Φ-MODULES AND COEFFICIENT SPACES
Φ-MODULES AND COEFFICIENT SPACES
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Φ-模和系数空间
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发表时间：
2008
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通讯作者：
M. Rapoport
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作者：
G. Pappas;M. Rapoport
本文受Kisin的文章[Ki 1]的启发，研究了由有限平坦群格式定义的局部p-adic域的Galois表示的变形。结果Kisin最相关的我们的文件是他的建设一种决议的正式变形空间的给定伽罗瓦表示，通过构建一个计划，其中分类所有有限的平坦群计划产生的变形伽罗瓦表示。我们的目的是使Kisin的建筑全球化。设K是Qp的有限扩张，其剩余域为k.设K0是Qp包含在K中的极大非分歧扩张。则K0是Witt向量环W = W（k）的分数域。设π是K的一个单化子，E（u）∈ W [u]是π所满足的Eisenstein多项式.设GK = Gal（K/K）是K的绝对Galois群。设S为形式幂级数W [[u]]的环。设φ：S → S使得φ| W是Frobenius自同构，φ（u）= up。Kisin的构造是基于从一个合适的S-模范畴M到Spec（OK）上p-幂秩的有限平坦（交换）群概型范畴的完全忠实正合函子的存在性。这反过来又受到了工作的Breuil [Br]谁给了类似的，但更复杂的描述，这种集团计划。这个函子的一个变体也适用于系数：如果R是一个有n个元素的Z代数，则存在一个类似的函子从一个合适的具有φ-线性自同态Φ的S <$Zp R-模M的范畴到SpecOK上具有R-作用的p-幂秩的有限平坦群概型的范畴。设K ∞/K是通过邻接π的pn-幂根的相容系而获得的扩张，设GK ∞ = Gal（K/K ∞）是其绝对伽罗瓦群。设OE是完备离散赋值环S [1/u]的p-adic完备化，其一致化子为p，剩余域为k（（u））= k [[u]][1/u].然后，具有同构Φ：φ ∈（M）→ M的n-生成OE-模M的范畴与GK ∞在Zp-模中的连续表示范畴之间存在范畴等价性，并且通过从GK-表示到GK ∞表示的限制函子与前面的函子相容.同样，对于值在有限系数Zp-代数R中的表示也有一个变体。
This paper is inspired by Kisin’s article [Ki1], in which he studies deformations of Galois representations of a local p-adic field which are defined by finite flat group schemes. The result of Kisin most relevant to our paper is his construction of a kind of resolution of the formal deformation space of the given Galois representation, by constructing a scheme which classifies all finite flat group schemes giving rise to the deformed Galois representation. Our purpose here is to globalize Kisin’s construction. Let K be a finite extension of Qp, with residue field k. Let K0 be the maximal unramified extension of Qp contained in K. Then K0 is the fraction field of the ring of Witt vectors W = W (k). Let π be a uniformizer of K and E(u) ∈ W [u] the Eisenstein polynomial that π satisfies. Let GK = Gal(K/K) be the absolute Galois group of K. Set S for the ring of formal power series W [[u]]. Let φ : S → S be such that φ|W is the Frobenius automorphism and with φ(u) = up. Kisin’s construction is based on the existence of a fully faithful exact functor from a suitable category of S-modules M equipped with a φ-linear endomorphism Φ to the category of finite flat (commutative) group schemes of p-power rank over Spec (OK). This in turn was inspired by work of Breuil [Br] who gave a similar but more complicated description of such group schemes. A variant of this functor also works with coefficients: if R is a Zpalgebra with finitely many elements, then there is a similar functor from a suitable category of S ⊗Zp R-modules M with φ-linear endomorphism Φ to the category of finite flat group schemes of p-power rank with R-action over SpecOK . Let K∞/K be the extension obtained by adjoining a compatible system of pn-power roots of π, and let GK∞ = Gal(K/K∞) be its absolute Galois group. Let OE be the p-adic completion of S[1/u], a complete discrete valuation ring, with uniformizer p and residue field k((u)) = k[[u]][1/u]. Then there exists an equivalence of categories between the category of finitely generated OE -modules M equipped with an isomorphism Φ : φ∗(M) → M and the category of continuous representations of GK∞ in Zp-modules, and this is compatible with the previous functor via the restriction functor from GK-representations to GK∞representations. Again there is also a variant for representations with values in a finite coefficient Zp-algebra R.