普通变形环的奇异性
Singularities of ordinary deformation rings
摘要：
设R^univ是局部伽罗瓦群的剩余表示的泛变形环. Kisin证明了MaxSpec（R^univ[1/p]）中的许多位点都是Zebrski闭的，并给出了研究R^univ中相应商的通有纤维的方法.然而，他的方法给出的关于p反演前的商环的信息很少.精确地说，假设V_0是平凡的二维表示，设R是R^univ的唯一Z_p-平坦和约化商，使得MaxSpec（R[1/p]）由Hodge-Tate权为0和1的常表示组成。我们描述的函子的点（略有修改的版本）R，并表明，不可约组件的规范（R）是正常的，科恩-麦考利，而不是Gorenstein。这在整体变形环上有两个众所周知的应用：第一，使用这些局部条件的整体变形环是无挠的;第二，基辛的R[1/p]=T[1/p]定理可以在这种情况下升级为R=T定理。
英文摘要：
Let R^univ be the universal deformation ring of a residual representation of a local Galois group. Kisin showed that many loci in MaxSpec(R^univ[1/p]) of interest are Zariski closed, and gave a way to study the generic fiber of the corresponding quotient of R^univ. However, his method gives little information about the quotient ring before inverting p. We give a method for studying this quotient in certain cases, and carry it out in the simplest non-trivial case. Precisely, suppose that V_0 is the trivial two dimensional representation and let R be the unique Z_p-flat and reduced quotient of R^univ such that MaxSpec(R[1/p]) consists of ordinary representations with Hodge--Tate weights 0 and 1. We describe the functor of points of (a slightly modified version of) R and show that the irreducile components of Spec(R) are normal, Cohen--Macaulay and not Gorenstein. This has two well-known applications to global deformation rings: first, a global deformation ring employing these local conditions is torsion-free; and second, Kisin's R[1/p]=T[1/p] theorem can be upgraded in this setting to an R=T theorem.
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