]m_j$, we have an exact sequence $$\mathbb{E}_p^n\rightarrow \mathbb{E}_p^n\rightarrow 0$$ then by the equivalence of category we get $\widehat{W}\otimes_{\mathbb{E}_p}\mathbb{E}_p^n\rightarrow\widehat{W}\otimes_{\mathbb{E}_p}\mathbb{E}_p^n\rightarrow M_G\rightarrow 0.$ If we choose the curvilinear coordinate system of $G$ by $m_1,\cdots ,m_n$, how can we compute the group laws from the Witt vectors?
Add a comment
|