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变形环的反问题

Inverse Problems for deformation rings

Frauke M. Bleher, Ted Chinburg, Bart de Smit

备注：
17 pages; this paper is closely related to arXiv:1003.3143. In the third version, we added a new Section 5 with further examples
发表信息：
Trans. Amer. Math. Soc. 365 (2013), 6149-6165
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摘要：

设$\mathcal{W}$是一个完全局部交换Noether环，其剩余域$k$具有正特征$p$ .本文研究了$k$上的有限群$Γ$的有限维表示$V$相对于$\mathcal{W}$的双变形环$R_{\mathcal{W}}(Γ,V)$的逆问题.我们证明了对于所有的$p$和$n \ge 1$，环$\mathcal{W}[[t]]/(p^n t,t^2)$是一个泛变形环.如果$p^n\mathcal{W}\neq\{0\}$是环，则该环不是完全交，从而得到了M的一个问题的答案. Flach在所有特征中。我们还研究了环$\mathcal{W}[[t]]/(p^n t,t^2)$的“逆逆问题”，这是为了确定所有对$(Γ, V)$，使$R_{\mathcal{W}}(Γ,V)$是同构的这个环。

英文摘要：

Let $\mathcal{W}$ be a complete local commutative Noetherian ring with residue field $k$ of positive characteristic $p$. We study the inverse problem for the versal deformation rings $R_{\mathcal{W}}(Γ,V)$ relative to $\mathcal{W}$ of finite dimensional representations $V$ of a profinite group $Γ$ over $k$. We show that for all $p$ and $n \ge 1$, the ring $\mathcal{W}[[t]]/(p^n t,t^2)$ arises as a universal deformation ring. This ring is not a complete intersection if $p^n\mathcal{W}\neq\{0\}$, so we obtain an answer to a question of M. Flach in all characteristics. We also study the `inverse inverse problem' for the ring $\mathcal{W}[[t]]/(p^n t,t^2)$; this is to determine all pairs $(Γ, V)$ such that $R_{\mathcal{W}}(Γ,V)$ is isomorphic to this ring.

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