Functional analysis on two-dimensional local fields Alberto C´amara∗ 2 1 October 11, 2012 0 2 t c O Abstract 0 We establish how a two-dimensional local field can be described as a 1 locally convex space once an embedding of a local field into it has been fixed. We study the resulting spaces from a functional analytic point of ] T view: in particular we characterize bounded, c-compact and compactoid submodulesof two-dimensional local fields. N . h Introduction t a m This work is concerned with the study of characteristic zero two-dimensional [ localfields. These are complete discrete valuation fields whose residue field is a 1 local field, either of characteristic zero or positive. v Followinganideaintroducedin[11],wedonotregardtwo-dimensionalfields 5 asfields inthe usualsense,but asanembedding offields K ֒→F, whereK is a 9 localfieldandF isa two-dimensionallocalfield. Givenatwo-dimensionallocal 9 field F, such field embeddings always exist and we are not assuming any extra 2 . conditions on F; we are only changing our point of view. 0 In the arithmetic-geometric context, such field embeddings arise in the fol- 1 lowing way (see [11] for details): suppose that S is the spectrum of the ring 2 1 of integers of a number field and that f : X → S is an arithmetic surface (for : our purposes it is enough to suppose that X is a regular 2-dimensional scheme v i and that f is projective and flat). Choose a closed point x ∈ X and an irre- X ducible curve {y}⊂X passing through x. Suppose for simplicity that x∈{y} r isregularandlets=f(x)∈S. Startingfromthelocalringofregularfunctions a O ,weobtainatwo-dimensionallocalfieldF throughaprocessofrepeated X,x x,y completions and localizations: \ F =Frac O[ . x,y X,xy (cid:18) (cid:19) This is analogue to the procedure of completion and localizationthat allows us to obtain a local field K =Frac(O ). The structure morphism O →O s S,s S,s X,x induces a field embedding K ֒→F . s x,y d ∗Theauthor issupportedbyaDoctoralTrainingGrantattheUniversityofNottingham. 1 The moral of the above paragraph is that if two-dimensional fields arise from an arithmetic-geometric context then they always come with a prefixed local field embedded into them. What we study in this work is the K-vector space structure associated to F via the embedding K ֒→ F. As such, we connect the topological theory of two-dimensional local fields with the theory of nonarchimedean locally convex vector spaces. In particular, for the fields K((t)) and K{{t}} (see §2 for the definition of the latter), we establish in Corollaries 3.2 and 3.7, a family of defining seminorms for the higher topology of the form x ti =sup|x |qni, i i (cid:13) (cid:13) i (cid:13)Xi (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) where {ni}i∈Z ⊂Z∪{−∞}(cid:13)is a sequ(cid:13)ence subject to certain conditions and q is the number of elements in the residue field of K. In particular, this provides us with a new way to describe higher topologies ontwo-dimensionallocalfieldswhichdoesnotrelyontakingalifting mapfrom theresiduefieldasin[9]. Thisalsointroducesanewconceptofboundedsubset. Although our description of higher topologies is valid for both equal and mixed characteristic two-dimensional local fields, the study of the functional theoretic properties in the two cases suggests that similarities stop here. Equal characteristic fields may be shown to be LF-spaces (a direct limit of Fr´echet spaces) and, as such, they are bornological, nuclear and reflexive. This charac- terization is unavailable for mixed characteristic fields such as K{{t}} and we show how these properties do not hold. One of the advantages of our point of view is that certain submodules of F arise as the families of c-compact and compactoid submodules, and there- fore have a property which is a linear-topological analogue of compactness. In particular,compactoidsubmodules coincide withbounded submodules inequal characteristic (this is a consequence of nuclearity but can be made very ex- plicit, cf. Proposition 5.8) and define a family strictly contained in that of bounded submodules in mixed characteristic. By using the associated bornol- ogyweachieveinTheorem6.3 averyexplicitself-duality resultwhichis hinted at in [3, §3]. We briefly outline the contents of this work. Sections §1 and §2 summarise certain bits of the theory of nonarchimedean locally convex vector spaces and the structure of two-dimensional local fields, respectively. We have included them in this work in order to be able to refer to certain general results in later parts of the work and in order to fix notations and conventions. Hence, we do notsupply prooffor anystatementinthese sections,but referthe readerto the available literature. Insection§3wedescribehowhighertopologiesinducethestructureoflocally convexK-vectorspacesonK((t))andK{{t}}. ThemainresultsarePropositions 3.1and3.6,alongwithCorollaries3.2and3.7,whichdescribehighertopologies in terms of seminorms. Sections §4, §5, §6 and §7 develop a systematic study of the locally convex vectorspacesintroducedin§3. Thosesectionsstudyboundedsetsandbornolog- 2 ical properties; complete, c-compact and compactoid submodules; duality and nuclearity, respectively. In §8, we extend the results of the previous sections to the case of a general embedding K ֒→F of a local field into a two-dimensional local field. Sections§9and§10explainhowtheresultsinthisworkcanalsobeappliedto archimedeantwo-dimensionallocalfields andpositive characteristiclocalfields, respectively. In the first case, we are dealing with LF-spaces and we deduce our results from the well-established theory of (archimedean) locally convex spaces. In the second case, we relate the locally convex structure of vector spaces over F ((u)) to the linear topological structure of vector spaces over F q q through restriction of scalars. In this case we recover already known results of Beilinson [1] and Kapranov [6]. Finally, we discuss some applications and further directions of research in §11. Notation. Whenever F is a complete discrete valuation field, we will denote by O ,p ,π and F its ring of integers, the unique prime ideal in the ring F F F of integers, an element of valuation one and the residue field, respectively. A two-dimensional local field is a complete discrete valuation field F such that F is a local field. Throughout the text, K will denote a characteristic zero local field, that is, a finite extension of Q for some prime p. The cardinality of the finite field K p willbedenotedbyq. TheabsolutevalueofK willbedenotedby|·|,normalised so that |π | =q−1. Due to far too many appearances in the text, we will ease K notation by letting O :=O , p:=p and π :=π . K K K The conventions p−∞ =K, p∞ ={0} and q−∞ =0 will be used. Acknowledgements. IamindebtedtoMatthewMorrowandOliverBra¨unling, withwhomIhadtheinitialdiscussionsthatlaterturnedintothispieceofwork; partoftheir ownworkonresidues wasaninitialmotivationto study these top- ics. IamalsoingreatdebtwithRalfMeyer,whoreadanearlydraftandshared manydiscussionsandcommentswithme. I amgratefulto ThomasOliver,who pointed out several mistakes and improvements to this text. Finally, I thank my supervisor Ivan Fesenko for his guidance and encouragement. 1 Locally convex spaces over K In this section we summarise some concepts and fix some notation regarding locally convex vector spaces over K. This is both for the reader’s convenience asmuchasforestablishingcertainstatementsandpropertiesforlaterreference. The theory of locally convex vector spaces over a nonarchimedean field is welldevelopedinthe literature,sowewillkeepaconciseexpositionofthe facts that we will require later. Details and proofs for results stated in this section may be found in [13]; we often use similar notations. Let V be a K-vector space. A lattice in V is an O-submodule Λ ⊆ V such that for any v ∈ V there is an element a ∈ K× such that av ∈ Λ. This is equivalent to having Λ⊗ K ≃V O 3 as K-vectorspaces. A subset of V is saidto be convexif it is of the form v+Λ forv ∈V andΛalatticeinV. AvectorspacetopologyonV issaidtobelocally convexif the filter of neighbourhoods ofzero admits a collectionof lattices as a basis. A seminorm on V is a map k·k:V →R such that: (i) kλvk=|λ|·kvk for every λ∈K, v ∈V, (ii) kv+wk≤max(kvk,kwk) for all v,w ∈V. These conditions imply in particular that a seminorm only takes non-negative values and that k0k=0. The gauge seminorm of a lattice Λ⊆V is defined by the rule: k·k :V →R, v 7→ inf |a|. (1) Λ v∈aΛ Given a family of seminorms {k·k } on V, there is a unique coarsest j j∈J vector space topology on V making the maps k·k : V → R continuous for j every j ∈J. Such topology is locally convex,as the open balls centered at zero for the defining seminorms supply a basis of neighbourhoods of zero consisting of open lattices. A locally convex topology can be described in terms of lattices or in terms of seminorms; passing from one point of view to the other is a simple matter of language. AsubsetB ⊂V is bounded if forany openlattice Λ⊂V,there is ana∈K suchthatB ⊆aΛ. Alternatively,B isboundedifforeverycontinuousseminorm k·k on V we have supkvk<∞. v∈B A locally convex K-vector space V is bornological if any seminorm which is bounded on bounded sets is continuous. ThecollectionofboundedsetsofV defines abornology,thatis: acollection of subsets of V which is stable under finite unions and hereditary by inclusion [5]. Justlikeatopologyonasetistheminimumamountofinformationrequired in order to have a notion of open set and continuous map, a bornologyon a set is the minimum amount of information required in order to have a notion of bounded set and bounded map. Bornological locally convex spaces are the ones for which topology is de- termined by bornology: a linear map from such a space into another space is continuous if and only if it is bounded. On an arbitrary vector space, if a convex bornology is specified, we get a bornological locally convex space by considering the strongest locally convex topology that gives rise to the specified family of bounded sets. Openlatticesinanon-archimedeanlocallyconvexspacearealsoclosed. The space V is said to be barrelled if any closed lattice is open. Among many general ways to construct locally convex spaces [13, §5], we will require the use of products. Proposition 1.1. Let {V } be a family of locally convex K-vector spaces, i i∈I and let V = V . Then the product topology on V is locally convex. i∈I i Q 4 If{Λ } denotesthe setofopenlatticesofV fori∈I,thenthesetofopen i,j j i lattices of V is given by finite intersections of lattices of the form π−1Λ . i i,j Equivalently,theproducttopologyonV isthe onedefinedbyallseminorms of the form v 7→supkπ (v)k , i i,j i,j where{k·k } isadefiningfamilyofseminormsforV foralli∈I,π :V →V i,j j i i i isthecorrespondingprojectionandthesupremumistakenoverafinitecollection of indices i,j. Another construction which we will require is that of inductive limits. Let V be a K-vector space and {V } be a collection of locally convex K-vector i i∈I spaces. Let, for each i ∈I, f : V →V be a K-linear map. The final topology i i for the collection {f } is not locally convex in general. However, there is a i i∈I finest locally convex topology on V making the map f continuous for every i i∈I. That topology is called the locally convex final topology on V. Inductive limits and direct sums of locally convex spaces are particular examples of such construction. Definition 1.2. Suppose that V is a K-vector space and that we have an increasing sequence of vector subspaces V ⊆ V ⊆ ··· ⊆ V such that V = 1 2 ∪n∈NVn. Suppose that for each n ∈ N, Vn is equipped with a locally convex topology such that V ֒→ V is continuous. Then the final locally convex n n+1 topology on V is called the strict inductive limit topology Next, we need to discuss completeness issues. We require to deal not only with sequences, but arbitrary nets. Let I be a directed set and V a locally convex K-vector space. A net in V is a family of vectors (v ) ⊂ V. A i i∈I sequence is a net which is indexed by the set of natural numbers. Thenet (v ) convergestoa vectorv, andweshallwrite v →v,if forany i i∈I i ε>0 and continuous seminorm k·k on V, there is an index i∈I such that for every j ≥i we have kv −vk≤ε. j Similarly,thenet(v ) issaidtobeCauchyifforanyε>0andcontinuous i i∈I seminorm k·k on V there is an index i ∈ I such that for every pair of indices j,k ≥i we have kv −v k≤ε. j k Definition 1.3. A subset A ⊆ V is said to be complete if any Cauchy net in A converges to a vector in A. Example 1.4. K is complete for nets, since it is a normed K-vector space. The usual topological notion of compactness is not very powerful for the study of infinite dimensional vector spaces over non-archimedean fields. In our case,there isanO-linear conceptofcompactnesswhichis a goodsubstitute for compactness. Definition 1.5. Let A be an O-submodule of V. A is said to be c-compact if, for any decreasingly filtered family {Λ } of open lattices of V, the canonical i i∈I map A→limA/(Λ ∩A) ←− i i∈I is surjective. 5 Example 1.6. The base field K is c-compactas a K-vectorspace. This shows that a c-compact module need not be bounded. This property may be phrased in a more topological way. Proposition 1.7. An O-submodule A⊆ V is c-compact if and only if for any family {C } of closed convex subsets C ⊆A such that C =∅ there are i i∈I i i∈I i finitely many indices i ,...,i ∈I such that C ∩...∩C =∅. 1 m i1 Tim Proof. See [13, Lemma 12.1.ii and subsequent paragraph]. Proposition1.8. Let{V } beacollectionoflocallyconvexK-vectorspaces, h h∈H and for each h ∈H let A ⊆ V be a c-compact O-submodule. Then A h h h∈H h is c-compact in V . h∈H h Q Proof. [13, PropQ. 12.2]. Another notion which is used in this setting is that of a compactoid O- module; it is a notion which is analogous to that of relative compactness. Definition 1.9. Let A ⊆ V be an O-submodule. A is compactoid if for any open lattice Λ of V there are finitely many vectors v ,...,v ∈V such that 1 m A⊆Λ+Ov +···+Ov . 1 m Let A ⊆ V be an O-submodule. If A is c-compact, then it is closed and complete. Similarly, if A is compactoid then it is bounded. [13, §12]. Proposition 1.10. Let A ⊆ V be an O-submodule. The following are equiva- lent. (i) A is c-compact and bounded. (ii) A is compactoid and complete. Proof. [13, Prop. 12.7]. The collection of compactoid O-submodules of V generates a bornology which is a priori weaker than the one given by the locally convex topology. If V,W are two locally convex K-vector spaces, a linear map f :V →W is continuous as soon as the pull-back of a continuous seminorm is a continuous seminorm. We denote the K-vectorspace ofcontinuous linear maps betweenV and W by L(V,W). The space L(V,W) may be topologized in the following way. Let B be a collection of bounded subsets of V. For any continuous seminorm k·k on W and B ∈B, consider the seminorm k·k :L(V,W)→R, f 7→ supkf(v)k. B v∈B Definition 1.11. We write L (V,W) for the space of continuous linear maps B fromV toW endowedwiththelocallyconvextopologydefinedbytheseminorms k·k , for every continuous seminorm k·k on W and B ∈B. B In the particular case in which B consists of all bounded sets of V, we write L (V,W) for the resulting space, which is then said to have the topology b 6 of uniform convergence. If B consists only of the singletons {v} for v ∈ V, we denotetheresultingspacebyL (V,W)andsaythatithasthetopologyofpoint- s wise convergence. Finally, if B is the collectionof compactoidO-submodules of V, we denote the resulting space by L (V,W). c There are two cases of particular interest: the topological dual space V′ = L(V,K), and the endomorphism ring L(V) = L(V,V). We denote F′,F′,F′, s b c L (V),L (V) and L (V) for the corresponding topologies of point-wise conver- s b c gence, uniform convergence and uniform convergence on compactoid submod- ules, respectively. The notion of polarity plays a role in the study of duality, as it provides us with a way of relating O-submodules of V to O-submodules of V′. Definition 1.12. If A⊆V is an O-submodule, we define its pseudopolar by Ap ={l∈V′; |l(v)|<1 for all v ∈A}. The pseudobipolar of A is App ={v ∈V; |l(v)|<1 for all l ∈Ap}. Wehavethat,l ∈Ap ifandonlyifl(A)⊆p. Notethatthetraditionalnotion of polar relaxes the condition in the definition of pseudopolar to |l(v)| ≤ 1 or, equivalently, l(A) ⊆ O. Introducing the distinction is an important technical detail, as pseudo-polarity is a better-behaved notion in the nonarchimedean setting. Pseudopolarity provides with a way to relate O-submodules of V to O- submodules in V′. Proposition 1.13. Let A⊆V be an O-submodule. We have (i) If A⊆B ⊆V is another O-submodule, then Bp ⊆Ap. (ii) Ap is closed in V′. s (iii) If A∈B, then Ap is an open lattice in V′. B Proof. This is part of [13, Lemma 13.1]. Inordertoconcludethissectionwedefinenuclearspaces. Foranysubmodule A ⊆ V, denote V := A ⊗ K, endowed with the locally convex topology A O associatedto thegaugeseminormk·k . V maywellnotbe aHausdorffspace, A A but its completion V := limV /πnA A ←− A n∈Z is a K-Banachspace. c Definition 1.14. V is said to be nuclear if for any open lattice Λ ⊆ V there exists another open lattice M ⊆ Λ such that the canonical map V → V is M Λ compact, that is: there is an open lattice in V such that the closure of its M image is bounded and c-compact. d c d Proposition 1.15. We have: 7 (i) An O-submodule of a nuclear space is bounded if and only if it is com- pactoid. (ii) Arbitrary products of nuclear spaces are nuclear. (iii) Strict inductive limits of nuclear spaces are nuclear. Proof. (i) is [13, Proposition 19.2], (ii) is [13, Proposition 19.7] and (iii) is [13, Corollary 19.8]. 2 Our point of view on two-dimensional local fields We consider the category whose objects are field inclusions K ֒→F where K is our fixed characteristic zero local field and F is a two-dimensional local field. In such case, we shall say that F is a two-dimensional local field over K. A morphism in this category between K ֒→ F and K ֒→ F is a 1 2 commutative diagram of field inclusions F // F OO1 ⑤⑤>> 2 ⑤ ⑤ ⑤ ⑤ ⑤ K where F ֒→F is an extension of complete discrete valuation fields. 1 2 The classification of two-dimensional local fields follows from Cohen struc- ture theory of complete local rings and was established in [9]. The particular case with which we are dealing is very well described in [11, §2.2 and 2.3]. Bythisclassification,givenatwo-dimensionallocalfieldF itisalwayspossi- bletoexhibitalocalfieldcontainedinit,soourassumptiondoesnotimplyany further structure on F. Let us briefly recall the structure of two-dimensional local fields, which depends on the relation between the characteristicsof F and F. IfcharF =charF, the choice ofa uniformizert forthe discrete valuationof F determines an isomorphism F ≃ F((t)). Such an isomorphism is not unique, as it depends on the choice of a coefficient field F ֒→F. Besides fields of Laurent series, there is another construction which is key in order to work with two-dimensional local fields, and higher local fields in general. For any complete discrete valuation field L, consider L{{t}}= x ti; x ∈L, infv (x )>−∞, a →0(i→−∞) , i i L i i (i∈Z ) X with operations given by the usual addition and multiplication of power series. Note that we need to use convergence of series in L in order to define the product. With the discrete valuation given by ∞ v a ti :=infv (a ), L{{t}} i L i ! i=−∞ X 8 L{{t}} turnsinto acomplete discretevaluationfieldandthe extensionL{{t}}|L is an unramified extension of complete discrete valuation fields. When L is a characteristic zero local field, the field L{{t}} turns into a 2- dimensionallocalfieldwhichwecallthestandardmixedcharacteristicfieldover L. Its first residue field is L t . WeviewelementsofLaselementsofL{{t}}intheobviousway. Inparticular, (cid:0)(cid:0) (cid:1)(cid:1) if π is a uniformizer of O , it is also a uniformizer of O ; the element L L L{{t}} t∈L{{t}} is such that t∈L t is a uniformizer. Suppose now that F is any two-dimensional local field such that charF 6= (cid:0)(cid:0) (cid:1)(cid:1) charF, and that an embedding K ֒→ F is given. In this case, F contains a subfield which is K-isomorphic to K{{t}}, and this extension is finite. RegardlessofcharF,theexistenceofafieldinclusionK ⊂F forcesacertain compatibilitybetweentherank-twovaluationofF andthediscretevaluationof K. Namely, we have an inclusion of abelian groups K× ⊂ F×. The structure of these abelian groups is well known, and implies that one of the components of the rank-two valuation of F restricts to the discrete valuation of K and the other one restricts trivially. Example 2.1. Consider K = Q ⊂ Q {{t}} = F. In such case, the rank-two p p valuation of F is (v ,v ):F× →Z⊕Z, a ti 7→ infv (a ),inf{i; a ∈/ pZ } . 1 2 i p i i p i∈Z i∈Z (cid:18) (cid:19) X The restriction of v to K is v , while v restricts trivially. 1 p 2 Example 2.2. Consider K = Q ⊂ Q ((t)) = F. In such case, the rank-two p p valuation of F is (v ,v ):F× →Z⊕Z, a ti 7→(i ,v (a )), 1 2 i 0 p i0 iX≥i0 where we suppose that a is the first nonzero coefficient in the power series. i0 The restriction of v to K is trivial while the restriction of v to K is v . 1 2 p Remark 2.3. There are two particular local fields which play a very distin- guished role when these objects are to be studied from a functional analytic point of view. Those are the fields K((t)) and K{{t}}. As we will see, most topological properties which hold in these particular cases will hold in general aftertakingrestrictionsofscalarsorabasechangeoverafinite extensionwhich topologically is equivalent to taking a finite cartesian product. It is for this reason that we will work from now on with these two particular examples of two-dimensional local fields. We will explain how our results extend to the general case in §8. Notation. When working with the two-dimensionallocal fields F =K{{t}} or F =K((t)), for any collection {A } of subsets of K, we will denote i i∈Z A ti = x ti ∈F; x ∈A for all i∈Z . i i i i i∈Z ( i ) X X We will also denote O = O{{t}}. After all, this ring consists of all K{{t}} power series in K{{t}} all of whose coefficients lie in O. 9 3 Higher topologies are locally convex In this section we will explain how the higher topology on K((t)) and K{{t}} is a locally convex topology. We are forced to study both cases separately. 3.1 Equal characteristic The higher topology on K((t)) is defined as follows. Let {U } be a collection i i∈Z of open neighbourhoods of zero in K such that, if i is large enough, U = K. i Then define U = U ti. (2) i i∈Z X ThecollectionofsetsoftheformU definesthesetofneighbourhoodsofzero of a group topology on K((t)). Proposition3.1. ThehighertopologyonK((t))definesthestructureofalocally convex K-vector space. Proof. AsK isalocalfield,thecollectionofopenneighbourhoodsofzeroadmits a collection of open subgroups as a filter, that is: the basis of neighbourhoods of zero for the topology is generated by the sets of the form pn ={a∈K; v (a)≥n}, K where n ∈ Z ∪ {−∞}. These closed balls are not only subgroups, but O- fractional ideals. This in particular means that the sets of the form Λ= pniti ⊆K((t)), (3) i∈Z X where n = −∞ for large enough i, generate the higher topology. Moreover, i they are not only additive subgroups, but also O-modules. If x = x ti ∈ K((t)) is an arbitrary element, and i is such that i≥i0 i 1 n =−∞ for all i>i then we have the possibilities: i 1 P (i) i <i , in which case x∈Λ. 0 1 (ii) i ≤i . In such case, let 1 0 n=max max n ,0 . i (cid:18)i0≤i≤i1 (cid:19) Then πn ∈O satisfies πnx∈Λ. Thus, Λ is a lattice and the higher topology is locally convex. As a consequence of the previous proposition, it is possible to describe the higher topology in terms of seminorms. Corollary 3.2. For any sequence (n ) ⊂ Z∪{−∞} such that there is an i i∈Z integer k satisfying n =−∞ for all i>k, define i k·k:K((t))→R, x ti 7→max|x |qni. (4) i i i≤k i≫−∞ X 10