The Image of the Coefficient Space In the Universal Deformation Space of a Flat Galois Representation of a P-Adic Field

Determining the image of the coefficient space in the universal deformation space

by Anuj Kumar Singh*, Dr. K. K. Jain , Dr. Yogesh Sharma,

- Published in Journal of Advances in Science and Technology, E-ISSN: 2230-9659

Volume 1, Issue No. 1, Feb 2011, Pages 0 - 0 (0)

Published by: Ignited Minds Journals

ABSTRACT

The coefficient space is a kind of resolution of singularities of theuniversal flat deformation space for a given Galois representation of some local field. It parametrizes(in some sense) the finite flat models for theGalois representation. The aim of this note is to determine the image of the coefficient space in the universal deformation space.

KEYWORD

coefficient space, universal deformation space, flat Galois representation, P-adic field, resolution of singularities

. Introduction

n the theory of deformations of Galois representations one is often interested in a subfunctor of he universal deformation functor consisting of those deformations that satisfy certain extra onditions, so called deformation conditions . If we deal with a representation of the bsolute Galois group of a finite extension K of in a finite dimensional vector space in haracteristic p, there is the deformation condition of being flat, which means that there is a finite lat group scheme over the ring of integers of K such that the given Galois representation is somorphic to the action of the Galois group on the generic fiber. The structure of the ring pro- epresenting this deformation functor is of interest for modularity lifting theorems (see [Ki1] for xample). To get more information about this structure, Kisin constructs some kind of "resolution f singularities" of the spectrum of the flat deformation ring. This resolution is a scheme arametrizing modules with additional structure that define possible extensions of the epresentation to a finite flat group scheme over the ring of integers. In [PR2] Pappas and apoport globalize Kisin’s construction and define a so called coefficient space parametrizing all isin modules that give rise to the given representation. ollowing the presentation in [PR2] we want to determine here the image of the coefficient space n the universal deformation space. This question was raised by Pappas and Rapoport in [PR2, .c]. Further we show how to recover Kisin’s results from the more abstract setting in [PR2]. The ain result of this note is as follows. Let K be a finite extension of where p is an odd prime, nd be a continuous flat representation of the absolute Galois group on some d-dimensional vector space over a finite field of characteristic p. If is a deformation of we write for the coefficient space of (locally free) isin modules over Spec A that are related to the flat models for the deformation (see also the efinition below). heorem 1.1. Assume that the flat deformation functor of is pro-representable by a complete ocal noetherian ring We write for the universal flat deformation. Then the orphism Spec R is topologically surjective. orollary 1.2. If it exists, the flat deformation ring R is topologically flat, i.e. the generic fiber pecis dense in Spec If the ramification index of the local field K over is smaller han p-1, then this implies the following result, already contained in [PR2] orollary 1.3. Denote by e the ramification index of K over . Assume that the flat deformation unctor of is pro-representable and that Then is the scheme theoretic image of the oefficient space.

. Notations

et p be an odd prime and K be a finite extension of with ring of integers , uniformizer and residue field Denote by K0 the maximal unramified extension of in and by W = W(k) its ring of integers, the ring of Witt vectors with coefficients in k. Fix an lgebraic closure of K and denote by the absolute Galois group of K. Further e choose a compatible system roots of the uniformizer in and denote by the ubfield of We write for its absolute Galois group. et d > 0 be an integer and a finite field of characteristic p. Let be a continuous epresentation of GK and denote by the restriction of e consider the deformation functors on local Artinian - algebras with esidue field . For a local Artinian ring (A,m) we have here two lifts are said to be equivalent if they are conjugate under some The functor is the flat deformation functor , i.e. the ubfunctor of consisting of all deformations that are (isomorphic to) the generic fiber of some inite flat group scheme over Spec . Here "isomorphic to" means isomorphic as as the action of the coefficients in the generic fiber does not need to extend to the roup scheme. f are pro-representable, the pro-representing ring will be denoted by ecall that d > 0 denotes an integer and consider the following stacks on -algebras, defined in PR2]. For a -algebra R, write RW for W and where the competed tensor products are the ompletions for the u-adic topology. Further we denote by the endomorphism of that is he identity on R, the Frobenius on W and that maps . e define an fpqc-stack R on the category of -schemes such that for a -algebra R the roupoid R(R) is the groupoid of pairs , where M is an RW((u))- module that is fpqc-locally n Spec R free of rank d as an RW((u))-module, and is an isomorphism . urther we define a stack C as follows. The R-valued points are pairs , where M is a locally ree of rank d and . For consider the substacks given y pairs satisfying

(2.1)

or we write for the substack of C consisting of all satisfying ere is the minimal polynomial of the uniformizer over K0. In the following we ill only consider the case h = 1 and just write CK for C1;K. We will write resp. for the estrictions of the stacks CK (resp. R) to the category Nilp of -schemes on which p is locally ilpotent. See also [PR2, §2] for the definitions. he motivations for these definitions are the following equivalences of categories . roposition 2.1. Let A be a local Artin ring with residue field a finite field of characteristic p. hen the category of -representations on free A-modules of rank d is equivalent to the ategory of that are free of rank d. heorem 2.2. Let p > 2. Then there is an equivalence between the groupoid of finite flat group chemes over Spec and the groupoid of pairs , where of rojective dimension 1 and is a map such that the cokernel of the linearisation f is killed by E(u). Under this equivalence the restriction of the Tate twist of the GK- epresentation on to corresponds to the . urther we will use the following notations: Let (A,m) be a complete noetherian W(F)-algebra and be an A-valued point of Write for the reduction of By [PR2, orollary 2.6; 3.b] the fiber product s representable by a projective that is a closed subscheme of some affine rassmannian over These schemes give rise to a formal scheme ver Spf A. Using the very ample line bundle on the affine Grassmannian this formal scheme is lgebraizable. The resulting projective scheme over Spec A will be denoted by emark 2.3. Note that this does not give an arrow For example the module together with the does not define a -valued point f CK but rather a "formal" point owever if B is some -algebra killed by some power of p, then nd hence any locally free with semi-linear map satisfying efines a B-valued point of CK.

. The image of the coefficient space

n the following we will assume that the representation is flat (i.e. is the generic fiber of some inite flat group scheme over Spec ) and that is representable. This is the case if, for xample, . We write for the universal flat deformation. By roposition 2:1 we have a map

(3.1)

ee also [PR2, 4.a] and [Ki1, 1.2.6, 1.2.7]. For some local Artinian ring A and some for the corresponding -module. More precisely, this map dentifies . The latter functor is given by all deformations in of the . Especially we find that the map in (3:1) is formally smooth. emma 3.1. The restriction of . Composing the canonical rojection with this morphism we obtain a 2-cartesian diagram of stacks on local rtinian W(F)-algebras: roof. Let A be a local Artinian W(F)-algebra such that pnA = 0 for some n > 0. We have to show hat there is a natural equivalence of categories irst it is clear that induces a natural map from the left to the right which is fully aithful. We have to show that it is essentially surjective. et be an A-valued point of the right hand side. Then and by PR2, Proposition 4.3] there is an associated flat representation of GK such that his shows that defines a unique point in It follows from the onstruction that this point maps to x. emark 3.2. Note that it is not clear whether is representable, even if is absolutely rreducible, since does not satisfy Mazur’s p-finiteness condition . As here is an isomorphism ach open subgroup of finite index is isomorphic to the absolute Galois group of some ocal field in characteristic p, here l is a finite extension of k and t is an indeterminate. Hence by Artin-Schreier theory for example), there is an isomorphism nd the latter group is infinite. f one restricts the attention to , then the eformation functor is representable if (see [Kim, Theorem 11.1.2]). The E- eight of a p-torsion -representation is defined as the minimal h such that the associated to the representation admits an W[[u]]- lattice with cokernel of the inearisation of (see [Kim, Definition 5.2.8] for the precise definition). roposition 3.3. Let denote the projective obtained from by algebraization. hen is reduced, normal and Cohen- Macaulay. The reduced subscheme nderlying the special fiber is normal and with at most rational singularities. Further he scheme is topologically flat, i.e. its generic fiber is dense. roof. This is similar to [Ki1, Proposition 2.4.6]. Denote by y : Spec the F-valued point efined by . Let x be a closed point of . Extending scalars if necessary, we may assume hat x is defined over F. Denote by the defined by the defined by x. We want to compare the structure of the local ring (resp. its ompletion) to the structure of a local model MK defined in [PR2, 3.a]. By loc. cit. Theorem 0.1. here is a "local model"-diagram

(3.2)

ith and formally smooth. Here the B-valued points of the stack are the together with an isomorphism for a B. e consider the following groupoids on local Artinian W(F)-algebras: Denote by Dx and Dy the roupoids ixing a basis of we may view x as an F-valued point of . Denote by the groupoid of eformations of x in . nder the morphism in (3:2), the point x maps to a point of MK. This point defines an be the groupoid of deformations of i.e. This groupoid is pro- epresented by the completion of the local ring . Now we have the following commutative iagram. here the lower left square is cartesian by Lemma 3:1. As remarked above is formally smooth nd hence so is . As is pro-represented by the complete local ring at some closed point of the ocal model MK and as are formally smooth, the assertion of the Proposition is true if it s true for MK. But if follows from the definitions (using the notation of [PR2, 3.c]) that or some cocharacters uch that here V is a local model in he sense of [PR1] (compare [PR2, Remark 3.3]). Hence, by [PR1, Theorem 5.4], the generic fiber f the local model MK is normal, reduced and Cohen-Macaulay. The special fiber ecomposes as follows: here _ runs over all cocharacters nd where is the maximal dominant cocharacter (for the dominance rder) such that the composition quals _. Now the claim again follows from [PR1, Theorem 5.4]. _ emark 3.4. We need to formulate the result on the local structure of the special fiber as a result bout the underlying reduced scheme as the local models are in general not defined over ut over a ramified extension and hence there are nilpotent elements in the special fiber roposition 3.5. The map becomes an isomorphism in the generic fiber over (F), i.e. roof. Using the result on the local structure of the proof is the same as in [Ki1, roposition 2.4.8]. The main point is to check that the map is a bijection on points. _ heorem 3.6. Suppose that the universal flat deformation ring exists and denote by the niversal flat deformation. Then the morphism is topologically surjective. e will prove this theorem in section 4 below. We will conclude this section with some onsequences of Theorem 3:6. orollary 3.7. Assume that exists, then Spec is topologically flat. Proof. This follows from heorem 3.6 and the corresponding result on roposition 3.8. Assume that the universal deformation ring R of __ exists with universal eformation factors over SpecR and is (canonically) isomorphic to roof. the reduction of modulo We consider the following diagram with all rectangles cartesian. roof. It is enough to show that is an isomorphism. We show that both objects ro-represent the same functor, i.e. pro-represents the deformation functor . Let A be a ocal Artinian ring and a flat deformation of By a result of Raynaud (cf. [Ra, roposition 3.3.2]) there is a unique flat model for this deformation. Denote by associated with this group scheme by Kisin’s classification. This is a [[u]]-submodule of the corresponding to the (twist of) the estriction of Replacing M by its AW[[u]]-span inside M, we may assume that it is an W[[u]]-submodule of M. Applying the argument of [PR2, Remark 4.4] we find that is free ver AW[[u]] This defines the unique point in CK(A) above . We have shown that the functor orphism is bijective on A-valued points. The claim follows. emark 3.10. All the above results also apply to framed deformation rings. We need to replace the eformation functors by deformation groupoids and the fiber products by 2-fiber products. For the orresponding result on the local structure one only needs that the morphism is smooth, here denotes the groupoid of framed deformations of The result for framed eformations (or for deformation stacks) can be stated as follows: Given a field F of characteristic and a morphism Spec There exists an fpqc-cover and a integral complete ocal ring (A;m) with char(FracA) = 0 and such that the composition ifts to a morphism SpecA ! SpecR. emark 3.11. If the prime p equals 2, then there is a similar classification of finite flat group chemes as in Theorem 2:2, but it only applies to connected group schemes (see [Ki2]). Hence the ame results hold in the case p = 2, if one considers deformations that are the generic fiber of a onnected finite flat group scheme.

. Proof of Theorem 3.6

n this direct sum the summand for i = 0 is the free part in the quotient and the i-th summand is the ontribution of the elements in to the he Lemma now follows from the following claim: e denote by j the minimal integer such that Then pj is the minimal integer such that But we have ence and the claim follows. emma 4.2. Let Then there are at most finitely many finitely generated n[[u]]-submodules and roof. The module M is vector space. Every finitely generated An[[u]] ubmodule Hence the argument of [Ki1, roposition 2.1.7] shows that there exists a lattice satisfying the properties of the Lemma satisfy hese are only finitely many lattices. _ roposition 4.3. Let the reduction modulo Assume that there exist finitely generated n[[u]] submodules such that and hen there exists such that roof. We denote by the set of all finitely generated An[[u]]-submodules such that and y assumption these sets are non empty and by Lemma 4:2 they are finite. Further if then the image of under the map efines an element of denoted by As the sets are non empty and finite we can nductively construct a sequencefor We denote this equence again by By Lemma 4:1 there are only finitely many possibilities for he isomorphism class of the u-torsion in Hence there exists a strictly increasing equence ni 2 N such that

(4.1)

or Now there is an isomorphism here the last summand is the u-torsion part of the left hand side. We find that sing (4:1) and Nakayama’s Lemma it follows that here all arrows are surjective. In the limit we get morphisms he modules in question have finite length. Note that we do not claim that the linearisation of e_ is n isomorphism after inverting u. ow the image of the vertical arrow defines (after inverting u) a free - submodule such that he image of defines a finitely generated submodule N such that urther N is by construction. We claim that N is free. As we have isomorphisms hen there exists the diagonal arrow in the diagram .e. is an F-valued point of the stack Cr defined in (2:1). By [PR2, Corollary 2.6] and the aluative criterion of properness, the diagonal arrow in the diagram below exists, his means that eN extends to an such that e denote by the reduction of By assumption there are finitely generated such that and y the same argument as in the proof of Proposition 4:3 we can assume that Mn maps onto nder the projection Now the argument of [Ki1, Proposition 2.1.7] shows hat there is an integer s only depending on r and e such that f we write then this shows ence is finitely generated over and contains an of M. Further it till satisfies is free over Hence we obtain he following commutative diagram here is the reduction of CK modulo p (compare [PR2, .b.]). By loc. cit. the stack C1 K is a closed substack of sing the valuative criterion of properness again we obtain the desired arrow. roof of Theorem 3.6. By Proposition 3:5 the morphism is an isomorphism in the eneric fiber over W(F). Especially it is surjective. We write for the special iber of . et be a point of that is not the unique closed point x0. We mark the specialization by a morphism here Spec is a complete discrete valuation ring and the morphism maps the generic point of pec and the special point to x0. By a Zariski density argument is suffices to assume that he residue field of contains k (recall that is a quotient of a power series ring in finitely any variables over W(F)). The morphism (4.2) nduces modules By Kisin’s classification of finite flat group schemes Theorem 2:2) there exist finitely generated k[[u]]-submodules such that and eplacing that it generates, we may assume that is table under the action of By Proposition 4:3 the arrow Spf is algebraizable to a orphism Spec and by Proposition 4:4 we obtain a commutative diagram his yields a commutative diagram e have to show that the arrow Spf factors over and that this morphism coincides ith the arrow in (4:2). y [PR2, Proposition 4.3] we obtain from the morphisms Spec flat GK- epresentations such that the restriction to induces the objects under the morphism 3:1). By [Br, Theorem 3.4.3] the restriction to is fully faithful on the category of flat p- orsion GK-representations and hence the two morphisms Spf coincide. This yields he claim.

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