I'm trying to read Kisin's paper about the Integral model of Shimura varieties. In section five he discusses versal deformation ring of a p-divisible group. Assume that $K$ is a number field with residue field $k$ and consider p_divisible group $G_0$ defined on $k$ and consider the cocharacter $\mu$ of $GL(D(G_0)(\mathbb{W}(k)))$ defined by the filtration on the $D(G_0)(\mathbb{W}(k))$ and. $R$ versal deformation ring of $G_0$ is defined as the completion of the local ring of the identity section of the nilpotent group $U^0(\mu)$

consider a reductive group $H\subset GL(D(G_0)(\mathbb{W}(k)))$ defined by tensors $\{s_\alpha\}$ such that cocharacter $\mu$ of the natural filtration on $D$ factors through $H$. consider $U_H^0(\mu)$ the unipotent subgroup defined by $\mu$ and define the $R_H$ as the completion of the local ring of the natural section of $U_H^0(\mu)$ define $S=\widehat{W[u,\frac{E[u]^n}{n!}}]$. we want to know when $O_K$ points of $Spf(R)$ come from $Spf(R_H)$. theorem says that a point comes from $R_H$ if and only if for the associated p-divisible group $G$ you can lift $\{s_\alpha\}$ to $D(G)(S)$ satisfying certain condition.

I have some problems understanding this theorem. I want to know why we need the ring $S$ in this theorem instead of saying an $O_K$ point of $R$ is a point of $R_H$ if and only if you can lift $S_\alpha$ to an element of $D(G)(O_K)$ satisfying those condition.

I think the point is that because $O_K$ isn't in the crystal site so p_divisible group on it doesn't come from $R(O_K)$ and so if you have a p-divisible group on $O_K$ you only get a point in $R(S)$ is this true?

I'm trying to read Kisin's paper about the Integral model of Shimura varieties. In section five he discusses versal deformation ring of a p-divisible group. Assume that $K$ is a number field with residue field $k$ and consider p_divisible group $G_0$ defined on $k$ and consider the cocharacter $\mu$ of $GL(D(G_0)(\mathbb{W}(k)))$ defined by the filtration on the $D(G_0)(\mathbb{W}(k))$ and. $R$ versal deformation ring of $G_0$ is defined as the completion of the local ring of the identity section of the nilpotent group $U^0(\mu)$

consider a reductive group $H\subset GL(D(G_0)(\mathbb{W}(k)))$ defined by tensors $\{s_\alpha\}$ such that cocharacter $\mu$ of the natural filtration on $D$ factors through $H$. consider $U_H^0(\mu)$ the unipotent subgroup defined by $\mu$ and define the $R_H$ as the completion of the local ring of the natural section of $U_H^0(\mu)$ define $S=\widehat{W[u,\frac{E[u]^n}{n!}}]$. we want to know when $O_K$ points of $Spf(R)$ come from $Spf(R_H)$. theorem says that a point comes from $R_H$ if and only if for the associated p-divisible group $G$ you can lift $\{s_\alpha\}$ to $D(G)(S)$ satisfying certain condition.

I have some problems understanding this theorem. I want to know why we need the ring $S$ in this theorem instead of saying an $O_K$ point of $R$ is a point of $R_H$ if and only if you can lift $S_\alpha$ to an element of $D(G)(O_K)$ satisfying those condition.

I'm trying to read Kisin's paper about the Integral model of Shimura varieties. In section five he discusses versal deformation ring of a p-divisible group. Assume that $K$ is a number field with residue field $k$ and consider p_divisible group $G_0$ defined on $k$ and consider the cocharacter $\mu$ of $GL(D(G_0)(\mathbb{W}(k)))$ defined by the filtration on the $D(G_0)(\mathbb{W}(k))$ and. $R$ versal deformation ring of $G_0$ is defined as the completion of the local ring of the identity section of the nilpotent group $U^0(\mu)$

consider a reductive group $H\subset GL(D(G_0)(\mathbb{W}(k)))$ defined by tensors $\{s_\alpha\}$ such that cocharacter $\mu$ of the natural filtration on $D$ factors through $H$. consider $U_H^0(\mu)$ the unipotent subgroup defined by $\mu$ and define the $R_H$ as the completion of the local ring of the natural section of $U_H^0(\mu)$ define $S=\widehat{W[u,\frac{E[u]^n}{n!}}]$. we want to know when $O_K$ points of $Spf(R)$ come from $Spf(R_H)$. theorem says that a point comes from $R_H$ if and only if for the associated p-divisible group $G$ you can lift $\{s_\alpha\}$ to $D(G)(S)$ satisfying certain condition.

I have problem understanding proof of this theorem. in particular I want to see if the proof become simpler in the case the $K$ is unramified and so $S=W$. is there any good reference for this theorem?

I'm trying to read Kisin's paper about the Integral model of Shimura varieties. In section five he discusses versal deformation ring of a p-divisible group. Assume that $K$ is a number field with residue field $k$ and consider p_divisible group $G_0$ defined on $k$ and consider the cocharacter $\mu$ of $GL(D(G_0)(\mathbb{W}(k)))$ defined by the filtration on the $D(G_0)(\mathbb{W}(k))$ and. $R$ versal deformation ring of $G_0$ is defined as the completion of the local ring of the identity section of the nilpotent group $U^0(\mu)$

consider a reductive group $H\subset GL(D(G_0)(\mathbb{W}(k)))$ defined by tensors $\{s_\alpha\}$ such that cocharacter $\mu$ of the natural filtration on $D$ factors through $H$. consider $U_H^0(\mu)$ the unipotent subgroup defined by $\mu$ and define the $R_H$ as the completion of the local ring of the natural section of $U_H^0(\mu)$ define $S=\widehat{W[u,\frac{E[u]^n}{n!}}]$. we want to know when $O_K$ points of $Spf(R)$ come from $Spf(R_H)$. theorem says that a point comes from $R_H$ if and only if for the associated p-divisible group $G$ you can lift $\{s_\alpha\}$ to $D(G)(S)$ satisfying certain condition.

I have some problems understanding this theorem. I want to know why we need the ring $S$ in this theorem instead of saying an $O_K$ point of $R$ is a point of $R_H$ if and only if you can lift $S_\alpha$ to an element of $D(G)(O_K)$ satisfying those condition.

I think the point is that because $O_K$ isn't in the crystal site so p_divisible group on it doesn't come from $R(O_K)$ and so if you have a p-divisible group on $O_K$ you only get a point in $R(S)$ is this true?