Computation of framed deformation functors Pietro Ploner Abstract 3 1 In this work we compute the framed deformation functor associated 0 to a reducible representation given as direct sum of 2-dimensional repre- 2 sentations associated to elliptic curves with appropriate local conditions. n Suchconditionsarise in theworks of Schoofand correspond toreduction a properties of modular elliptic curves. J 2 2 ] T N 1 Introduction . h t a m [ 2 This work originates from the articles of Schoof about classification of abelian v varieties [15]. There he examines the case of abelian varieties over Q with 3 semistable reduction in only one prime ℓ and good reduction everywhere else 7 and proves that they do not exist for ℓ=2,3,5,7,13,while for ℓ=11 they are 8 3 given by products of the Jacobian variety J0(11) of the modular curve X0(11). . In [16] he makes some generalisations of this result when ℓ is not a prime and 1 1 the base field is not Q, but a quadratic field. Some similar results, given in 2 terms of p-divisible groups, were also previously obtained by Abrashkin in [1]. 1 Themainpurposeofthisworkistotranslatesomeresultsofthosearticlesin : v termsofdeformationtheoryofrepresentationsassociatedtoelliptic curves. We i examine the following setting: let p 6= ℓ be distinct primes and S = {p,ℓ,∞}. X Letρ¯ :G →GL (k) i=1,...,nbeGaloisrepresentations,wherek isafinite i S 2 r fieldofcharacteristicpandG istheGaloisgroupofthemaximalextensionofQ a S unramified outside S. We can suppose that there are exactly r non-isomorphic representations among them and, up to reordering indexis, suppose that they are ρ¯ ,...,ρ¯ . Then we can write 1 r ρ¯=ρ¯ ⊕···⊕ρ¯ =⊕r ρ¯ei :G →GL (k) (1) 1 n i=1 i S 2n r where e =n. The main result is the following i=1 i TheorPem 1.1. : Suppose that: 1. the k-vector space Ext1 (V ,V ) of killed-by-p extensions is trivial for D,p ρ¯i ρ¯j every i,j =1,...,r; 1 1 INTRODUCTION 2 k if i=j 2. Hom (V ,V )= . G ρ¯i ρ¯j (0 if i6=j (cid:3) Let F be the framed deformation functor associated to ρ¯ with the local D conditions: • ρ is p-flat over Z[1/ℓ]; • ρ satisfies (ρ (g)−Id)2 =0 for every g ∈I ; i ℓ • ρ is odd. (cid:3) (cid:3) Then F is represented by a framed universal ring R which is isomorphic D D to W(k)[[x ,...,x ]], where N =4n2− r e 2. 1 N i=1 i P The setting works in particular when ρ¯ is the representation associated to i the p-torsion points of an elliptic curve E over Q having semistable reduction i in ℓ and good supersingular reduction at p, as the varieties described in [15]. Moreover the local condition in ℓ corresponds to the condition of semistable actiondescribedin[15,Section2],whiletheconditioninpistheclassicalflatness condition introduced in [13]. The final result want to express that the framed deformation ring turns out to be the “simplest” possible, giving an analog for deformation of [15, Th. 8.3]. The work is structured as such: in section 2 we recall the main features of deformationtheory,followingmainlyMazur’soriginalformulation(see[11,12]) and we introduce framed deformation functors, following Kisin’s approach[9], but avoidingthe use ofthe formalismof groupoids. Then inthe following three chapters we describe the local conditions we have used in our theorem: first we deal with the flatness condition in the residual prime p, which is treated according the initial results of Ramakrishna [13] and expanded by the work of Conrad [3, 4, 5]; then we pass to examine the semistability condition, which is treatedasaparticularcaseofrepresentationofSteinbergtype: thecomputation of the universal ring in this part is pretty indirect and passes through the use of formal schemes [10]. Finally, for the archimedean prime, the computation of the deformation ring is performed directly. Chapters 6 and 7 are dedicated to local-to-globalarguments, which consent to build up a presentation of a global deformation ring using the computations on the local ones we did in the pre- vious chapters. The definition of geometric deformation functor is introduced, too. It is due to Kisin [10] and the name comes form the “geometric”represen- tationsdefinedinSerre’sconjecture. Inthe finalchapterTheorem1.1isproved with the use of the technical instruments introduced and through a final direct computation of the universal framed deformation, using matrix algebras. 2 BASICS NOTIONS OF DEFORMATION THEORY 3 2 Basics Notions of Deformation theory Let k be a finite field of characteristic p and let S be a finite set of rational primes containing p and the archimedean prime. Denoting by Q the maximal S extensionofQunramifiedoutsideS andbyG =Gal(Q /Q),consideraGalois S S representation ρ¯: G →GL (k) and denote by V the associated G -module. S N ρ¯ S Let A be a complete noetherian local W(k)-algebra with residue field k and denote by Aˆr the category of such algebras. A lift of ρ¯to A is a representation ρ :G →GL (A) such that the diagram A S N GL (A) vvρvAvvvvvv;; N(cid:15)(cid:15)πA G ρ¯ //GL (k) S N commutes, where π is the natural projection on the residue field. Two lifts A ρ ,ρ are said to be equivalent if there exists a matrix M ∈Ker(π ) such that 1 2 A Mρ (g)M−1 =ρ (g) for every g ∈G . 1 2 S A deformation of ρ¯to A is an equivalence class of lifts. The starting repre- sentation ρ¯is called residual. Deformations can be better understood via a categoricalapproach. Given a Galois representation ρ¯, we can define the deformation functor F :Aˆr →Sets, (2) ρ¯ which associates to an element A∈Aˆr the set of deformation classesof ρ¯to A. Kisinhasintroducedavariantofthedeformationfunctor. Letβ beak-basis of the Galois module V . A framed deformation of the couple (V ,β) to a ring ρ¯ ρ¯ A∈Aˆr is a couple (V ,β ), where V is a free N-dimensional A-module with A A A continuous G -action lifting the action of V and β is an A-basis of V lifting S ρ¯ A A β. We can then define the framed deformation functor F(cid:3) : Aˆr → Sets which ρ¯ associatesto analgebraA∈Aˆr the setof frameddeformationclassesof(V ,β) ρ¯ to A. Theorem 2.1. 1. Theframeddeformation functorisrepresentablebyaring R(cid:3) =R(cid:3) ∈Aˆr. ρ¯ 2. If ρ¯satisfies the trivial centralizer condition End (V )≃ k, then F is k[G] ρ¯ ρ¯ representable by a ring R=R ∈Aˆr. ρ¯ (cid:3) The rings R and R are called the universal deformation ring and the uni- versal framed deformation ringofρ¯respectively. Theyareuniversalinthesense that any deformation of ρ¯to an element A∈Aˆr can be recovered via a unique homomorphism R→A. AmongthealgebrasinAˆrthereisonewithparticularproperties. Letk[ǫ]be theringofpolynomialsinǫwiththeconditionǫ2 =0andletF beadeformation functor. The tangent space of F is the set F(k[ǫ]). It has a natural structure of k-vector space. 2 BASICS NOTIONS OF DEFORMATION THEORY 4 Proposition 2.2. 1. F (k[ǫ]) is a finite dimensional k-vector space; ρ¯ 2. F (k[ǫ])≃H1(G ,Ad(ρ¯))≃Ext1 (V ,V ) as k-vector spaces; ρ¯ S k[G] ρ¯ ρ¯ 3. dim F(cid:3)(k[ǫ])=dim F (k[ǫ])+N2−dim H0(G ,Ad(ρ¯)). k ρ¯ k ρ¯ k S Let F be a deformation functor and let P be the category of pairs (A,V ) ρ¯ A with A ∈ Aˆr and V ∈ F (A). Let D be a full subcategory of P. We say that A ρ¯ D is a deformation condition if the following conditions hold: 1. if(A,V )→(B,V )isamorphisminP and(A,V )∈D,then(B,V )∈ A B A B D; 2. if(A,V )→(B,V )isaninjectivemorphisminP and(B,V )∈D,then A B B (A,V )∈D; A 3. if(A× B,V)liesinD,thenalsothe projections(A,V )and(B,V )do. C A B Given a deformation condition D, we can consider the functor F :Aˆr → ρ¯,D Sets that sends a ring A∈Aˆr to the set of deformations ρ of ρ¯to A such that (A,V ) ∈ D. If F is representable by a ring R , then F is representable, ρ ρ¯ ρ¯ ρ¯,D too, by a quotient of R . Moreover the tangent space F (k[ǫ]) is a k-vector ρ¯ ρ¯,D subspace of F (k[ǫ]). ρ¯ Given a Galois representation ρ¯: G → GL (k) a natural way to attach a S N deformation condition is the following: for every ℓ ∈S we consider the restric- tion ρ¯ : G → GL (k) together with a local deformation condition D . Then ℓ ℓ N ℓ wecandefineaglobaldeformationconditionDgivenbytheobjects(A,V )∈P A whose local restriction to ℓ lies in D for every ℓ∈S. ℓ Another important example of deformation condition is given by the fixed determinant. Let χ : G → W(k)∗ be a linear character. We say that a S representation ρ : G → GL (A) has determinant χ if the G -action induced S N S on the wedge product ΛN(V ) is given by the character ρ χ :G →χ W(k)∗ →φ A∗, (3) A S whereφistherestrictionoftheW(k)-algebrastructuremorphism. Thesubcate- goryDofpairs(A,V )suchthatρhasdeterminantχisadeformationcondition ρ and the corresponing deformation functor is denoted as Fχ. Moreoverwe have ρ¯ that Fχ(k[ǫ])≃H1(G ,Ad0(ρ¯)), (4) ρ¯ S where Ad0 denotes the vectorspaceofmatriceswith tracezeroprovidedby the adjoint G -action. S 3 THE LOCAL FLAT DEFORMATION FUNCTOR 5 3 The local flat deformation functor Inthissectionwewanttodealwithalocaldeformationconditionwhichrefersto the prime p, characteristic of the finite base field k. This condition was mainly studied by Ramakrishna in [13] and then generalised by Conrad in [4],[5] and Kisin in [9]. From now on, we will only deal with representations of degree 2. Lemma 3.1. (Ramakrishna) Let C be a full subcategory of Rep (G) closed k under passage to subobjects, direct products and quotients and let D be the full subcategory of P of pairs (A,V ) such that V ∈ C. Then D is a deformation A A condition. Proof. We need to prove that D satisfies the three properties of deformation conditions. Property 1 comes from the fact that, if (A,V ) → (B,V ) is a A B morphisminD,thenV ⊗ B ≃V andthetensorproductcanberecoveredvia A A B directsumsandquotients. Property2comesfromtheclosureundersubobjects. Property 3 comes from the fact that the fiber product and its projections can be constructed via direct sums and quotients, too. Let F be a finite extension of Q and ρ¯: G → GL (k) be a Galois repre- p F 2 sentation. If ρ is a deformation of ρ¯to a coefficient ring A, we say that ρ is flat if there exists a finite flat group scheme X over the ring of integers O such F that V ≃X(F¯), that is, V is the generic fiber of X. ρ ρ Proposition 3.2. The subcategory D of flat deformations is a deformation condition. Proof. it suffices to show that D satisfies lemma 3.1. Let 0 → T → U → V → 0 be a sequence of G-modules such that U is the generic fiber of a finite flat group scheme X over O . Then we can take the F schematicclosureX ofT inX (see[13,Lemma2.1]fordetails)andX =X/X 1 2 1 to see that also T and V are generic fibers of finite flat group schemes. This argument and the fact that a direct sum of finite flat group schemes is still a finite flat group scheme show that the subcategory of flat deformations is a deformation condition. If ρ¯ satisfies the trivial centralizer condition End (V ) = k, then the k[GF] ρ¯ deformation functor which assigns to a coefficient ring the set of deformations of ρ¯ which are flat, called the flat deformation functor and denoted as Ffl, is representable by a noetherian ring Rfl, which is called the local flat universal p deformation ring. Wewanttogiveaproofofthemainresultofrepresentability for this condition, which was proven by Ramakrishna for p6=2 and by Conrad for all cases. First we need some technical data Definition 3.3. Let φ denote the absolute Frobenius morphism. A Fontaine- Lafaille module is a W(k)-module M provided with a decreasing, exhaustive, separated filtration of W(k)-submodules {M } such that, for every index i, there i exists a φ-semilinear map φ :M →M with the property that φ (x)=pφ (x) i i i i+1 for every x∈M. 3 THE LOCAL FLAT DEFORMATION FUNCTOR 6 We denote by MF the category of Fontaine-Lafaille modules over W(k). Moreover we denote by MFf the full subcategory of objects such that M has tor finitelengthand Im(φ )=M andbyMFf,j the subcategoryofobjectssuch i tor that M = M and M = 0. Finally we say that a Fontaine-Lafaille module 0 j P is connected if the morphism φ is nilpotent The main result about Fontaine- 0 Lafaille modules is the following Theorem 3.4 (Fontaine-Lafaille). For every j ≤p there exists a faithful exact contravariant functor MFf,j →Repf (G), (5) tor Zp which is fully faithful if j < p and becomes fully faithful when restricted to the subcategory of connected Fontaine-Lafaille modules if j = p. Morevoer MFf,2 tor is antiequivalent to the category of finite flat group schemes over W(k) Proof. See [8, Ch.8-9] for a proof and description of the functor. We say that a representation ρ has weight j if it comes from a Fontaine- Lafaillemodule lying inMFf,j andwedenote byF the subfunctor ofF given tor j ρ¯ by deformations of ρ¯which are of weight j. It follows that if ρ¯is flat, then the functors F and F are the same, therefore we will identify them in the rest of 2 fl the chapter. We can now prove the main result for flat deformation functor. The proof is due to Ramakrishna for the case p>2 (see [13, section 3]); then Conrad has shown (see [3]) that the proof works also in the case p=2, since the Fontaine- Lafaille module used is connected. Theorem 3.5. (Ramakrishna) Let ρ¯:G →GL (k) be a flat residual Galois Qp 2 representation with trivial centralizer and such that det(ρ¯) = χ, where χ is the cyclotomic character. Then Rfl(ρ¯)≃W(k)[[T ,T ]]. (6) p 1 2 Proof. We split the proof in two parts. Suppose first that k = F and ρ¯is the p representation attached to the p-torsion points of an elliptic curve E over Q p with good supersingular reduction. We prove the theorem in this particular case,where computations are relativelyeasy,andthen pass to the generalcase. In the particular case we have chosen, we know that ρ¯ satisfies the triv- ial centralizer hypothesis and is of weight 2. We calculate the tangent space F (F [ǫ]). Viewing F [ǫ]2 as a 4-dimensional F -vector space, we can see an 2 p p p element ρ∈F (F [ǫ]) as a matrix 2 p ρ¯(g) 0 ρ(g)= (7) R ρ¯(g) g (cid:18) (cid:19) and such a representationgives clearly an element of Ext1 (V ,V ), the exten- 2,p ρ¯ ρ¯ sions in the category of weight 2 representations which are killed by p. It is immediate to check that equivalence of litings correspond to equivalent exten- sions. 3 THE LOCAL FLAT DEFORMATION FUNCTOR 7 Let M be the Fontaine-Lafaille module associated to V via Theorem 3.4. ρ¯ By fullfaithfulness ofthe functor, wehavethatExt1 (V ,V )=Ext1 (M,M) 2,p ρ¯ ρ¯ 2,p and that End (M)=F . MF p We want to write the module in a compactified manner in terms of a 2×2- matrixX . ForthatweusethefactthatM is1-dimensional(itwillbeproved M 1 shortly) and that φ (M )=0. Then we write 0 1 α 0 ∗ γ α γ φ = , φ = , X = . (8) 0 β 0 1 ∗ δ M β δ (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) ThematrixX encodesallthe informationsofthe structureofM. Wealso M wantto write the elements of Ext1 (M,M)via these matrices. If N is such an 2,p element, we have X C X = M , C ∈M (F ). (9) N 0 X 2 p M (cid:18) (cid:19) The matrix C corresponds to an element of Hom(M,M). If N′ is another elementofExt1 (M,M)andD isthe2×2matrixinitsuppertriangularpart, 2,p then it represents the same extension of N if and only if there exist a matrix Id R ∈M (F ) such that 0 Id 4 p (cid:18) (cid:19) Id R X C X D Id R M = M (10) 0 Id 0 X 0 X 0 Id M M (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) and this happens if and only if C −D = [R,X ]. Moreover R must preserve M the filtration of M, because the isomorphism between N and N′ does so. Let H be the set of such matrices R. It follows that Ext1 (M,M)≃Hom(M,M)/{[R,X ]: R∈H} (11) 2,p M Now we know that dim M = 2 and dim M = 0. If dim M 6= 1 Fp 0 Fp 2 Fp 1 then any endomorphism of M does not need to respect any filtration structure and therefore the centralizerof X in M (F ), which has at least dimension 2, M 2 p wouldbelongtoEnd (M);thisisimpossiblebecausetheendomorphismring MF is 1-dimensional. Therefore dim M =1. Fp 1 Now we can compute the dimension of the tangent space: observe that Hom(M,M)hasdimension4,thesetofmatricesRwhichpreservesthefiltration of M has dimension 3 and the kernel of the map R → [R,X ] has dimension M 1 (it is isomorphic to End (M)). Therefore the tangent space has dimension MF 4−(3−1)=2. Now we have that R (ρ¯) = Z [[T ,T ]]/I. We count the number of Z /pl- 2 p 1 2 p valued points of the universal ring, which is the number of objects N ∈MFf,2 tor which are free Z /pl-modules of rank 2. If N denotes the kernel of multiplica- p p tion by p in N, then we need N ≃ M, in terms of matrices, since X ≡ X p N M (mod p). Since X ∈ M (Z /pl) and we have to consider modulo p, there are N 2 p p4(l−1) such matrices. We have to consider them modulo isomorphism. Now if X ≃ X , then there exists a matrix R ∈ M (Z /pl) which respects the N1 N2 2 p 3 THE LOCAL FLAT DEFORMATION FUNCTOR 8 filtration of M such that RX = X R; there are p3(l−1) such matrices and N1 N2 pl−1 lie in the center of M (Z /pl), therefore commute with allthe X . So the 2 p N number of Z /pl-valued points is p4(l−1)/(p3(l−1)/pl−1)=p2(l−1). Observethat p this is the same number of Z /pl-valued points of Z [[T ,T ]]. p p 1 2 Let now f ∈ I and (x,y) ∈ (Z /pl)2, then f(x,y) ≡ 0 (mod pl) for every p positiveintegerl. Itfollowsthat,takingliftingstocharacteristiczero,f(x,y)= 0 for all (x,y)∈(pZ )2 and therefore f =0. So I =0 and R (ρ¯)=Z [[T ,T ]]. p 2 p 1 2 Nowwecanpasstothe proofofthe theoreminthe generalcaseandremove the hypothesis that k = F and that ρ¯ is the representation coming from an p elliptic curve. A lemmaofSerre(whose proofcanbe foundin[13])tells us that ρ¯has restriction to inertia given by ψ 0 ρ¯| = (12) I 0 ψp (cid:18) (cid:19) whereψ isafundamentalcharacteroflevel2. Forsucharepresentationwewill compute both the “unrestricted” universal ring and the flat one. First of all we want to show that H2(G,Ad(ρ¯)) = 0. By Tate local duality we have that H2(G,Ad(ρ¯)) = H0(G,Ad(ρ¯)∗) = (Ad(ρ¯)∗)G. Let φ ∈ (Ad(ρ¯)∗)G, we want to show that its kernel is 4-dimensional and therefore φ = 0. Let R ∈ Ad(ρ¯), we have φ(gR)=gφ(R), where the G-action is given by conjugacy composed with ρ¯ on the left and by determinant on the right. It follows that, if g ∈ I, then ρ¯(g)Rρ¯(g)−1−det(ρ¯(g))R∈Ker(φ). Then, if we define the map T :R→ρ¯(g)Rρ¯(g)−1−det(ρ¯(g))R, (13) g it suffices to show that there exists g ∈I such that Ker(T )=0. We choose a g g such that ψ(g)=α where α is an element of order p2−1 in k∗. Then, taking explicit formulas x y x(1−αp+1) y(α1−p−αp+1) R= , T (R)= (14) z w g z(αp−1−αp+1) w(1−αp+1) (cid:18) (cid:19) (cid:18) (cid:19) and the last matrix is zero if and only if R=0. Then our claim is proved. Nowwe usethe formulaforthe Euler-Poincar´echaracteristicforAd(ρ¯). Let hi =dim(Hi(G,Ad(ρ¯))). We have c (Ad(ρ¯))=h0−h1+h2 =−dim Ad(ρ¯). (15) EP k We have that h2 = 0, h0 = 1 (it is the trivial centralizer condition) and dim Ad(ρ¯) = 4, therefore h1 = 5. It follows that the unrestricted universal k ring for such a representation is isomorphic to W(k)[[T ,T ,T ,T ,T ]]. 1 2 3 4 5 The flat deformationring canbe computed by means of calculationssimilar to the ones performed in the case k = F , except that we have to consider p Fontaine-Lafaille modules overW(k) instead of Z and all the dimensions have p to be computed over k. We obtain again that R (ρ¯) = W(k)[[T ,T ]]. In 2 1 2 particularR (ρ¯)is aquotientofR(ρ¯)andthe surjectivemapbetweenthemhas 2 a 3-dimensional kernel. The theorem is therefore proved. Now we give a refinement of this result, which is due to Conrad [4]. 4 STEINBERG REPRESENTATIONS AT PRIMES ℓ6=P 9 Theorem 3.6. Letρ¯beas in theprevious theorem andlet Ffl,χ bethesubfunc- tor of flat deformations of ρ¯which have fixed determinant χ. Then this functor is representable by the ring Rfl,χ(ρ¯)≃Z [[T]]. (16) p p Proof. For the proof see [4, Ch. 4, Th.4.1.2]. EXAMPLE: Let E be an elliptic curve over Q that has supersingular re- p duction in p. Let ρ¯: G → GL (F ) be the representation coming from the Qp 2 p Galois action on the p-torsion points of E. Then, by the results of [5], ρ¯ is absolutelyirreducibleandthereforethefunctorFfl isrepresentable. Therefore, ρ¯ applying Ramakrishna’s theorem, we have that the flat universal deformation ring is Z [[T ,T ]]. p 1 2 4 Steinberg representations at primes ℓ 6= p Now we want to analyse local conditions at finite primes which are different from p. We continue to assume that the representationspace V has dimension ρ¯ 2. Definition 4.1. : A 2-dimensional representation ρ¯: G → GL (k) is called ℓ 2 of Steinberg type if it is a non-split extension of a character λ:G →k∗ by the ℓ twist λ(1)=λ⊗χ of λ by the p-adic cyclotomic character χ . p p A representation of Steinberg type has the matricial form λ(1)(g) ∗ ρ¯(g)= ∀g ∈I (17) 0 λ(g) ℓ (cid:18) (cid:19) Observethatsince ℓ6=p the modp cyclotomiccharacteris unramifiedand, if p is not a square mod ℓ, it also happens that χ and its twists are trivial. We do p notimposeanyramificationrestrictiononthecharacterλ. Uptotwistingbythe inversecharacterofλ,wemayassumethatdet(ρ¯)=χp andthatV(ρ¯)(−1)Gℓ 6= 0, which means that there is a subrepresentation of dimension 1 on which G ℓ acts via χ . p We define a subfunctor Lχp :Aˆr →Sets (18) ρ¯ of the deformation functor Fχp as ρ¯ Lχp(A)=(V ,L ) (19) ρ¯ A A where • V is a deformation of ρ¯to A. A 4 STEINBERG REPRESENTATIONS AT PRIMES ℓ6=P 10 • L is a submodule of rank 1 of V on which G acts via χ . A A ℓ p We define in the same way the framed subfunctor Lχp,(cid:3) :Aˆr →Sets as ρ¯ Lχp,(cid:3)(A)=(V ,β ,L ) (20) ρ¯ A A A where • (V ,β ) is a framed deformation of ρ¯to A. A A • L is a submodule of rank 1 of V on which G acts via χ . A A ℓ p This is the subfunctor corresponding to liftings of Steinberg type. In the following we work with the framed setting to avoid representability problems. Inorderto dealwithrepresentabilityofdeformationsfunctorsofSteingberg type, we need to recall the main definitions of formal schemes. Let R be a noetherian ring and I an ideal and assume that R is I-adically complete, so that we have R=limR/In. (21) ← We define a topologycal space Spf(R) in the following way: given an element f ∈Randf¯itsreductionmoduloI,wedefineD(f¯)tobethesetofprimeideals of R/I not containing f¯. Then the set Spec(R/I) with the induced topology is called the formal spectrum of R, with respect to I, and denoted by Spf(R). The sets D(f¯) are a basis for the topology of Spf(R). For each f ∈R we define Rhf−1i=limR[f−1]/In (22) ← Then the assignment D(f¯)7→Rhf−1i defines a structure sheaf on Spf(R). Definition 4.2. The affine formal scheme Spf(R) over R with respect to I is the locally ringed space (X,O ), where X = Spec(R/I) and O (D(f¯)) = X X Rhf−1i for each f ∈R. A noetherian formal scheme is a locally ringed space (X,O ), where X is a X topological space and O is a sheaf of rings over X such that each point x∈X X admits a neighborhood U such that (U,O | ) is isomorphic to an affine formal X U scheme Spf(R). A morphism of formal schemes is a pair (f,f∗):(X,O )→(Y,O ), where X Y f : X → Y is a continuous map of topologycal spaces and f∗ : O → f O is Y ∗ X a morphism of sheaves. If (X,O ) is a scheme, we can obtain a formal scheme Xˆ by the following X construction: let I ⊆ O be an ideal sheaf and consider Xˆ the completion of X X along I. Its underlying topological space is given by the subscheme Z of X defined by I and the structure sheaf is defined as before. A formal scheme obtained in this way is called algebrizable.