A short and elementary proof of the structure theorem for finitely generated modules over PIDs (Q670729)

Let \(F\) be a finite extension of \({\mathbb Q}_p\), \(k\) a finite field of characteristic \(p\) and \(V_0\) a finite dimensional \(k\)-vector space with a continuous representation of the absolute Galois group. Let \(\mathcal{O}\) be a finite totally ramified extension of \(W(k).\) In this situation there exists a universal ring \(R^{\mathrm{univ}}\) that parametizes framed deformations of \(V_0\) to Artinian \(\mathcal{O}\)-algebras. Let \(X\subset {\mathrm{MaxSpec}}(R^{\mathrm{univ}}[1/p])\) be the locus corresponding to ordinary representations, i.e., those that fulfill the following condition: \[ V|_{I_F}=\begin{pmatrix} \chi *\\ 0 1 \end{pmatrix}, \] where \(\chi\) is the cyclotomic character. Then it is known that \(X\) is Zariski closed and therefore equal to \( {\mathrm{MaxSpec}}(R[1/p])\) for a unique \(\mathcal O\)-flat reduced quotient of \(R^{\mathrm{univ}}.\) Some work devoted to understanding \(R\) was done by \textit{M. Kisin} [Invent. Math. 178, No. 3, 587--634 (2009; Zbl 1304.11043)]. In the paper under review, the author presents an original method which allows him to describe the functor of points of \(R\) as well as obtain some new results about \(R.\) For example, he shows that a minor modification of \(R\) is normal and Cohen-Macaulay but usually not Gorenstein.