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Locally convex structures on higher local fields Alberto Cámara∗ October 31, 2012 2 1 0 2 Abstract t c We establish how a higher local field can be described as a locally O convex vector space once an embedding of a local field into it has been fixed. This extends previous results that had been obtained in the two- 0 dimensional case. In particular, we study and characterize bounded and 3 compactoid submodules of these fields and establish a self-duality result once a suitable topology on thedual space has been introduced. ] T N 1 Introduction . h t In [1] we explained how characteristic zero two-dimensional local fields may be a m regardedas locally convex vectorspaces once an embedding of a localfield into them has been fixed. [ This note, designed as a natural continuation of that work, explains how 1 the locally convex approach to higher topologies works for a higher local field v of arbitrary dimension. 8 It is perhaps necessary to explain the need to treat the arbitrary dimen- 6 0 sional case separately. The two-dimensionalcase often supplies the first step of 8 induction and therefore it is a goodidea to treatit first. At the same time, the 0. casesfordimensiongreaterthantwoquicklyturnintoaratherinvolvedexercise 1 in notation and the application of arguments which are familiar from the case 2 n=2. As such, the proof of many results in this note often refers to [1] for the 1 case n=2 and then indicates how to proceed by induction. : v Furthermore, there are many relevant functional analytic properties which i may be shown to hold in the two-dimensional case and which fail in greater X dimensionor whichone couldonlyexpect to holdin few particularcases;being r a bornological,reflexiveornuclearisanexampleofpropertieswhichdo nothold, at least in the general case. However, it is possible to describe the families of bounded and compactoid O-submodules of F explicitly. We also show an explicit self-duality result in Theorem 7.4 which generalises [1, Theorem 6.3]. We notonly often refer to concepts and results explained in [1], but alsoas- sumethereadertobefamiliarwithit. Inparticular,wereviewedthedefinitions and results of the theory oflocally convex vectorspaces overa localfield which are needed for this work in [1, §1]. Besides that, [6] contains an introductionto the topic; our notations closely resemble notations therein. ∗Theauthor issupportedbyaDoctoralTrainingGrantattheUniversityofNottingham. 1 We outline the contents of this work. §2 is shaped very much after [1, §2] and summarizes certain results from the structure theory of higher local fields. At the end of this section we focus our attention on a higher local field of the form F =K{{t }}···{{t }}((t ))···((t )) 1 r r+1 n−1 and deduce our results first in this case. Section §3 explains how the higher topology on F is locally convex. §4 ex- posesfactswhichareeitherveryeasyconsequencesoftheresultsintheprevious section or well-known facts about higher topologies. Sections §5 and §6 deal with the characterizations of bounded submodules and compactoid submodules of F, respectively. We study duality issues in §7. We prove self-duality after topologizing the dualspaceadequately(Theorem7.4),andalsodescribepolarsandpseudopolars of relevant submodules of F. We explain how the results obtained in the previous sections may be ex- tended from F to an arbitrary n-dimensional local field in §8 and we dedicate a few words to the positive characteristic and archimedean cases in section §9. Finally, we discuss some interesting questions and directions of work specif- ically related to this note in §10. Notation. Whenever F is a complete discrete valuation field, we will denote by O ,p ,π ,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 (one-dimensional) 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 sothat|π |=q−1. Due tofartoofrequentapparitionsinthe text,wewillease K notation by letting O :=O and p:=p . K K The conventions p−∞ =K, p∞ =0 and q−∞ =0 will be used. The main object of study of this workis a field inclusion K ⊂F where F is an n-dimensional local field. See §2 for details. Acknowledgements. I thank Thomas Oliver for carefully reading this note andpointingoutseveralimprovements. IalsothankmysupervisorIvanFesenko forhisguidanceandencouragement. Finally,IamalsogratefultoCristinaPérez Garc´ıa for several useful conversations regarding the theory of locally convex nonarchimedean spaces. 2 Higher local fields arising from arithmetic con- texts A zero-dimensional local field is a finite field. An n-dimensional local field, for n ≥ 1, is a complete discrete valuation field F such that F is an (n−1)- dimensionallocalfield. Thus,alocalfieldintheusualsenseisaone-dimensional local field. 2 An n-dimensional local field F determines then a collection of fields F , i i∈{0,...,n} with F =F, F a finite field and F =F for 1≤i≤n. n 0 i i−1 We refer to [4] for an excellent introduction to this topic and to [2] for a collection of surveys on the theory of higher local fields. In [1, §2], we explained how, as it was first introduced in [5], it is a good idea to regard two-dimensional local fields arising from an arithmetic context as vector spaces over a local field. The constructionmaybe generalisedto anarithmetic scheme ofanydimen- sion as follows. Let S be the spectrum of the ring of integers of a number field and f : X → S be an arithmetic scheme of dimension n (that is, X is an n-dimensional regular and f is projective and flat). Given a complete flag of irreducible subschemes η ∈ {η } ⊂ ··· ⊂ {η } = X, and assuming for n n−1 0 simplicity that all the {η } are regular at η , define An =O\ and i n X,ηn Ai =A[i+1, i∈{0,...,n−1}. ηi It can be shown [4, Remark 6.12] that F = A0 is an n-dimensional local field. The ring homomorphism O → O induces a field embedding K ֒→ F, S,f(x) X,x whereK =Frac O\ . Ourconclusionisthatwhenevern-dimensionallocal S,f(x) fieldsarisefroma(cid:16)narithm(cid:17)etic-geometricsettingtheyalwayscomeequippedwith a prefixed embedding of a local field into them. This justifies our decision to fix a characteristic zero local field K and to study n-dimensional local fields not as fields F, but as pairs of a field F and a field embedding K ֒→ F. We shall refer to such a pair as an n-dimensional local field over K. A morphism of higher local fields over K is a commutative diagram of field embeddings F // F OO1 ⑤⑤>> 2 ⑤ ⑤ ⑤ ⑤ ⑤ K whereF andF arehigherlocalfieldsandF →F isanextensionofcomplete 1 2 1 2 discrete valuation rings. The structure of higher local fields is explained in [3], [2, §1] or [4, Theorem 2.18]. They may be classified using Cohen structure theory for complete rings. In particular, we have the following possibilities: (i) If charF is positive, then it is possible to choose t ,...,t ∈F such that 1 n F ≃ F ((t ))···((t )). In this work we assume charF = 0, and only treat 0 1 n the positive characteristic case in §9. (ii) If charF =0, then there are t ,...,t ∈F such that 1 1 n−1 F ≃F ((t ))···((t )). 1 1 n−1 Moreover,if an embedding of fields K ֒→F has been fixed, then we have a finite extension K ֒→F . 1 (iii) If none of the above holds, then there is a unique r ∈{1,...,n−1} such that charF 6=charF . Then there is a characteristic zero local field L r+1 r and elements t ,...,t ∈F such that F is a finite extension of 1 n−1 L{{t }}···{{t }}((t ))···((t )). 1 r r+1 n−1 3 Moreover, if charF = p, L may be chosen to be the unique unramified 0 extensionofQ withresiduefieldF . Inthisworkwewillnotrequirethis p 0 fact, but simply use the fact that given an embedding K ֒→F, there is a finite subextension K{{t }}···{{t }}((t ))···((t ))֒→F. 1 r r+1 n−1 Notation. We let from now on, and until the beginning of §8, F =K{{t }}···{{t }}((t ))···((t )) 1 r r+1 n−1 with 0 ≤ r ≤ n − 1. The extremal case r = 0 (resp. r = n − 1) stands for F = K((t ))···((t )) (resp. F = K{{t }}···{{t }}). We also let L = 1 n−1 1 n−1 K{{t }}···{{t }}((t ))···((t )), by which we simply mean that L is the sub- 1 r r+1 n−2 field of F consisting of power series in t ,...,t . 1 n−2 Itwillbeextremelyconvenienttousemulti-indexnotation. Forthispurpose, letI =Zn−1 andJ =Zn−2. Forl∈{1,...,n−1},thenotation(∗,i ,...,i ) l n−1 refers to any element in I whose last entries agree with (i ,...,i )∈Zn−l−1. l n−1 In this fashion, denote I(i ,...,i )={α∈I; α=(∗,i,...,i )}. l n−1 l n−1 Any element in x∈F can be written uniquely as a power series x= ··· ··· x ti1···tin−1, i1,...,in−1 1 n−1 in−X1≫−∞ ir+1X≫−∞iXr∈Z iX1∈Z with x ∈K. We will abbreviate such an expression to i1,...,in−1 x= x tα, α α X for α ∈ I and x ∈ K. Finally, for α = (i ,...,i ) ∈ I, denote −α = α 1 n−1 (−i ,...,−i ). 1 n−1 Severalproofswilluseinductionarguments. Forsuch,itwillbeconvenientto denoteelementsofLas x tβ forβ ∈J andx ∈K,withthisnotationbeing β β β analogoustothatadoptedforelementsinF. Thestatementα=(β,i)forα∈I, P β ∈J and i∈Z means that if β =(i ,...,i ), then α=(i ,...,i ,i). 1 n−2 1 n−2 3 Local convexity of higher topologies Theconstructionofthe highertopologyonF isexplainedin[4, §4]and[3,§1]). It revolves around two basic constructions. First suppose that a L is a field on which a translation invariant and Haus- dorff topology has been defined. Let {U } be a sequence of neighbourhoods i i∈Z of zero of L, with the property that there is an index i ∈ Z such that U = L 0 i for all i≥i . The sets of the form 0 U ti := x ti; x ∈U for all i (1) i i i i i∈Z (i≫−∞ ) X X 4 describe a basis of neighbourhoods of zero for a translation invariantHausdorff topology on L((t)). Second, suppose that L is a complete discrete valuation field with charL6= charL,sothatitmakessensetoconsiderthefieldL{{t}}. Supposethatatrans- lationinvariantandHausdorfftopologyhasbeendefinedonL. Let{V } bea i i∈Z sequenceofneighbourhoodsofzeroofLsatisfyingthe followingtwoconditions: (i) There is c∈Z such that pc ⊂V for every i∈Z. L i (ii) Foreveryl ∈Z there is i ∈Zsuchthat for everyi≥i we havepl ⊂V . 0 0 L i This condition simply means that as i → ∞ the neighbourhoods of zero V become bigger and bigger. We will denote this condition by V →L as i i i→∞. The sets of the form V ti := x ti ∈L{{t}}; x ∈V for all i (2) i i i i (i∈Z ) X X constitute the basis of neighbourhoods of zero for a translation invariant and Hausdorff topology on L{{t}}. The procedure for topologizing F is an inductive application of the two constructionsspecifiedabove. Namely,foreveryk ∈{1,...,r},applythesecond construction inductively on E{{t }}, with E =K{{t }}···{{t }}. k 1 k−1 Fork ∈{r+1,...,n−1},applyinductivelythefirstconstructiononE((t )), k with E =K{{t }}···{{t }}((t ))···((t )). 1 r r+1 k−1 The resulting topology on F is called the higher topology. Itisawell-knownfactthatifr <nthehighertopologyonF dependsonthe choice of a coefficient field, that is, a field inclusion F ⊂ F. Since in this case charF =0andF istranscendentaloverQ,there areinfinitely manychoicesfor suchan embedding. In our description ofthe higher topology,we are implicitly choosing an isomorphism F ≃ K{{t }}···{{t }}((t ))···((t )). In the two- 1 r r+1 n−2 dimensional equal characteristic case, there is a unique coefficient field which factors the field embedding K ֒→ F, namely the algebraic closure of K in F. Such a choice is not possible whenever n≥3. Proposition 3.1. The higher topology on F is locally convex. It may be described as follows. Consider a collection {n } ⊂Z∪{−∞} subjected to the α α∈I conditions: (i) For any l ∈ {r+1,...,n−1} and fixed indices i ,...,i ∈ Z, there l+1 n−1 is a k ∈Z such that for every k≥k we have 0 0 n =−∞ for all α∈I(k,i ,...,i ). α l+1 n−1 (ii) For any l ∈{1,...,r} and fixed indices i ,...,i , there is an integer l+1 n−1 c∈Z such that n ≤c for every α∈I(i ,...,i ), α l+1 n−1 and we have that n →−∞, α∈I(k,i ,...,i ), as k →∞. α l+1 n−1 5 Then, the open lattices of F are those of the form Λ= pnαtα, (3) α X where {n } ranges through the sequences specified above. α α∈I Remark 3.2. Let us clarify what the second equation in condition (ii) above standsfor. Theconditionisthatforanyl ∈{1,...,r},fixedindicesi ,...,i l+1 n−1 andd∈Z,thereisanintegerk suchthatforeveryk ≥k andα∈I(k,i ,...,i ) 0 0 l+1 n−1 we have n ≤d. α Proof. We will prove the result by induction on n. For n = 2, see [1, §3]. Suppose n > 2. Then write L = K{{t }}···{{t }}((t ))···((t )), with r ∈ 1 r r+1 n−2 {0,n−2}. By induction hypothesis, the higher topology on L is described by lattices of the form M = pnβtβ, (4) β∈J X with n ∈Z∪{−∞} satisfying the conditions in the statement of the proposi- β tion. Now we need to distinguish two cases. First, if r ≤ n−2, we must apply the construction in which neighbourhoods of zero are of the form (1), as F = L((t )). So we let n−1 Mi = pn(β,i)tβ, i∈Z β∈J X with the property that there is an i ∈Z such that for all i≥i , M =L. This 0 0 i last condition is equivalent to setting n =−∞ for all β ∈J and i≥i . As (β,i) 0 the M describe a basisofneighbourhoodsofzerofor the higher topologyonL, i the higher topology on F admits a basis of neighbourhoods of zero formed by sets of the form Λ= M ti . i n−1 i∈Z X Byinductionhypothesis,theM areallO-lattices,whichisenoughtoshowthat i ΛisanO-lattice. So,inthiscase,werearrangenotationbylettingα=(β,i)so that a basis of neighbourhoods of zero for the higher topology is described by sets Λ = α∈Ipnαtα. On top of the conditions which the indices n(β,i) satisfy by the induction hypothesis for β ∈J, we must add the further condition that P there is an integer i such that for all i ≥ i , n = −∞ for all α ∈ I(i). This 0 0 α shows that our claim holds in this case. The second case is the one in which r > n − 2 and we must apply the construction in which neighbourhoods of zero are givenby sets of the form (2), as F =L{{t}}. So we set Mi = pn(β,i)tβ, i∈Z, β∈J X subject to the properties: (i) There is an integer c such that for every i ∈ Z, pc ⊂ M . By induction L i hypothesis, this means that n ≤c for every β ∈J and i∈Z. (β,i) 6 (ii) M →Lasi→∞. Thisisequivalentton →−∞forβ ∈J asi→∞. i (β,i) As M describe a basis of neighbourhoods of zero of the topology of L, the sets i of the form Λ= M ti i n−1 i∈Z X describe a basis of neighbourhoods of zero for the higher topology on F. Since theM areO-lattices,wegetthatΛisanO-lattice. Sowearrangenotationsby i lettingα=(β,i),sothattheO-latticeΛmaybedescribedasΛ= pnαtα. α∈I On top of the conditions satisfied by the n which are inherited by induction, α P there are the two new conditions: (i) There is an integer c such that n ≥c for all α∈I. α (ii) n →−∞ for α∈I(i), as i→∞. α This shows that the proposition also holds in this case. Proposition3.3. Thehigher topologyonF isthelocallyconvexK-vectorspace topology defined by the seminorms of the form k·k:F →R, x tα 7→sup|x |qnα (5) α α α α X as {n } ranges through the collections described in the statement of Propo- α α∈I sition 3.1 above. Proof. It is necessary to show that the gauge seminorm associated to the open lattice Λ as in (3) is the one given by (5). The gauge seminorm defined by Λ is precisely kxk= inf |a|, for x∈F. x∈aΛ Let x = αxαtα. We have that x ∈ aΛ if and only if xα ∈ apnα for every α∈I. That is, if and only if P |x |qnα ≤|a| for all α∈I. (6) α The infimum ofthe values |a| for which(6)holds is the supremumofthe values |xα|qnα as α∈I. Definition 3.4. The seminorms on F defined in the previous proposition will be referred to as admissible seminorms. Anadmissibleseminormk·kisattachedtoacollection{n } ⊂Z∪{−∞}. α α∈I If we have chosennotation not to reflect this fact it is in the pursue of a lighter reading and understanding that the collection {n } , when needed, will be α α∈I clear from the context. 7 4 First properties Wesummarisesometopologicalpropertieswhichwereknownalreadyforhigher topologies, or which are deduced from the fact that these topologies are locally convex. We also state some properties which do not hold in general because they are known not to hold already for n=2. The field F, equipped with a higher topology, becomes a locally convex K-vector space. As such, it is a topological vector space. It is a previously known fact that higher topologies are Hausdorff. In order to show that this property holds in our setting it is enough to show that, given x∈ F×, there is anadmissible seminormk·k for which kxk6=0. If the coefficientcorresponding to α ∈ I is nonzero, any admissible seminorm for which n > −∞ suffices. α Moreover, the reduction map O → F is open when O is given the subspace F F topologyandF a highertopologycompatible withthe choiceofcoefficientfield if charF 6=charF. MultiplicationF →F byafixednonzeroelementinducesahomeomorphism of F, but multiplication µ:F ×F →F is not continuous. Highertopologiesarenotfirst-countableandthereforenotmetrizable. More- over, in general, F is not bornological, barrelled, reflexive nor nuclear and its rings of integers are not c-compact or compactoid. Remark 4.1. One of the advantages of higher topologies is that power series inasystemofparametersconverge. Toexplainitconcisely,equipthesetI with theinverseofthelexicographicalorder,forwhich(i ,...,i )<(j ,...,j ) 1 n−1 1 n−1 if and only if for an index l ∈ {1,...,n−1} we have i < j and i = j for l l m m l<m≤n−1. Let x = x tα ∈ F. We define a net in F by taking (x ) ⊂ F. If α∈I α α α∈I k·k is an admissible seminorm on F, then for α ∈I large enough the value 0 P x− x tα α (cid:13) (cid:13) (cid:13)(cid:13) αX≤α0 (cid:13)(cid:13) (cid:13) (cid:13) is arbitrarily small. (cid:13) (cid:13) (cid:13) (cid:13) 5 Bounded O-submodules Let us study the bounded O-submodules of F. Proposition 5.1. Let {k } ⊂Z∪{∞} subject to the conditions: α α∈I (i) For every l ∈ {r+1,...,n−1} and indices i ,...,i ∈ Z there is l+1 n−1 an index j ∈ Z such that for every j < j we have k = ∞ for all 0 0 α α∈I(j,i ,...,i ). l+1 n−1 (ii) For every l ∈{1,...,r} and i ,...,i ∈Z, there is an integer d such l+1 n−1 that k ≥d for all α∈I(i ,...,i ). α l+1 n−1 The bornology of F admits the collection of sets B = pkαtα (7) α∈I X as a basis. Moreover, these are the only bounded O-submodules of F. 8 Proof. First, let us showthat the sets B arebounded. As we will use induction on n, the case n=2 is thoroughly explained in [1, §4]. Let k·k be an admissible seminorm attached to {n } ⊂Z∪{−∞}. α α∈I Let x= x tα. We have α α P kxk=sup|x |qnα ≤supqnα−kα α α α and therefore it is enough to prove that the set {n −k } ⊂ Z∪{−∞} is α α α∈I bounded above. We distinguishtwocases. Supposefirstthatr≤n−2. ThenF =L((t )). n−1 On one hand, there is an index j ∈ Z such that k = ∞ for every α ∈ I(j), 0 α j < j . On the other hand, there is an index j ∈ Z such that n = −∞ for 0 1 α every α∈I(j), j >j . It is therefore enough to show that the sets 1 N(j)={n −k ; α∈I(j)}, j ≤j ≤j α α 0 1 areboundedabove. Butourhypothesisofinductionimpliesthatforeachj ∈Z, {nα}α∈I(j) definesanadmissibleseminormand β∈Jpk(β,j)tβ ⊂Lisabounded O-submodule; this implies the boundedness of N(j). P The case in which r = n−1, and therefore F = L{{t }}, is simpler: we n−1 have that all the n are bounded above and all the k are bounded below; α α therefore the differences n −k are bounded above. α α Second,wehavetoshowthatanyboundedO-submoduleofF isofthe form (7). If C is an O-submodule of F, we may assume without loss of generality that C = C ti with C ⊂ L bounded O-submodules. By induction i∈Z i n−1 i hypothesis, let us write, for each i∈Z, P Ci = pk(β,i)tβ, β∈J X with k ∈Z∪{∞} satisfying the conditions exposed in the statement of the (β,i) proposition. By letting α = (β,i) ∈ I, we may write C = pkαtα and we only have α∈I to show that the indices k satisfy the conditions exposed in the proposition. α P Let us consider separate cases again. First, if r ≤n−2 and F =L((t )), the indices k satisfy condition (ii) by n−1 α induction hypothesis. Condition (i) is also satisfied by induction hypothesis for every l ∈ {r+1,...,n−2}. So we only have to show condition (i) in the case l = n−1, that is: there is an index j ∈ Z such that k = ∞ for all α ∈ I(j), 0 α j < j . We show that if this last condition does not hold, then C cannot be 0 bounded. Iftheconditiondoesnothold,thenthereisadecreasingsubsequence(j ) h h≥0 and one index α ∈I(j ) for each h for which k <∞. Let h h αh −j +k , if α=α ∈I(j ), n = h αh h h α −∞ otherwise. (cid:26) Then the collection {n } defines an admissible seminorm k·k. Consider α α∈I xh =πkαhtαh ∈C. Then kxhk=q−jh and ourseminormtakesarbitrarilylarge values on C. 9 Second,supposethatr =n−1andF =L{{t }}. Byinductionhypothesis, n−1 we know that condition (ii) holds for every l ∈ {1,...,n−2} and we need to check that the condition also holds in the case l=n−1, that is: there is d∈Z such that k ≥ d for all α ∈ I. We show that if this is not the case, then C is α not bounded. So we assume that the sequence k is not bounded below. By condition α (ii) for l = n − 2, we have that for each j ∈ Z there is d ∈ Z such that j k ≥d foreveryα∈I(j). Ourassumptionimpliesthatthed arenotbounded α j j below. Supposethereisadecreasingsubsequence(j ) forwhichd →−∞. h h≥0 jh Without loss of generality,there is an index α ∈I(j ) such that k =d for h h αh jh every h≥0. In this case, let 0, if α=α and j <0, n = h h α −∞ otherwise. (cid:26) The collection {n } defines an admissible seminorm k·k. Now, for h ≥ 0, α α∈I the elementsxh =παhtαh ∈C haveseminormkxhk=q−djh andhenceC isnot bounded. Finally, if there is no decreasing subsequence (j ) for which d →−∞, h h≥0 jh there exists an increasing one with this property. Again without loss of generality, there is an index α ∈ I(j ) for which k = d for every h ≥ 0. h h αh jh Let (k −1)/2, if α=α ,j >0 and k odd. αh h h αh n = k /2, if α=α ,j >0 and k even. α  αh h h αh −∞ otherwise.  The collection {n } defines an admissible seminorm. Moreover, we have α α∈I nαh −kαh → ∞ as h → ∞. The elements xh = πkαhtαh ∈ C satisfy kxhk = qnαh−kαh and, therefore, C is not bounded. Proposition 5.2. The submodules B in Proposition 5.1 are complete. Proof. Let B = α∈Ipkαtα with {kα}α∈I ⊂ Z∪{∞} satisfying conditions (i) and (ii) in the statement of Proposition 5.1. P Let H be a directed set and (x ) ⊂B a Cauchy net. Let us denote, for h h∈H each h∈H, xh = αxα,htα with xα,h ∈pkα for α∈I. We havethat, for a fixed α∈I, (xα,h)h∈H ⊂pkα is a Cauchy net. As pkα is complete, the net cPonverges to an element xα ∈pkα. If the power series x tα defines an element x in F, then x ∈ B and α α x →x. This is easy to check by induction on n (the case n=2 may be found h P in [1, §5]). Proposition 5.3. Multiplication µ:F ×F →F is a bounded map. Proof. The argument for the proof is by induction on n. The case n = 2 is dealt with in [1, Proposition4.8], and the same argumentapplies when looking at F =L{{t }} or L((t )) n−1 n−1 6 Compactoid O-submodules The result below outlines which bounded submodules of F are compactoid. 10