On the kernel and the image of the rigid analytic regulator in positive characteristic

ON THE KERNEL AND THE IMA GE OF THE RIGID ANAL YTIC REGULA TOR IN POSITIVE CHARA CTERISTIC Ambrus P ´ al May 19, 2009. A bs tr act . W e wi ll formulate and pr o v e a certain recipro cit y law relating certain residues of the differen tial symbol dl og 2 from the K 2 of a Mumford curve to the rigid analytic regulator constructed b y the author in a previous pap er. W e will use this result to deduce some consequences on the k ernel and image of the rigid analytic regulator analogous to some old conjectures of B ei l inson and Blo c h on the complex analytic regulator. W e also relate our construction to the symbol de fined by Con tou- Carr` ere and to Kato’s r esidue homomorphism, and we sho w that W eil’s recipro cit y law di rectly implies the recipro cit y la w of Anderson and Romo. 1. Introduction a n d annou n cement of resul ts Motiv ation 1. 1. In the pap e r [17] the author introduced the c o ncept o f the rigid analytic reg ulator. This is a homomorphism from the motivic cohomolo gy gro up H 2 M ( X, Z ( 2)) o f a Mumford curve X ov er a lo cal field F in to the F ∗ -v alued har- monic co chains of the graph of components of the special fib er. It is defined t hrough non-archimedean integration, hence it is elementary in na ture and it is a menable to computation. In par ticula r the author was able to compute its v alue on so me explicit e le men ts o f the K 2 of Dr infeld mo dular c urv es c o nstructed using mo dula r units and r elate it to s pecial v alues of L -functions in the pap er [1 8]. It is quite rea- sonable to consider this r esult as a function field analogue of Beilinson’s class ical theorem on the K 2 of elliptic mo dular curves as well a s the rigid a na lytic regula- tor is a non-archimedean ana logue of the co mplex analy tic Beilinson-B loch-Deligne regulator . An old conjecture of Blo ch and Beilinson predicts that the complex analytic regulator is injective on the motivic co homology gro up H 2 M ( X , Q (2)) of a regula r int egra l mo del X of a smo oth, pro jective curve X defined ov er a num b er field and the image of H 2 M ( X , Z (2)) is a Z -lattice of a co njecturally desc ribed r ank. Hence it is de s irable to understand the basic prop erties of the r igid a na lytic regulato r s uc h as its image and kernel, partially b ecause any a nalogous res ult would be evidence tow ar ds the conjecture men tioned ab ov e . W e o ffer the following result: let X b e a Mumford curve which is the g eneral fibe r of a regular quas i- pro jectiv e surface X fibred ov er a smo oth affine curve defined ov er a finite field. W e s ho w that the kernel of the rigid a nalytic regula tor in H 0 ( X , K 2 , X ) is a p -div isible group and the image of H 0 ( X , K 2 , X ) is a finitely generated Z -mo dule who se clo sure in the p - completion 2000 Mathematics Subj ect Classific ati on. 19F27 (primary), 19D45 (secondary). Ty p eset by A M S -T E X 1 2 AMBRUS P ´ AL of the tar get group is a Z p -mo dule of the same r a nk which is at mos t as large as conjectured by Beilinson. In pa rticular the kernel of this ma p is tor sion if Parshin’s conjecture ho lds. As a key ingredient of the pr oo f we compute ce rtain residues of the logarithmic differentials o f element s of H 2 M ( X , Z (2)) in terms of the r igid analytic regulator , generalis ing a formula of O sipov, prov ed for fields of zero characteris tic in [16], to any characteristic. (A closely rela ted result has b een obtained by M. Asakura in a recent preprint [1] for c e r tain tw o dimensio nal lo cal fields o f zero characteristic using simila r metho ds.) O f course the case of p o sitiv e characteris tic is the most in tricate, due to the lack o f log arithm. This formula can b e considered as a rela tiv e of the explicit recipro city law o f Kato and Be sser’s theor e m expressing the Coleman- de Sha lit reg ulator in terms o f the synt omic re g ulator (see [19 ] a nd [4]), although it is simpler to pr o ve. Then our ma in theorems follow from the Blo c h-Gabb er-Ka to theor em, Deligne’s purity theor em and the degenera tion of the slop e sp ectral sequence. As an imp ortant intermediate step we als o rela te the rigid analytic regulato r to K a to’s residue homomorphism for higher lo cal fields and the Contou-Carr` ere symbol. The symbols mentioned ab ov e are esse n tially just different formalizations o f the same phenomenon whic h w as disc overed indep endent ly at least three times. As an e asy application of our r esults we show that the la tter is bilinear a nd satisfies the Steinberg relation. A t the s ame stro ke we show that W eil’s recipro city law directly implies the recipro city law o f Anderson and Romo. Notation 1 .2. By s ligh tly extending the usual ter minology we will call a scheme C defined ov er a field a curve if it is r e duced, lo cally of finite type and of dimensio n one. A cur v e C is said to have normal crossing s if every singular p oint of C is an ordinary double point in the usual sense. W e say that a curve C ov er a field f is totally degener ate if it has nor mal cro ssings, every o rdinary double p oint is defined ov er f a nd its irreducible comp onen ts are pro jective lines over f . F or a n y curve C with norma l cros sings let e C denote its no rmalization, and le t e S ( C ) denote the pre-image of the set S ( C ) of singular p oints o f C . W e denote by Γ( C ) the or ie n ted graph whose set of vertices is the se t of irreducible co mponents of e C , and its set of edges is the set e S ( C ) such that the initial v ertex o f any edge x ∈ e S ( C ) is the irreducible comp onent of e C whic h contains x and the terminal vertex of x is the irreducible comp onent which c on tains the other element of e S ( C ) which maps to the same singular p oin t with res p ect to the nor malization map as x . The normalizatio n map identifies the ir reducible comp onents of C and e C whic h we will use without further notice. Definition 1 .3. F or any (or ien ted) gr aph G let V ( G ) and E ( G ) denote its set of vertices a nd edges, r espec tively . Let G b e a lo cally finite or ien ted g r aph which is equipp e d with a n inv olution · : E ( G ) → E ( G ) such that for e a c h edge e ∈ E ( G ) the original and ter mina l vertices of the edge e ∈ E ( G ) ar e the terminal a nd o riginal vertices of e , res p ectively . The edge e is ca lle d the edg e e with r eversed orientation. Let R b e a commutativ e group. A function φ : E ( G ) → R is called a harmo nic R -v a lue d co chain, if it satisfies the following conditions: ( i ) W e hav e: φ ( e ) + φ ( e ) = 0 ( ∀ e ∈ E ( G )) . ( ii ) If for an edge e we introduce the notation o ( e ) and t ( e ) for its o riginal and ON THE KERNEL AND THE IMAGE OF THE RIGID ANA L YTIC REGULA TOR 3 terminal vertex r espectively , X e ∈E ( G ) o ( e )= v φ ( e ) = 0 ( ∀ v ∈ V ( G )) . W e deno te by H ( G, R ) the gro up of R -v alued harmonic co chains on G . Notation 1.4 . Let k be a p erfect field and let B b e a smo oth irr educible pr o jective curve over k . Let ∞ b e a clos ed p oint of B and let F denote the co mpletion of the function field of B at ∞ . Let O denote the v a luation ring of F and let π : X → B be a regular irre ducible pro jective surfac e fibre d ov er B such that the fib er X ∞ of X ov er ∞ is to ta lly deg enerate. Then the base change X of X to F is a Mumford curve ov er F which has a regula r , semistable mo del ov er the sp ectrum of O who se sp ecial fiber is X ∞ . The rig id ana lytic r egulator intro duced in [17] is a ho momorphism: {·} : H 2 M ( X, Z (2)) → H (Γ( X ∞ ) , F ∗ ) . Let U ⊂ X b e an o pen subv ariety s uch that its complement is a normal crossing s divisor D which is the preimage of a finite se t of closed po in ts of B containing ∞ . Let the symbol {·} a lso denote the comp osition of the functoria l map H 0 ( U , K 2 , U ) → H 2 M ( X, Z (2)) and the r ig id a nalytic regulator. The main result of this pap er is the following: Theorem 1 .5. A ssume t hat k has char acteristic p . Then t he map: {·} : H 0 ( U , K 2 , U ) /p n H 0 ( U , K 2 , U ) → H (Γ( X ∞ ) , F ∗ / ( F ∗ ) p n ) induc e d by t he r e gulator { ·} is inje ctive for every natura l n umb er n . Notation 1.6. W e call a Z -submo dule Λ of a Hausdorff top ological group G which is the dir e ct sum of a discrete group and a pro- p gr oup p -saturated, if it is finitely generated and the map Λ ⊗ Z p → b G is a n injection, where b G is the p -completion o f G . Note that every discrete, finitely ge ne r ated Z -submo dule is p -sa turated. Assume now that k is a finite field of characteristic p . The result ab ov e, Delig ne’s purity theorem and the degeneration of the slop e s p ectral s e q uence imply the following: Corollary 1. 7. The image of the r e gulator {·} : H 0 ( U , K 2 , U ) → H (Γ( X ∞ ) , F ∗ ) is p -satur ate d and its r ank is at most as lar ge as the r ank of the gr oup H (Γ( D ) , Z ) . This lattic e is discr ete if D = X ∞ . The kernel of this r e gulator is a p -divisible gr oup. In p articular it is torsion if Parshin ’s c onje cture holds for X , and it is fin ite if the Bass c onje ct ur e holds for X . F urther Re sults 1.8. L e t A b e a lo cal Artinian r ing with res idue field k which is again allowed to b e an arbitra ry p erfect field. In [7] a map: h· , ·i : A (( t )) ∗ × A (( t )) ∗ → A ∗ was defined, called the Co n tou-Carr` er e symbo l in [2], where t is a v ar ia ble. The Contou-Carr` ere symbol is equa l to the tame symbol if A is a field. In [2] G. Ander- son and F. P . Ro mo proved that the Contou-Carr` ere symbol is bilinear and prov ed a recipro city law for it. Their pro of is the g eneralization of the pro of o f the residue 4 AMBRUS P ´ AL theorem by T a te a nd the W eil recipro city law by Arab ello, De Concini a nd Kac. They work directly over Ar tinia n r ings, so they have to develop an elab orate theory generalizing a ll concepts app earing in the pro ofs quo ted ab ov e for Artinian rings. In this article we will present a different pro of of the bilinearity of this symbol and their r ecipro cit y law. It is based on the o bserv ation that if A is the quotient of a discrete v aluation ring , then the Co n tou-Carr` er e symbo l h f , g i of any f and g in A (( t )) ∗ is just the reductio n of Kato’s residue (see [12] a nd [13 ]) of some lifts of f a nd g . The existence and the basic pr oper ties of the latter follows at o nce from the prop erties of the rigid a nalytic r e g ulator. Hence the a forementioned r e sults follow immediately from some well-kno wn facts s uc h a s the defor mation theory of smo oth pro jectiv e cur v es is unobstructed and W eil’s r ecipro cit y law. One may even give a new pro of of the W eil recipro city law using the observed contin uity of the tame symbol by degener ating the c ur v e to a stable curve with r a tional comp onents in its s pecial fib er. Since the Anderso n-Romo recipr o city law implies them, the recipro city laws of T ate and Witt a lso follow from W eil’s law. Con ten ts 1.9. The go al of the next chapter is to re view the cons tr uction of the rigid analytic regula tor and to list its basic prop erties for rationa l sub domains of the pro jectiv e line without pro ofs, mainly for the sa k e of the reader. The relations hip with K ato’s residue homomor phism is established in the third chapter. The fourth chapter is conce r ned with the Contou-Carr` ere symbol and the Ander son-Romo recipro city law. W e review the general co ns truction of the res idue homomorphis m for K¨ ahler differe ntials in the fifth chapter which we r elate to the Contou-Carr` ere symbol in ca se of lo cal Ar tinian rings, and to Kato ’s residue ho momorphism in case of lo cal fields of dimension tw o. This re c ipr oc ity law is used to deduce Theorem 1.5 and Coro llary 1.7 in chapter six. Ac knowledgmen t 1. 10. I wish to thank the IH ´ ES for its warm hospitality and the pleasa n t environmen t they created for pro ductiv e r esearch, where this ar ticle was written. 2. Review of the rigid anal ytic regula tor Notation 2.1 . In this chapter all c la ims are stated without pro of. The in terested reader is kindly as k ed to c o nsult [17]. Let F be a lo cal field and let C deno te the completion of the alg ebraic closur e of F with res pect to the unique ex tension o f the abs o lute v a lue on F . Recall that C is a n algebr aically c lo sed field complete with r espec t to an ultr a metric abso lute v alue which will be denoted by | · | . Let | C | denote the set of v alues o f the la tter. Let P 1 denote the pro jective line ov er C . F o r any x ∈ P 1 and any tw o ra tional non-z e r o functions f , g ∈ C (( t )) on the pro jectiv e line let { f , g } x denote the tame symbol o f the pair ( f , g ) at x . Recall that a subset U of P 1 is a co nnected ra tional sub domain, if it is non-empty and it is the complement of the union o f finitely many pa ir-wise disjoint op en discs. Let ∂ U denote the set of these complementary op en discs. The elements o f ∂ U are called the b oundary comp onen ts of U , by s ligh t abuse o f languag e. Let O ( U ), R ( U ) denote the alg ebra of holomo rphic functions on U and the subalg ebra of restrictions of r ational functions, resp ectively . Let O ∗ ( U ), R ∗ ( U ) denote the gro up of inv ertible elements of these algebra s. The group R ∗ ( U ) consists of ra tional functions which do not hav e po les or zer os lying in U . F or each f ∈ O ( U ) let k f k denote sup z ∈ U | f ( z ) | . This is a finite num b e r, and makes O ( U ) a Banach alg ebra ov er C . W e say that the ON THE KERNEL AND THE IMAGE OF THE RIGID ANA L YTIC REGULA TOR 5 sequence f n ∈ O ( U ) converges to f ∈ O ( U ), denoted by f n → f , if f n conv er ges to f with resp ect to the top olog y of this Banach algebra , i.e. lim n →∞ k f − f n k = 0. F or every real num b er 0 < ǫ < 1 we define the sets O ǫ ( U ) = { f ∈ O ( U ) |k 1 − f k ≤ ǫ } , and U ǫ = { z ∈ C || 1 − z | ≤ ǫ } . Theorem 2 .2. Ther e is a unique map {· , ·} D : O ∗ ( U ) × O ∗ ( U ) → C ∗ for every D ∈ ∂ U , c al le d the rigid analytic r e gulator, with the fol lowing pr op erties: (i) F or any two f , g ∈ R ∗ ( U ) their r e gulator is: { f , g } D = Y x ∈ D { f , g } x , (ii) the r e gu lator { · , ·} D is biline ar in b oth variables, (iii) the r e gulator {· , ·} D is alternating: { f , g } D · { g , f } D = 1 , (iv) if f , 1 − f ∈ O ( U ) ∗ , then { f , 1 − f } D is 1 , (v) for e ach f ∈ O ǫ ( U ) and g ∈ O ∗ ( U ) we have { f , g } D ∈ U ǫ .  Remark 2 .3. It is an immediate consequence of prop erty ( v ) that the rigid analytic regulator is co ntin uous with r espect to the supremum top ology . Explicitly , if f and g are elements of O ∗ ( U ), D ∈ ∂ U is a b oundary comp onent, and f n ∈ O ∗ ( U ), g n ∈ O ∗ ( U ) are sequences such that f n → f and g n → g , then the limit lim n →∞ { f n , g n } D , exists, and it is equal to { f , g } D . Let M ( U ) denote the field of mer omorphic functions o f U a nd let M ∗ ( U ) denote the multiplicativ e g roup of non-zero elements o f M ( U ). Theorem 2.4 . The r e is a unique set of homomorphisms deg D : M ∗ ( U ) → Z wher e U is any c onne ct e d r ational sub domain and D ∈ ∂ U is a b oundary c omp onent with the fol lowing pr op erties: ( i ) the homomorph ism deg D is zer o on O 1 ( U ) , ( ii ) for every f ∈ R ∗ ( U ) the inte ger deg D ( f ) is the numb er of zer os z of f with z ∈ D c oun te d with multiplicities minus the nu mb er of p oles z of f with z ∈ D c ounte d with multiplicities, ( iii ) for every f ∈ M ∗ ( U ) we have deg D ( f | Y ) = deg D ( f ) wher e Y ⊆ U is any c onne ct e d r ational sub domain satisfying the pr op ert y D ∈ ∂ Y .  Definition 2.5. If U is still a co nnected r ational sub domain of P 1 , and f , g a re t wo meromorphic functions on U , then for all x ∈ U the functions f and g hav e a power ser ies expansion ar o und x , so in pa rticular their tame symbol { f , g } x at x is defined. The tame symbols extends to a homomorphism {· , ·} x : K 2 ( M ( U )) → C ∗ . W e define the gr o up K 2 ( U ) as the kernel of the direct s um of tame symbols: M x ∈ U {· , ·} x : K 2 ( M ( U )) → M x ∈ U C ∗ . Let k = P i f i ⊗ g i ∈ K 2 ( U ), where f i , g i ∈ M ( U ), and let D ∈ ∂ U . Le t more o ver Y be a connected rational sub domain of U such that f i , g i ∈ O ∗ ( Y ) fo r all i and ∂ U ⊆ ∂ Y . Define the rig id analytica l r egulator { k } D by the formula: { k } D = Y i { f i | Y , g i | Y } D . 6 AMBRUS P ´ AL Theorem 2.6. ( i ) F or e ach k ∈ K 2 ( U ) the rigid analytic al r e gulator { k } D is wel l- define d, and it is a homomorphism { ·} D : K 2 ( U ) → C ∗ , ( ii ) for any two functions f , g ∈ O ∗ ( U ) we have { f ⊗ g } D = { f , g } D , ( iii ) for every k ∈ K 2 ( U ) the pr o duct of al l r e gulators on the b oun dary c omp onent s of U is e qual to 1: Y D ∈ ∂ U { k } D = 1 .  Definition 2. 7. F o r every U ⊂ P 1 connected rational sub domain le t Z ∂ U denote the free ab elian gro up with the elements of ∂ U as free gener a tors. Let H 1 ( U ) denote the quo tien t of Z ∂ U b y the Z -mo dule g enerated by P D ∈ ∂ U D . F or every D ∈ ∂ U we let D denote the c lass of D in H 1 ( U ) as well. Let A b denote the categor y of ab elian gro ups. Let C rs denote the categor y who se ob jects are connected ra tional sub do mains of P 1 and who s e mo rphisms ar e holomorphic maps b etw e e n them. Let D ( w, r ) deno te the op e n disc of radius r a nd center w , that is D ( w, r ) = { z ∈ C || z − w | < r } where 0 < r ∈ | C | . Finally for every pa ir a ≤ b of num b ers in | C | let A ( a, b ) denote the closed a nnulus P 1 − D (0 , a ) − D ( ∞ , 1 /b ). Of course it is a co nnected rational sub do main. Theorem 2.8. Th er e is a unique funct or H 1 : C rs → A b with the fol lowing pr op- erties: ( i ) for every U ⊂ P 1 c onne ct e d r ational su b domain H 1 ( U ) is the gr oup define d in 3.7, ( ii ) for every map U → Y which is t he r estriction of a pr oje ctive line ar tr ans- formation f and D ∈ ∂ U b oundary c omp onent we have: H 1 ( f )( D ) = f ( D ) ∈ H 1 ( Y ) , ( iii ) for every f : U → A ( a, b ) holomorphi c map and D ∈ ∂ U b oundary c omp o- nent we have: H 1 ( f )( D ) = deg D ( f ) D (0 , a ) ∈ H 1 ( A ( a, b )) .  Definition 2.9. Let U ⊂ P 1 be a connected rational sub domain. F or every class c ∈ H 1 ( U ) and element k ∈ K 2 ( U ) we define the regulator { k } c as { k } c = Y D ∈ ∂ U { k } c ( D ) D , where P D ∈ ∂ U c ( D ) D is a lift of c in Z ∂ U . By claim ( iii ) of Theo rem 2.6 this regulator is w ell-defined. F or every holomor phic map h : U → Y b et ween t wo connected ra tio nal sub domains let h ∗ : K 2 ( M ( Y )) → K 2 ( M ( U )) b e the pull- back homomo rphism induced by h . By restriction it induces a homo mo rphism K 2 ( Y ) → K 2 ( U ). ON THE KERNEL AND THE IMAGE OF THE RIGID ANA L YTIC REGULA TOR 7 Theorem 2 .10. F or any k ∈ K 2 ( Y ) and c ∈ H 1 ( U ) we have: { h ∗ ( k ) } c = Y E ∈ ∂ Y { k } H 1 ( h )( c ) .  Definition 2. 11. Let U be now a connected rationa l sub domain o f P 1 defined ov er F . This means that U = { z ∈ P 1 || f i ( z ) | ≤ 1 ( ∀ i = 1 , . . . , n ) } as a s et fo r some natura l n umber n a nd rationa l functions f 1 , . . . , f n ∈ F ( t ). Let O F ( U ), R F ( U ), O ∗ F ( U ), R ∗ F ( U ) and M F ( U ) denote the alg ebra o f holo mo rphic functions, the subalg ebra of restrictions of F -r ational functions, the gr oups of in- vertible elements of these algebr as a nd the field o f meromo rphic functions on the rigid analytic space U , r espec tively . Let U ′ denote the underlying r ational sub do- main over C . Let K 2 ( U ) denote the lar gest subgro up of K 2 ( M F ( U )) whic h maps int o K 2 ( U ′ ) under the restrictio n homo morphism K 2 ( M F ( U )) → K 2 ( M ( U ′ )). An F -rational boundar y comp onent of U is a set D ∈ ∂ U suc h that D is the im- age of the open disc o f radius 1 and cent er 0 under an F -linear pro jectiv e linear transformatio n of P 1 . Prop osition 2.12. L et D b e an F -r ational b oundary c omp onent of U , and let k ∈ K 2 ( U ) . Then { k } D ∈ F ∗ .  3. Ka to’s residue hom o morphism Definition 3.1. In this chapter next we will contin ue to use the notation o f the second chapter. In this c hapter we will a lso assume that the a bsolute v alue on F is induced by a discrete v alua tion. Let D denote the op en disc D (0 , 1). Let M b e the field of fractions of O [[ z ]] and let c M deno te the completio n of M with r espect to the discrete v aluation of M defined by the prime ideal m O [[ z ]] of heig h t one, wher e m is the unique maximal ideal of O . The field c M is just the field of bidirectional formal Laurent series of the form P n ∈ Z a n z n ov er F suc h that | a n | is b ounded ab o ve and lim n →−∞ | a n | = 0. It is a lo cal field equipp ed with the absolute v alue k X n ∈ Z a n z n k s = max n ∈ Z | a n | . Every elemen t of the formal Laurent ser ies ring O [[ z ]] defines a holomorphic function on the r ig id ana ly tic space D , hence every element M g iv es a mer omorphic function on D . By W eier strass’ prepa ration theorem each element of O [[ z ]] is the pro duct of a p olynomial and a unit of this r ing, hence it has only a finite nu mber o f zero s in D . Therefore the limit { f , g } D = lim ǫ → 0 0 <ǫ< 1 ǫ ∈| C | { f , g } D (0 , 1 − ǫ ) = Y x ∈ D { f , g } x bec omes stationa ry for any pair of elements f , g ∈ M ∗ and defines an F ∗ -v alued bilinear ma p sa tisfying the Steinberg rela tion by Theorem 2.2 a nd Pro pos ition 2.1 2. Therefore it induces a ho momorphism {·} D : K 2 ( M ) → F ∗ . Note that the rigid analytic regulato r denoted by the s ame symbo l has the same v alue as this pairing for those pair s o f functions for which bo th of them ar e defined b y Theo rem 2 .6. Hence our nota tion will not cause confusion. 8 AMBRUS P ´ AL Prop osition 3.2 . Ther e is a un ique homomorphism {·} D : K 2 ( c M ) → F ∗ , c al le d Kato’s r esidue homomorphism, su ch that ( i ) the c omp osition of the natu r al homomorphism K 2 ( M ) → K 2 ( c M ) and Kato’s r esidue homomorphism is the homomorphi sm {·} D define d ab ove, ( ii ) for e ach f ∈ c M ∗ and g ∈ O [[ z ]] with k 1 − g k s < ǫ < 1 we have { f , g } D ∈ U ǫ . Pro of. Clea rly Kato’s res idue homo morphism is unique if it exits. W e claim that for each f , g ∈ M ∗ with k 1 − g k s < ǫ < 1 we have { f , g } D ∈ U ǫ . W e first show that this cla im implies the prop osition. In this case we may define { f , g } D for any two elements f and g of c M ∗ as the limit lim n →∞ { f n , g n } D , where f n ∈ M ∗ , g n ∈ M ∗ are sequences such that f n → f and g n → g . This limit exits b ecause the seque nce ab ov e is Ca uc hy by the claim ab ov e. Its v a lue is non-zero as 1 = lim n →∞ { f n , g n } D · { g n , f n } D = lim n →∞ { f n , g n } D · lim n →∞ { g n , f n } D . It is also indep endent of the s equences chosen as any t wo sequences may b e combed together to show that they give the same limit. The map { · , ·} D defined this wa y is a utomatically a bilinea r map sa tisfying claim ( ii ) and the Steinberg relatio n, hence the existence follows. F or every 1 > δ ∈ | C | sufficien tly c lo se to 1 the holomorphic functions f a nd g ar e elements of O ∗ ( A δ ), where A δ is the annulus A δ = { z ∈ C || z | = δ } . W rite 1 − g = P n ∈ Z a n z n as an element of c M . This p ow er series will conv erge for all z ∈ A δ when δ sufficiently close to 1 , hence there is a nu mber 0 < ρ < 1 and a neg ativ e int eger N such that | a n | ≤ ǫρ − n for all n < N . F or all δ ∈ | C | s uch that ρ < δ < 1 we hav e the following estimate fo r the supremum norm k 1 − g k on the annulus A δ : k 1 − g k ≤ max( k X n 0 and let W ( k ) denote the ring of Witt vectors of k of infinite length. F or every pair of natural num be rs n and m let B n,m denote the lo cal Artinia n algebra : B n,m = W ( k )[[ x 1 , . . . , x n ]] / ( { p J (0) · n Y j =1 x J ( j ) j | J : { 0 , 1 , . . . , n } → N , n X j =0 J ( j ) > m } ) . F or every A ∈ C there is a sur jectiv e lo cal homomorphism B n,m → A for some n and m . By r epeating the arg ument ab ov e we may reduce the pro of of the pr opo sition to show that there is a ho momorphism φ : B n,m → B 0 ,m such that φ ( x ) 6 = 1 for every x ∈ B ∗ n,m with x ≡ 1 mo d m k − 1 , but x 6≡ 1 mo d m k for some p ositive int eger k ≤ m , where m is the maximal ideal o f B n,m . Every lo cal homomorphism φ : B n,m → B 0 ,m induces a k -linea r homomo rphism T l φ : m l / m l +1 → n l / n l +1 for any po sitiv e l ≤ m , wher e n is the maximal prop er idea l of B 0 ,m . Let T p , T ⊥ denote the k -linear subspace o f m / m 2 generated by p and the element s x 1 , x 2 , . . . , x n , resp ectively . The m ultiplication induces a natural isomor phism: m k / m k +1 = k M j =0 S ym j ( T p ) ⊗ S y m k − j ( T ⊥ ) and another b etw een S y m k ( n / n 2 ) and n k / n k +1 . Moreo ver there is a c a nonical isomorphism ι : T p → n / n 2 betw een these o ne-dimensional vector spaces. Under these identifications w e hav e T k φ = k M j =0 S ym j ( ι ) ⊗ S y m k − j ( T 1 φ | T ⊥ ) . Since every k -linear ma p h : T ⊥ → n / n 2 ∼ = k is induced by a lo cal homomor phism φ : B n,m → B 0 ,m , the pro positio n follows fro m the lemma ab ov e .  Definition 4.6. Let A ∈ C be a lo cal Artinian r ing with maximal idea l m and let f b e any element of A (( t )) ∗ . Then there is an integer w ( f ) ∈ Z , and a seq uenc e of elements a i ∈ A indexed by the int egers such that a 0 ∈ A ∗ , a − i ∈ m for i > 0, a − i = 0 for i sufficiently lar g e, and f = a 0 · t w ( f ) · ∞ Y i =1 (1 − a i t i ) · ∞ Y i =1 (1 − a − i t − i ) , and these are uniquely deter mined by f . The integer w ( f ) is called the winding nu mber of f and the elements a i are called the Witt co ordinates of f . Let f , g ∈ A (( t )) ∗ be arbitrary with winding num b ers w ( f ), w ( g ) a nd Witt co ordina tes a i , b j , resp ectively . By definition the Co n tou-Carr` er e symbol h f , g i is : h f , g i = ( − 1) w ( f ) w ( g ) a w ( g ) 0 Q ∞ i =1 Q ∞ j =1 (1 − a j / ( i,j ) i b i/ ( i,j ) − j ) ( i,j ) b w ( f ) 0 Q ∞ i =1 Q ∞ j =1 (1 − a j / ( i,j ) − i b i/ ( i,j ) i ) ( i,j ) ∈ A ∗ . ON THE KERNEL AND THE IMAGE OF THE RIGID ANA L YTIC REGULA TOR 13 Obviously a ll but finitely many terms are equal to one in the infinite pro duct ab ov e , hence the Co n tou-Carr` er e is a well-defined a lternating map: h· , ·i : A (( t )) ∗ × A (( t )) ∗ → A ∗ . It is a lso clea r from the formula tha t the Contou-Carr` ere s y m bo l is equal to the tame symbol if A is a field. Prop osition 4 .7. The Contou-Carr` er e symb ol is a bi line ar map satisfying the Steinb er g r elation. Pro of. Be c a use k is p e r fect for e very ob ject A of C ther e is a lo cal Artinian ring B with residue field k , where the latter is the algebra ic closure of k , and an injective lo cal homomor phism i : A → B . Indeed the algebra B = A ⊗ k k and B = A ⊗ W ( k ) W ( k ) will do, when k has characteris tic zero or p ositive characteristic, resp ectively , using the fact has A can b e equipp ed with the structure o f a k -a lgebra or W ( k )-algebr a, res pectively , by Lemmas 4.2 and 4.3 . Hence we may assume that k is alg ebraically closed. Let D denote the sub class of those lo cal Ar tinian rings with residue field k which s atisfy the claim of the pro positio n a b ov e. Clear ly we only hav e to show that this sub class satisfies the co nditions of P rop osition 4.4 . If A ∈ C is the quotient of a discrete v a lua tion ring R with res idue field k , we may assume that R is complete with resp ect to its v a luation. Let K b e the q uotien t field of R and let C be the completion of the algebr aic closure of K . The latter is an alg ebraically c losed field co mplete with resp ect to an ultrametric a bsolute v alue. F or every f ∈ A (( t )) ∗ there is a lift ˜ f ∈ R (( t )) ∗ whose ima ge is f under the functorial map R (( t )) → A (( t )). By Lemmas 3.4 and 3 .5 the Contou-Carr` ere symbol of f and 1 − f is the reduction o f rigid ana lytic r egulator { ˜ f , 1 − ˜ f } D ∈ R mo dulo the maximal ideal of R , hence the Contou-Carr` ere symbol sa tisfies the Stein b erg relation. A similar a rgument shows that it is a lso bilinea r, therefore ( i ) of Pro p osition 4.4 ho lds for D . Prop erty ( ii ) also follows from same reasoning, bec ause every f ∈ A (( t )) ∗ has a lift ˜ f ∈ B (( t )) ∗ if the map B → A is surjective. Finally let A ∈ C b e an alg ebra which satisfie s the condition in ( iii ) o f Pr opo sition 4.4. Assume that there is an 1 6 = f ∈ A (( t )) ∗ such that h f , 1 − f i 6 = 1. Then there is a B ∈ D a nd a homomor phism φ : A → B such that 1 6 = φ ( h f , 1 − f i ) = h φ ∗ ( f ) , 1 − φ ∗ ( f ) i 6 = 1 , wher e φ ∗ : A (( t )) → B (( t )) is the functorial map induced by φ , which is a contradiction. A similar argument shows that the Co n tou-Carr` er e symbol is bilinear ov er A , therefore pr oper t y ( iii ) also holds for D .  Let x ∈ A [[ t ]] b e an uniformizer, which means that x is of the form ct + t 2 h , where c ∈ A ∗ and h ∈ A [[ t ]]. In this cas e there is a unique A -a lgebra automor - phism φ : A [[ t ]] → A [[ t ]] such that φ ( t ) = x . On the o ther hand every A -algebra automorphism o f A [[ t ]] is of this form. The a utomorphism φ extends uniquely to an automorphism φ : A (( t )) → A (( t )) by lo calizing at the maximal ideal. Prop osition 4.8. The automorphism φ le aves the Contou - Carr ` er e symb ol invari- ant. Pro of. As in the pro of ab ov e we may assume that k is algebraica lly clos ed. Let D again deno te the sub class of those lo cal Artinia n rings with residue field k which satisfy the claim of the pr o pos ition ab o ve. W e need to show only that this sub class satisfies the conditio ns o f Prop osition 4 .4. Let ψ : B → A b e a s urjectiv e homo- morphism of lo cal algebr as with r esidue field k and let ψ ∗ : B (( t )) → A (( t )) b e 14 AMBRUS P ´ AL the functoria l map induced by ψ . If B is Ar tinia n or a discrete v aluatio n r ing then there is a B -a lgebra a utomorphism φ B : B (( t )) → B (( t )) s uc h that ψ ∗ ◦ φ B = φ ◦ ψ ∗ which is of type describ ed befor e Prop osition 3.3 if B is a discr ete v a lua tion ring. Hence ( i ) and ( ii ) of Prop osition 4.4 holds for D by Prop osition 3.3 . A similar argument as ab ov e shows that condition ( ii i ) a lso holds for D .  Notation 4 .9. Let A ∈ C b e a lo cal Artinian ring and let π : X → Sp ec( A ) b e a pro jectiv e fla t mor phism who se fib er X 0 ov er the unique closed p oint of Sp ec( A ) is a reduced, connected, regular curv e ov er k . Let S be a finite set o f closed po in ts of X (or, equiv alently , of X 0 ) and let f and g b e tw o r ational functions on X which ar e in vertible on the complement o f S . F or every s ∈ S cho ose an A -algebra iso mo rphism φ s betw een the co mpletion b O s,X of the stalk O s,X of the structure sheaf of X at s and A [[ t ]]. The la tter induces an iso morphism b etw een the lo calization o f b O s,X by the semigroup of thos e elements whose image under the canonic a l map b O s,X → b O s,X 0 is non-zero , wher e b O s,X 0 is the the co mpletio n of the stalk O s,X 0 , a nd A (( t )), which will be denoted b y φ s , by the usual abuse o f notation. Let h f , g i s denote the Contou-Carr` ere symbol of the ima ge of f a nd g under φ s for every s in S . By P rop osition 4 .8 the v alue of h f , g i s is indep enden t of the choice of the isomo r phism φ s , so the symbol h f , g i s is well-defined. The following re s ult is the r ecipro cit y law of Anderso n a nd Romo (see [2]). Prop osition 4.1 0. The pr o duct of al l the Contou-Carr ` ere symb ols of f and g is e qual to 1: Y s ∈ S h f , g i s = 1 . Pro of. W e a re going to use the sa me strategy for pro of as we used befor e: in par- ticular we assume that k is alg ebraically closed. Let D deno te the sub class of tho se lo cal Artinian rings with residue field k which sa tisfy the claim of the pro positio n ab o ve. W e will s ho w that this s ubclass satisfies the conditions of Pro pos ition 4.4 . If A ∈ C is the quotient o f a disc r ete v aluation ring R with residue field k , we may again assume that R is complete with resp ect to the v aluation. Let K denote the quotient field of R a nd let C denote the completion of the alg ebraic closur e of K as ab o ve. Because the deformation theo ry of regula r pro jective curves is unobstructed, there is a for ma l scheme X ov er the formal sp ectrum o f R who se fib er ov er Sp ec( A ) is X . By the a lgebraicity theorem of Grothendieck X is actually the formal com- pletion of a smo oth curve ov er Sp ec( R ) which will b e denoted by X by a buse of notation. By flatness there a re rationa l functions ˜ f and ˜ g on X whose res triction to the fib er ov er Sp e c( A ) is f and g , resp ectively . The r ig id a nalytic domain D s of C -v alued p oints of X which r educes to s is isomo rphic to the op en disc D by the formal inv erse function theor em. By Lemmas 3.4 and 3.5 the Co n tou-Carr` er e symbol o f f and g is the r e duction of rigid analytic r egulator the pro duct of the tame s ym b ols { ˜ f , ˜ g } x mo dulo the max imal ideal of R where x is running throug h the C -v alued p oints of set D s . The rationa l functions ˜ f and ˜ g has only p oles or zeros in the union of the sets D s hence the recipro city law o f Anderson and Romo holds by W eil’s recipro city law. Prop erty ( ii ) also follows fro m same rea soning. If the map B → A is surjective and X , f a nd g are as a b ov e, then there is a simila r triple ˜ X , ˜ f and ˜ g ov er Sp ec( B ) such that X is the fiber of ˜ X over Spec ( A ) is X a nd the re s triction o f ˜ f a nd ˜ g to ON THE KERNEL AND THE IMAGE OF THE RIGID ANA L YTIC REGULA TOR 15 X is f a nd g , re s pectively , b ecause the defor ma tion theory o f X is unobstructed. If B ∈ D then the claim holds fo r the triple ˜ X , ˜ f , ˜ g , s o it must ho ld for the triple X , f , g as well. Finally let A ∈ C b e an a lg ebra which satisfies the condition in ( i ii ) of Prop osition 4.4 . Assume that there are ra tional functions f and g as ab ov e such that Q s ∈ S h f , g i s 6 = 1 . Then there is a B ∈ D and a homomo rphism φ : A → B such tha t 1 6 = φ ( Q s ∈ S h f , g i s ) = Q s ∈ φ ∗ ( S ) h φ ∗ ( f ) , φ ∗ ( g ) i s , where φ ∗ ( f ), φ ∗ ( g ) and φ ∗ ( S ) are the base change o f the corres ponding ob jects on the curve φ ∗ ( X ) which is the base change of X with r espect to the map φ ∗ : Sp ec( B ) → Sp ec( A ). This is a contradiction, therefore prop erty ( iii ) als o holds for D .  Remark 4. 11. It is p ossible to push the metho ds o f this pap er a bit further to actually give a pro of of W eil’s re c ipro c it y law itself by r educing it to the cas e of Mumford curves, when it fo llo ws from ( iii ) of Theo rem 2 .6 at once. W e will only sketc h this ar gumen t be c ause it uses a c onsiderable a moun t o f ma c hinery compar e d to the rela tively element ary nature o f W eil’s recipro city law. F o r any s c heme S and any s table cur v e π : C → S of genus g let ω C /S denote relative dualizing sheaf. By Theorem 1.2 of [8], pag e 77 the functor which assig ns to each scheme S the set of stable curves π : C → S , and an isomorphism P ( π ∗ ( ω ⊗ 3 C /S )) ∼ = P 5 g − 6 S (mo dulo isomorphism) is represented by a fine mo duli scheme H g . By Cor ollary 1.7 of [8], page 8 3 and the main r esult of [8], page s 92-96, the scheme H g is s moo th ov e r the sp ectrum of Z a nd the base change ( H g ) Sp ec( k ) is irreducible for any algebr aically closed field k . Let X smo oth, pro jective curve ov er k and let f and g b e t wo non-zero rational functions on X . W e may as sume that the genus g o f X is at least t wo by ta king a cov er of X a nd pr o ving the recipr ocity law for the pull-back of f and g instead. Let x b e a k -v alued p oint o f H g such that the underly ing curve is X a nd let y be ano ther k -v alued po in t such that the underlying cur v e is tota lly degenerate. Since ( H g ) Sp ec( k ) is a n irr educible, smo oth quasi-pro jective v ar iet y , rep eated applica tion o f Bertini’s theorem shows that there is a smo oth, ir reducible curve S mapping to ( H g ) Sp ec( k ) whose image co n tains b oth x a nd y . L et π : C → S be the pull- bac k of the universal family . There are ra tional functions ˜ f and ˜ g o n C whose res triction to the fib er ov er x , which is X , are f and g , resp ectively . Since the bas e change of C to the sp ectrum of the lo cal field of S at y is a Mumford curve, W eil’s r ecipro cit y law holds for ˜ f a nd ˜ g , hence holds for f a nd g , to o. O ne may say that this pro of is close in spir it to the classical pro of of the rec ipro c it y law ov er the complex num b ers using triangulatio n, since it decomp oses the curve to small pieces in a suitable top olog y . 5. The differential reciprocity l a w Definition 5.1. W e will co n tin ue to use the no tation of the pr evious chapter. F or every k -a lg ebra A let Ω · A denote the graded differential algebra of k -linear K ¨ a hler differential forms of A and for every k -algebra homomorphism h : A → B let Ω k ( h ) : Ω · A → Ω · B induced by h by functor ialit y . Every ω ∈ Ω k A (( t )) can b e written uniquely in the form: ω = m X i =1 β i dt t i + ω 0 where m is a natural num b er, β i ∈ Ω k − 1 A and ω 0 ∈ Ω k A [[ t ]] + A (( t ))Ω k A . Let Res k ( ω ) ∈ Ω k − 1 A denote the element β 1 . W e get a map Res k : Ω k A (( t )) → Ω k − 1 A which is called 16 AMBRUS P ´ AL the residue. Prop osition 5 .2. The fol lowing holds: ( i ) we have Res k + i ( αω ) = α Res k ( ω ) for every α ∈ Ω i A and ω ∈ Ω k A (( t )) , ( ii ) we have Ω k − 1 ( h ) ◦ Res k = Res k ◦ Ω k ( h ′ ) wher e h : A → B is a k -algebr a homomorph ism and h ′ : A (( t )) → B (( t )) is the c orr esp onding k -al gebr a homomorph ism induc e d by functoriality, ( iii ) we have Res k ( ω ) = 0 for every ω ∈ Ω k A [[ t ]] and for every ω ∈ Ω k A [ 1 t ] , ( iv ) the map Res k do es not dep end on choic e of the uniformizer t . Pro of. O ur metho d of proving the firs t t wo claims is the sa me. Using the notation of Definition 5.1 we hav e: αω = m X i =1 αβ i dt t i + αω 0 . Because αβ i ∈ Ω k + i − 1 A and αω 0 ∈ Ω k + i A [[ t ]] + A (( t ))Ω k + i A we have Res k + i ( αω ) = αβ 1 by definition so cla im ( i ) is true. O n the o ther hand: Ω k ( h ′ )( ω ) = m X i =1 Ω k − 1 ( h )( β i ) dt t i + Ω k ( h ′ )( ω 0 ) where Ω k − 1 ( h )( β i ) ∈ Ω k − 1 B and Ω k ( h ′ )( ω 0 ) ∈ Ω k B [[ t ]] + B (( t ))Ω k B . Therefor e we hav e Res k (Ω k ( h ′ )( ω )) = Ω k − 1 ( h )( β 1 ) as claim ( ii ) says. The fir s t ha lf of c la im ( iii ) is immediate from the definition of the r esidue. In or der to prove the seco nd half we only need to show the identit y Res 1 ( dt − n ) = 0 for a ll n ≥ 1 by the Ω · A - linearity of the r esidue sp elled o ut in cla im ( i ). But the latter is o b vious. Claim ( iv ) means the following: let x ∈ A [[ t ]] b e a n unifor mizer, which means that x is of the for m tu , where u ∈ A [[ t ]] ∗ . In this case ther e is a unique A -algebra automorphism φ : A [[ t ]] → A [[ t ]] such that φ ( t ) = x as we already saw when w e prepared to for m ulate Prop osition 4.8. The automor phism φ e xtends uniquely to a n automorphism φ : A (( t )) → A (( t )) by lo calizing at the maximal ideal. Cla im ( i v ) means that the equation Res k ◦ Ω k ( φ ) = Res k holds. Because the homomorphis m H k ( φ ) maps Ω k A [[ t ]] and A (( t ))Ω k A int o itself we only need to s ho w that Res 1 ( dx x ) = 1 and Res 1 ( dx x n ) = 0 for a ll n ≥ 2 by Ω · A -linearity . These ident ities follow at once from the same t yp e of identit y in Prop osition 5’ in [2 0] on pag e s 20 -21 b y the pr inc iple of pr olongation of alg ebraic ident ities quoted in the pro of o f the prop osition just mentioned ab ov e. (Or one may use the functor ial argument explained in the r emark following the pro of o f Prop osition 5’ in [20 ] instea d.)  Notation 5.3. F or every k -algebra B let dl og : B ∗ → Ω 1 B denote the logar ithmic differential given b y the rule dl og ( u ) = u − 1 du for every u ∈ B ∗ . Let more over dl og 2 denote the Z -bilinear pairing: dl og 2 : B ∗ ⊗ B ∗ → Ω 2 B given by the rule: dl og 2 ( a, b ) = dl og ( a ) dlog ( b ) ( ∀ a ∈ B ∗ , ∀ b ∈ B ∗ ) . Recall that for every ob ject A of C the symbol h· , ·i denotes the Contou-Carr` er e symbol. ON THE KERNEL AND THE IMAGE OF THE RIGID ANA L YTIC REGULA TOR 17 Prop osition 5 .4. The fol lowing diagr am c ommutes: A (( t )) ∗ ⊗ A (( t )) ∗ dlog 2 − − − − → Ω 2 A (( t )) h· , ·i   y   y Res 2 A ∗ dlog − − − − → Ω 1 A for every obje ct A of C . Pro of. By bilinea r it y and a n tisymmetry o f the Contou-Carr` ere symbo l and the map dl og 2 it will b e sufficient to prove for every pair o f in tegers n , m a nd element a , b ∈ A the following identities: ( i ) Res 2 ( dlog 2 (1 − at n , 1 − bt m )) = 0, if n , m > 0 , ( ii ) Res 2 ( dlog 2 (1 − at − n , 1 − bt − m )) = 0, if a , b ∈ m and n , m > 0 , ( iii ) Res 2 ( dlog 2 ( at n , 1 − bt m )) = 0, if a ∈ A ∗ and m > 0 , ( iv ) Res 2 ( dlog 2 ( at n , 1 − bt m )) = 0, if a ∈ A ∗ , b ∈ m a nd m < 0, ( v ) Res 2 ( dlog 2 ( at n , bt m )) = m ( da/a ) − n ( db/b ), if a , b ∈ A ∗ , ( v i ) Res 2 ( dlog 2 (1 − at n , 1 − bt − m )) = dl og ((1 − a m/ ( n,m ) b n/ ( n,m ) ) ( n,m ) ), if b ∈ m and n , m > 0, ( v ii ) Res 2 ( dlog 2 (1 − f t M n +1 , 1 − bt − n )) = 0, if f ∈ A [[ t ]], b ∈ m and n > 0, where m is the ma x imal pr oper ideal of A and M is a po sitiv e integer s uc h that m M = 0. No te that dl og (1 − at n ) = − ( dat n + nat n − 1 dt )(1 + at n + a 2 t 2 n + · · · ) lies in Ω 1 A [[ t ]] , if a ∈ A and n > 0, and lies in Ω 1 A [ 1 t ] , if a ∈ m a nd n < 0. Hence the first tw o iden tities fo llo w fr om claim ( iii ) of Pr opo sition 5.2. F or every a ∈ A ∗ and b ∈ A we have: dl og 2 ( at n , 1 − bt m ) = − (1 + bt m + b 2 t 2 m + · · · )( da a + n dt t )( dbt m + mbt m − 1 dt ) =( ndb − mb a da )( t m − 1 dt + bt 2 m − 1 dt + b 2 t 3 m − 1 dt + · · · ) + ω 0 where ω 0 ∈ A (( t ))Ω 2 A when either m > 0 or when b ∈ m a nd m < 0. The first summand in the seco nd line lies in Ω 2 A [[ t ]] , if m > 0 , and lies in Ω 2 A [ 1 t ] , if b ∈ m and m < 0. Hence its residue is zero so identities ( ii i ) and ( iv ) are true. F or every a , b ∈ A ∗ we hav e: dl og 2 ( at n , bt m ) = ( da a + n dt t )( db b + m dt t ) = ( m da a − n db b ) dt t + ω 0 where ω 0 ∈ Ω 2 A so identit y ( v ) is clear . By definition: dl og ((1 − a m/ ( n,m ) b n/ ( n,m ) ) ( n,m ) ) = − ma m/ ( n,m ) b n/ ( n,m ) da a + na m/ ( n,m ) b n/ ( n,m ) db b 1 − a m/ ( n,m ) b n/ ( n,m ) for every a ∈ A , b ∈ m and n , m > 0. W e also hav e: (1 − at n ) − 1 (1 − b − m ) − 1 = X k ∈ Z X i,j ∈ N in − j m = k a i b j t k 18 AMBRUS P ´ AL for suc h a and b . Because i n − j m = m − n for any i, j ∈ N if and only if i + 1 = l m/ ( n, m ) and j + 1 = l n ( n, m ) for so me l ∈ N we hav e: (1 − at n ) − 1 (1 − b − m ) − 1 = ( ab ) − 1  ∞ X l =1 a lm/ ( n,m ) b ln/ ( n,m )  t m − n + ( r + s ) t m − n for some r ∈ 1 t A [ 1 t ] and s ∈ tA [[ t ]]. Hence w e have: dl og 2 (1 − at n , 1 − bt − m ) = ( dat n + nat n − 1 dt )( dbt − m − mbt − m − 1 dt ) (1 − at n )(1 − b t − m ) = − ( mbda + nadb ) t n − m − 1 dt (1 − at n )(1 − b t − m ) + ω 0 = − a m/ ( n,m ) − 1 b n/ ( n,m ) − 1 ( mbda + nadb ) t − 1 dt 1 − a m/ ( n,m ) b n/ ( n,m ) + ω 0 + ω 1 where ω 0 ∈ A (( t ))Ω 2 A and ω 1 ∈ Ω 2 A [ 1 t ] + Ω 2 A [[ t ]] . Identit y ( v i ) is now obvious. Finally consider the la st identit y . Note that dl og (1 − bt − n ) = − ( dbt − n − nbt − n − 1 dt )(1 + b t − n + b 2 t − 2 n + · · · + b M − 1 t (1 − M ) n ) bec ause b M = 0 by a ssumption, a nd dl og (1 − f t M n +1 ) = − ( d f t M n +1 + ( M n + 1) f t M n dt )(1 + f t M n +1 + f 2 t 2 M n +2 + · · · ) hence dl og 2 (1 − f t M n +1 , 1 − bt − n ) =( d f t + ( M n + 1) f dt )( db − n bt − 1 dt ) g =( td f db − nbd f dt + ( M n + 1) f dtdb ) g where g ∈ A [[ t ]]. The claim is now clear.  Definition 5.5 . Let L b e a field complete with resp ect to a discr ete v aluation and let R , m denote its discr ete v aluation ring and the maximal idea l of R , r espectively . Assume that the residue field of L is k and the quotient map R → k has a sec- tion which is a ring homomorphism. The latter equips L and R with a k -a lgebra structure. Let b Ω · L denote the g r aded differential alg e bra which is the quotient of the complex Ω · L by the homoge ne o us ideal generated by ∩ n ≥ 1 m n Ω · R and let b Ω k R denote the ima g e o f Ω k R in b Ω d L under the quo tien t map. F or every na tural n umber n let R n denote the truncated ring R / m n +1 and for every pair m ≤ n o f natural nu mbers let π n : R → R n and π n,m : R n → R m denote the c anonical pro jection. The sy s tem mo dules { Ω k R n } n ∈ N form a co mpatible system with resp ect to the mor - phisms Ω k ( π n,m ) ( m ≤ n ) hence it has a pro jectiv e limit lim ← − n →∞ (Ω k R n ). The maps Ω k ( π n ) : Ω k R → Ω k R n factor through b Ω k R and their limit induces an identification: b Ω k R ∼ = lim ← − n →∞ (Ω k R n ) which we will use without further notice. Let dl og : L ∗ → b Ω 1 L and dl og 2 : K 2 ( L ) → b Ω 2 L also deno te the compo sition of dlog , dl og 2 an the quotient map Ω 1 L → b Ω 1 L , Ω 2 L → b Ω 1 L , resp ectively . ON THE KERNEL AND THE IMAGE OF THE RIGID ANA L YTIC REGULA TOR 19 Definition 5.6. Let π b e a uniformizer of L and let b Ω k L ( log ) denote the subgr oup π − 1 b Ω k R of b Ω k L . Clea rly the group b Ω k L ( log ) is independent of the choice of the uni- formizer π . L e t O de no te the dis crete v aluation ring k [[ x ]] a nd let F denote its quotient field. Let c M denote the field attached to F which was introduced in Defi- nition 3.1 and let R denote the v a luation ring o f c M . The unifor mizer x of F is als o a unifomizer in c M . There is a natural is omorphism R n ∼ = O n (( z )) for every n ∈ N , where z denotes als o the reduction o f z in R n for every n by slig htly extending the nota tion introduce d in Definition 3.6, therefore for every ω ∈ Ω k R n the r e sidue Res k ( ω ) ∈ Ω k − 1 O n is well-defined. F or every ω ∈ b Ω k c M ( log ) let Res k ( ω ) ∈ b Ω k − 1 F be given by the rule: Res k ( ω ) = 1 x lim ← − n →∞ (Res k ( b Ω k ( π n )( xω ))) where the map b Ω k ( π n ) : b Ω k R → Ω k R n is induced by Ω k ( π n ). The system: { Res k ( b Ω k ( π n )( xω )) } n ∈ N satisfies the co mpatibilit y descr ibed ab ov e by cla im ( ii ) of P rop osition 5.2 hence Res k ( ω ) is well-defined. B e cause o f the O n -linearity of the res idue it is obvious that Res k ( ω ) is indep enden t of the choice of x a s the notation indicates. Remark 5. 7. Let φ : c M → c M b e a v aluation-pr e serving F -a lgebra automorphism. Then ther e is a unique map b Ω k ( φ ) : b Ω k c M → b Ω k c M such that b Ω k ( φ ) ◦ q k = q k ◦ Ω k ( φ ) where q k : Ω k c M → b Ω k c M is the quotient map. The automo rphism b Ω k ( φ ) of b Ω k c M preserves the subgroup b Ω k c M ( log ) and it co mm utes with the residue map Res k by claim ( iv ) of P rop osition 5.2 . Theorem 5 .8. We have dl og 2 ( k ) ∈ b Ω 2 c M for every k ∈ K 2 ( c M ) and the diagr am: K 2 ( c M ) dlog 2 − − − − → b Ω 2 c M ( log ) {· , ·} D   y   y Res 2 F ∗ dlog − − − − → b Ω 1 F is c ommutative wher e { · , ·} D denotes Kato’s r esidue homomorph ism. Pro of. By the linearity of the dl og 2 map we only have to verify the first cla im of the theorem as well as the ide ntit y e xpressed by the commutativ e dia g ram ab o ve for the elements o f any set of generators of K 2 ( c M ). Hence we may assume that k = u ⊗ v whe r e either u , v ∈ R ∗ or u = x and v is a rbitrary . In the first case we have dlog 2 ( k ) ∈ b Ω k R obviously and the identit y holds by Prop osition 5 .4. In the second case we may write v in the form v = x n w for some n ∈ Z and w ∈ R ∗ . Because { x, x } D = 1 and dl og 2 ( x ⊗ x ) = 0 by definition we may ass ume that v = w . The first claim is now obvious. Moreover in this ca se we may write v in the form v = z deg( v ) t for some t ∈ R ∗ such that the reduction t k of t mo dulo x k R lies in O k [[ z ]] ∗ ⊂ R k for every k ∈ N . There fo re dl og ( t k ) ∈ Ω 1 O k [[ z ]] and we have: Res 2 (Ω 2 ( π k )( dx dv v )) =Res 2 (deg( v )Ω 2 ( π k )( dx dz z )) + Res 2 (Ω 2 ( π k )( dx ) dt n t n ) = deg ( v )Ω 1 ( π k )( dx ) for every k ∈ N . The claim now follows from Lemma 3.7.  20 AMBRUS P ´ AL 6. The image and k ernel of the rigid anal ytic regula tor in positive characteristic Notation 6.1. F or every scheme X let K 2 ,X denote the sheaf on X a s so c iated to the pr esheaf U 7→ K 2 ( H 0 ( U, O X )) for the Z ariski-top ology where K 2 ( A ) denotes Milnor’s K -group o f any ring A . Let W n Ω ∗ X denote the de Rham-Witt pro -complex of a n y ringed top os X of F p -algebra s. Moreov er we let F denote the F rob enius morphism o f the de Rham-Witt pro- c omplex. Recall that the log arithmic differ- ent ial dlog 1 : O ∗ X → W n Ω 1 X is defined as the comp osition o f the T eic hm ¨ uller lift O ∗ X → W n Ω 0 X and the differe ntial d : W n Ω 0 X → W n Ω 1 X , where X is the same as ab o ve. The bilinear map of sheav es: dl og 2 : O ∗ X × O ∗ X − → W n Ω 2 X given by the formula: dl og 2 ( f ⊗ g ) = dl og 1 ( f ) dlog 1 ( g ) also satisfies the Steinber g relation dl og 2 ( f ⊗ (1 − f )) = 0 for a ll f ∈ O ∗ X with 1 − f ∈ O ∗ X , hence it induces a map dl og 2 : K 2 ,X → W n Ω 2 X . Moreov er let ν n ( k ) denote the kernel of 1 − F on the degr ee k term W n Ω k X of the de Rham-Witt pro- complex on the top os X . Let W n Ω i X,log denote the ab elian sub-sheaf generated by the image o f dl og i , where i = 1 , 2. It is ea sy to see using the defining rela tions of the de Rham-Witt pr o-complex that W n Ω i X, log lies in ν n ( i ). W e w ill need the following result which is a s p ecial ca se o f the celebrated theor em in [6] due to Blo ch, Gabb er and Kato. Theorem 6 .2. L et F b e a field of char acteristic p . Then the map K 2 ( F ) / p n K 2 ( F ) dlog 2 − − − → H 0 ( F et , W n Ω 2 F et ,log ) is an isomorphism, wher e H 0 ( F et , W n Ω 2 F et ,log ) denotes the gr oup of glob al se ctions of the she af W n Ω 2 F et ,log on t he ´ etale site of the sp e ct rum of F . Pro of. The map is well-defined as W n Ω ∗ is a nnihilated by p n . The map is an isomorphism by Co rollary 2 .8 of [6], page 117 -118.  Notation 6.3. Let k b e a pe rfect field as in the previous tw o chapters. F or every k -scheme X let Ω · X denote the complex of gr aded differential O X -algebra s of k - linear K¨ a hle r differential for ms o n X . Note that the co mplex Ω · X is canonica lly isomorphic to the complex W 1 Ω · X . In particular there is a map dl og 2 : K 2 ,X → Ω · X . F or e v ery k ∈ N a nd for every Car tier divisor D o n X let Ω k X ( D ) denote the shea f Ω k X ⊗ O X O X ( D ). Let i : X − D → X denote the o pen immers ion of the co mplemen t of the supp ort of D into X . Then the pull-ba c k i ∗ Ω k X ( D ) is c a nonically isomorphic to Ω k X − D . Let i ∗ : H 0 ( X, Ω k X ( D )) → H 0 ( X − D , Ω k X − D ) denote the comp osition o f the pull- bac k and this identification, to o. ON THE KERNEL AND THE IMAGE OF THE RIGID ANA L YTIC REGULA TOR 21 Lemma 6.4 . Assu m e that X is a smo oth surfac e over k and D is a n ormal cr oss- ings divisor. Then t he image of the map: dl og 2 : H 0 ( X − D , K 2 ,X ) → H 0 ( X − D , Ω 2 X − D ) lies in the image of the map i ∗ : H 0 ( X, Ω k X ( D )) → H 0 ( X − D , Ω k X − D ) intr o duc e d ab ove. Pro of. The cla im is clearly lo cal on X with resp ect to the Zariski topo logy hence we may ass ume that X is the spectr um of an integral regula r k -a lgebra A . W e may also assume that D has at mos t one singular p oint a nd its branches ar e the zer os of some elements of A . T he lo caliz a tion sequence for K -theory induces a complex: H 0 ( X, K 2 ,X ) − − − − → H 0 ( X − D , K 2 ,X ) T − − − − → L C ∈V (Γ( D )) H 0 ( X − S ( D ) , O ∗ C ) − − − − → L e ∈ S ( D ) Z e which is exa c t at the term H 0 ( X − D , K 2 ,S ), where the second map is the direct sum of tame symbols a long the irr educible c o mponents and the third map is the sum of the maps which as signs e very element of H 0 ( C − S ( D ) , O ∗ C ) to its divisor considered as a zero cycle supp orted on S ( D ) for every C ∈ V (Γ( D )). Let k b e an arbitrar y e lemen t of H 0 ( X − D , K 2 ,X ). Assume first that D is irreducible and let t ∈ A be an element whos e zero sch eme is D . Pick an element u ∈ A whose pull-back to D is equal to the tame symbol T ( k ). By s hrinking X further w e may assume that u ∈ A ∗ . Then T ( k ) = T ( t ⊗ u ) hence dl og 2 ( k − t ⊗ u ) is the pull- back of a differential form o n X by the lo caliza tion seque nce . O n the other hand dl og 2 ( t ⊗ u ) clea rly lies in the image o f the map i ∗ . Assume now tha t D ha s one ordinary double p oint s a nd let t 1 , t 2 ∈ A be tw o elemen ts whose zero s a r e the tw o branches of D . According to the complex a bove ther e is an n ∈ Z such that the v aluation of the restriction of T ( k ) o n to the zero s cheme of t 1 and t 2 at s is n a nd − n , re s pectively . Hence by shrinking X further we may as sume that ther e are u 1 , u 2 ∈ A ∗ such that the r estriction of T ( k ) onto the ze r o s cheme o f t 1 and t 2 is the restriction of u 1 t n 2 and u 2 t − n 1 , resp ectively . Then we have: T ( k ) = T ( t 1 ⊗ t 2 ) n · T ( t 1 ⊗ u 1 ) · T ( t 2 ⊗ u 2 ) and we may argue as ab ov e to conclude the pro of.  The lemma ab ove has the followin g imp ortant cor ollary: because X − D is Zariski- dense in X the map i ∗ : H 0 ( X, Ω k X ( D )) → H 0 ( X − D , Ω k X − D ) is injectiv e. Hence the map dl og 2 has a unique lift: dl og 2 : H 0 ( X − D , K 2 ,X ) → H 0 ( X, Ω 2 X ( D )) which will b e denoted by the same symbol by the usual a bus e o f notation. Prop osition 6.5. Assume that X is a smo oth irr e ducible pr oje ctive surfac e over k and t he field k is finite of char acteristic p . Th en the gr oup H 0 ( X , K 2 , X ) is the extension of a torsion gr oup by its maximal p -divisible su b gr oup. 22 AMBRUS P ´ AL Pro of. Using the notation o f [15] on pages 3 0 7 and 30 9 let H 2 ( X , Z (2)) de no te the pro jectiv e limit lim ← − ( H 0 ( X , ν n (2))). The lo garithmic differ e n tials dl og 2 : K 2 ( X ) → H 0 ( X , ν n (2)) satisfy the obvious co mpatibilit y hence they induce a map dl og 2 : K 2 ( X ) → H 2 ( X , Z (2)) . Let F ( U ) denote the function field of X . Let P denote the set o f prime divisors of X a nd for every P ∈ P let f P denote the function field o f the irr educible curve P . The lo calization sequence for K - theory furnishes an exact seq ue nc e : 0 − − − − → H 0 ( X , K 2 , X ) − − − − → K 2 ( F ( X )) T − − − − → L P ∈P f ∗ P where the second ma p is the direct sum of tame sy m bo ls along the irreducible com- po nen ts. E v ery element k ∈ H 0 ( X , K 2 , X ) of the kernel o f dlog 2 lies in M ( F ( X )) = ∩ n ∈ N p n K 2 ( F ( X )) by the Blo ch-Kato-Gabb er theore m. Since K 2 ( F ( X )) has no p - torsion by T he o rem 1.1 0 o f [23] on pag e 10, the gro up M ( F ( X )) is the ma ximal p -divisible subgroup of K 2 ( F ( X )). If the element l is in M ( F ( X )) ∩ H 0 ( X , K 2 , X ) and k ∈ M ( F ( X )) is its unique p n -th r oo t then T ( k ) is p n -torsion by the lo caliza- tion sequence. But the gro ups f ∗ P has no non-zer o p -tors ion s o k lies in the imag e of H 0 ( X , K 2 , X ). Therefore we ge t that M ( F ( X )) ∩ H 0 ( X , K 2 , X ), the kernel of the map dl og 2 in H 0 ( X , K 2 , X ), is p -div isible. Hence it will b e sufficien t to show that the gr oup H 2 ( X , Z (2)) is to rsion. This is proved in [15] (the c la im itself can b e found on page 335) a lthough the pro of is somewha t dis pers e d over the article. It is a n immediate conseque nc e of Pro positio n 5 .4 of the pap er c ited ab ov e on page 330-3 31, the v a lidit y of W eil’s conjecture s for cr ystalline coho mology (Remark 5.5 of [15] on pa ge 33 1 ), a nd the exact sequence on page 335 of the sa me pap er.  Notation 6.6 . Let A ∈ C be a lo cal Artinian k -algebra and let π : P 1 A → Sp ec( A ) be the pr o jective line over A . Let S b e a finite set of s ections s : Sp e c ( A ) → P 1 A and for every s ∈ S let s 0 denote the k -v alued p oint s 0 : Sp ec( k ) → P 1 k we get from s via base c ha nge. Assume that s 0 is differe nt from t 0 for every pair s , t ∈ S o f different s ections. F o r every s ∈ S cho ose a n A -algebra isomorphism φ s betw een the co mpletion b O s 0 , P 1 A of the stalk O s 0 , P 1 A of the structure shea f of P 1 A at s 0 and A [[ t ]]. The latter induces an isomo r phism b et ween the lo calization L s of b O s 0 P 1 A by the semigro up of those elements whos e imag e under the ca nonical map b O s 0 , P 1 A → b O s 0 , P 1 k is non-zer o, wher e b O s 0 , P 1 k is the completion of the stalk O s 0 , P 1 k , and A (( t )), which will b e denoted by φ s as well. The image of s is a lo cally principal closed subscheme of co dimension o ne in P 1 A for every element s o f S . Let S also denote Cartier divisor which is the sum of these divisors by slig h t a buse of notation. F or e v ery s ∈ S let Res k s denote the co mpositio n of the ma p: H k s : H 0 ( P 1 A − S, Ω k P 1 A − S ) − → Ω k L s Ω k ( φ s ) − − − − → Ω k A (( t )) , where the first arrow is induced by the tautological map P 1 A − S → Sp ec( L s ), and the residue Res k : Ω k A (( t )) → Ω k − 1 A . By claim ( iv ) of Prop osition 5.2 the map: Res k s : H 0 ( P 1 A − S, Ω k P 1 A − S ) − → Ω k − 1 A is indepe ndent of the choice of φ s . Recall that there is a canonica l inclus io n H 0 ( P 1 A , Ω 2 P 1 A ( S )) ⊂ H 0 ( P 1 A − S, Ω k P 1 A − S ). ON THE KERNEL AND THE IMAGE OF THE RIGID ANA L YTIC REGULA TOR 23 Prop osition 6 .7. The se quenc e: 0 → Ω 2 A π ∗ − − − − → H 0 ( P 1 A , Ω 2 P 1 A ( S )) ⊕ s ∈ S Res 2 s − − − − − − → L s ∈ S Ω 1 A Σ s ∈ S ( · ) − − − − − → Ω 1 A → 0 is exact wher e π ∗ is the pul l-b ack with r esp e ct t o t he map π : P 1 A → Sp ec( A ) . Pro of. By base change we may as sume that k is algebra ically closed, which implies that it is infinite. Let R ⊇ S b e any finite set. No te that for every ω ∈ 1 t Ω 2 A [[ t ]] we hav e ω ∈ Ω 2 A [[ t ]] if and only if Res 2 ( ω ) = 0. Therefore H 0 ( P 1 A , Ω 2 P 1 A ( S )) = { ω ∈ H 0 ( P 1 A , Ω 2 P 1 A ( R )) | Res 2 s ( ω ) = 0 ( ∀ s ∈ R − S ) } . Hence it is sufficient to prov e the prop osition a bove for R ins tea d of S . In par tic- ular w e may assume that the p oint at infinit y ∞ ∈ P 1 A lies in S after a suitable automorphism of the A -scheme P 1 A . Let x be the co ordinate function of the affine line A 1 A = P 1 A − ∞ . F or every ∞ 6 = s ∈ S let the same letter denote the unique element o f A such tha t the image of the sec tio n s is the zero scheme of x − s ∈ A [ x ]. Every ω ∈ H 0 ( P 1 A − S, Ω 2 P 1 A − S ) can b e written uniquely in the for m: ω = ω 0 + X s ∈ S −∞ n ( s ) X k =1 ω s,k ( x − s ) k dx + n ( ∞ ) X j =0 ω ∞ ,j x j dx where ω 0 ∈ Ω 2 A , n ( s ), n ( ∞ ) ∈ N and ω s,k , ω ∞ ,j ∈ Ω 1 A . F or every ∞ 6 = s ∈ S we may assume that x − s maps to t with r espect to φ s . Then it is obvious that H 2 s ( η x n dx ) , H 2 s ( η ( x − r ) − n dx ) ∈ Ω 2 A [[ t ]] for every η ∈ Ω 1 A , n ∈ N and s 6 = r ∈ S − ∞ . Ther efore we have ω s,k = 0 for every k > 1 when ω ∈ H 0 ( P 1 A , Ω 2 P 1 A ( S )). W e may assume also that x − 1 maps to t with resp ect to φ ∞ . In this case it is obvious that H 2 ∞ ( ω s, 1 dx x − s ) = − ω s, 1 dt t + η s for some η s ∈ Ω 2 A [[ t ]] for every s ∈ S − ∞ but H 2 ∞ ( ω ∞ ,j x j dx ) = − ω ∞ ,j t − j − 2 dt for every j = 0 , 1 , . . . , n ( ∞ ) s o w e must have: ω = ω 0 + X s ∈ S −∞ ω s, 1 x − s dx. By the a bove Res 2 s ( ω ) = ω s, 1 for every s ∈ S − ∞ and Res 2 ∞ ( ω ) = − P s ∈ S −∞ ω s, 1 so the claim is now obvious.  24 AMBRUS P ´ AL Notation 6 .8. No we a re going to consider the sa me situation that w e lo oked a t in the introduction. Let B b e a smo oth irreducible pro jectiv e curve ov er k a nd let π : X → B b e a reg ular ir reducible pro jective surface fibre d ov e r B such that the fiber X ∞ of X ov er the closed po in t ∞ of B is totally degenerate. Then the ba se change X of X to the completion F of the function field o f B with resp ect to the v aluation corr espo nding to ∞ is a Mumford curve ov er F . Let U ⊂ X be an op en subv a riet y such that its complement is a normal cros sings divisor D which is the preimage of a finite set of closed po in ts of B co n taining ∞ . The base ch ange o f X to the v aluation ring o f F is a semi-stable mo del o f X who se fibe r is X ∞ hence the rigid analytic regula tor { ·} introduce d in Definition 5.12 of [17] supplies a diagram: H 0 ( U , , K 2 , U ) − − − − → H 2 M ( X, Z (2)) {·} − − − − → H (Γ( X ∞ ) , F ∗ ) , where the fir st ho momorphism is induced b y functoriality . This comp osition will be denoted by the symbol {·} as w ell. Definition 6.9. F or e very ω ∈ H 0 ( X , Ω 2 X ( D )) we ar e going to de fine a function Res( ω ) : E (Γ( X ∞ )) → b Ω 1 F as follows. Fix an edge e ∈ E (Γ( X ∞ )) a nd let s ∈ S ( X ∞ ) denote the ima ge of e under the no r malization map. Le t C b e the ir reducible com- po nen t of X ∞ which corr esponds to the or ig inal vertex of e under the ident ification of Notation 1.2. Let b O s, X denote the completion of the sta lk O s, X of the structure sheaf o f X at s and let t ∈ O s, X be an element whose ze ro scheme is the germ of the curve C . Bec ause t generates a prime ideal in b O s, X the latter gives rise to a discrete v aluation o n the quotient field M e of b O s, X . Let c M e denote the completion of M e with resp ect to this v a luation and le t i e : Sp ec( c M e ) → X de no te the tauto logical map. Note that the closur e of the image of the stalk O ∞ ,B of the structure sheaf of B at ∞ in O s, X with re spect to the map induced by π : X → B in b O s, X is isomo r - phic cano nically to the v a lua tion ring O of F . Hence c M e is ca nonically equippe d with the structure of a n F - algebra. Let φ : c M e → c M b e the unique v aluatio n- preserving F -a lgebra homomorphism such that φ ( t ) = x where we contin ue to use the notation o f the previo us chapter. Note that q 2 (Ω( φ )( i ∗ e ( ω ))) ∈ b Ω k c M ( log ) where q k : Ω k c M → b Ω k c M is the quo tien t map. Hence the v a lue: Res( ω )( e ) = Res 2 ( q 2 (Ω( φ )( i ∗ e ( ω )))) ∈ b Ω 1 F is well-defined a nd it is indep enden t of the choice of the e le men t t by Rema rk 5.7 . F or every oriented gra ph G and commutativ e group R le t F ( G, R ) denote the group of functions f : E ( G ) → R . Theorem 6.1 0 . We have Res( dl og 2 ( k )) ∈ H (Γ( X 0 ) , b Ω 1 F ) for every k ∈ H 0 ( U , K 2 , U ) and the diagr am: H 0 ( U , K 2 , U ) {·} − − − − → H (Γ( X ∞ ) , F ∗ ) dlog 2   y   y dlog H 0 ( X , Ω 2 X ( D )) Res − − − − → F (Γ( X ∞ ) , b Ω 1 F ) is c ommutative. ON THE KERNEL AND THE IMAGE OF THE RIGID ANA L YTIC REGULA TOR 25 Pro of. W e are going to show that dlog ( { k } ( e )) = Res ( dlog 2 ( k ))( e ) for every edge e ∈ E (Γ( X ∞ )). Then the theorem will follow immedia tely be c a use { k } is a har - monic co c hain. The identit y a bove follows immediately from Theorem 5.8 and the following alternate descriptio n of the r igid ana lytic regula tor. The pull-back of k with resp ect to i e is an element i ∗ e ( k ) ∈ K 2 ( c M e ). Let φ ∗ : K 2 ( c M e ) → K 2 ( c M ) be the homomor phism induced by φ . Then we have { k } ( e ) = { φ ∗ ( i ∗ e ( k )) } D .  Prop osition 6. 11. L et k b e an element of H 0 ( U , K 2 , U ) such that Res( dlog 2 ( k )) = 0 . Then dl og 2 ( k ) = 0 , to o. Pro of. Let x ∈ F b e a uniformizer. The close d subscheme of B defined by the n -th power of the defining sheaf of ideals of the c losed subscheme ∞ is isomor phic canonically to Spe c( O n ) where O n = O /x n O as in chapter 5. Let i n : Sp ec( O n ) → B b e the closed immersion corr espo nding to this isomor phis m. F or every irreducible comp onen t C ∈ V (Γ( X ∞ )) let C n denote the clo sed subscheme of X defined by the n - th p ow er o f the defining sheaf o f idea ls of the c lo sed subscheme C . Let c n : C n → X be the closed immersion. Then there is a unique morphism p n : C n → Sp ec( O n ) such that c n ◦ π = p n ◦ i n . As an O n -scheme C n is iso morphic to the pro jectiv e line ov er Spec( O n ). Let S denote the Cartier divisor on C n which is the pull-ba c k of the diviso r on X that is the sum o f those irreducible comp onents of X ∞ which are in tersecting C with resp ect to the map c n and are different from C . Then S is the sum of images of sec tio ns of the map p n . Let C 0 be the divisor o f the element x ∈ O n ⊂ H 0 ( C n , O C n ). Mult iplication by x induces a map O ( S + C 0 ) → O ( S ). By o ur assumptions the res idues o f xc ∗ n ( dlog 2 ( k )) ∈ H 0 ( C n , Ω 2 C n ( S )) introduced in Definition 6.6 are all zero. Hence xc ∗ n ( dlog 2 ( k )) ∈ Ω 2 O n by Prop osition 6.7. But Ω 2 O n = 0 hence we get that the fo r mal co mpletio n o f dl og 2 ( k ) along the closed scheme X ∞ m ust b e zer o . The claim is now clear.  Assume now that k is a field o f characteristic p . Corollary 6.12. The map: {·} : H 0 ( U , K 2 , U ) /p n H 0 ( U , K 2 , U ) → H 0 (Γ( X ∞ ) , F ∗ / ( F ∗ ) p n ) induc e d by t he r e gulator { ·} is inje ctive for every natura l n umb er n . Pro of. W e are going to prove the claim by induction o n n . Le t F ( U ) denote the function field of U . Assume first that n = 1 and let k ∈ H 0 ( U , K 2 , U ) be an element such that { k } ∈ H (Γ( X ∞ ) , ( F ∗ ) p ). By Theor em 6.10 we hav e: Res( dl og 2 ( k )) = dl og ◦ { k } = 0 , hence dl og 2 ( k ) = 0 by Prop osition 6.11 . Therefore k = pl for some l ∈ K 2 ( F ( U )) by the Blo ch-Gabbe r -Kato theor em. Let P denote the set of prime divisor s of U and for e very P ∈ P let f P denote the function field o f the irr e ducible cur v e P . The lo calization sequence for K - theory furnis hes an exact sequence: 0 − − − − → H 0 ( U , K 2 , U ) − − − − → K 2 ( F ( U )) − − − − → L P ∈P f ∗ P where the second ma p is the dir ect sum of tame symbols along the irreducible comp onen ts. The image of l with res pect to the s econd map is p -torsio n. But the groups f ∗ P has no non-zer o p -to r sion so l is the image of an element of H 0 ( U , K 2 , U ). 26 AMBRUS P ´ AL Assume now that claim is proved for n − 1 and let k ∈ H 0 ( U , K 2 , U ) b e an element such that { k } ∈ H (Γ( X ∞ ) , ( F ∗ ) p n ). B y the induction hypothesis there is a n element l ∈ H 0 ( U , K 2 , U ) such tha t k = p n − 1 l . Because the group F ∗ has no p - torsion we hav e { l } ∈ H (Γ( X ∞ ) , ( F ∗ ) p ) therefor e l ∈ pH 0 ( U , K 2 , U ) by the ab ov e. Hence k ∈ p n H 0 ( U , K 2 , U ) as we wished to prov e.  Assume now that k is a finite field of c hara cteristic p . Theorem 6 .13. The fol lowing holds: ( i ) the quotient gr oup H 0 ( U , K 2 , U ) /H 0 ( X , K 2 , X ) is a finitely gener ate d ab elian gr oup whose r ank is at most as lar ge as the r ank of the gr oup H (Γ( D ) , Z ) . ( ii ) the kernel K er( {·} ) of the r e gulator {·} : H 0 ( U , K 2 , U ) → H (Γ( X ∞ ) , F ∗ ) has a sub gr oup of finite index which lies in H 0 ( X , K 2 , X ) , ( iii ) the kernel Ker( {·} ) ab ove is p -divisible. It is torsion if Parshin ’s c onje ctur e holds, and it is finite if the Bass c onje ctur e holds, ( iv ) the image Im ( {·} ) of the r e gulator {·} : H 0 ( U , K 2 , U ) → H (Γ( X ∞ ) , F ∗ ) is p -satur ate d, ( v ) the r ank of Im( {·} ) is at most as lar ge as the r ank of t he gr oup H (Γ( D ) , Z ) , ( v i ) the image Im( {·} ) is discr ete if D = X ∞ . Pro of. Let us recall that S ( D ), V (Γ( D )) denote the set of singular p oints and the set of irreducible comp onents of the curve D , resp e ctiv ely . The lo calizatio n sequence for K -theor y induces a complex: H 0 ( X , K 2 , X ) − − − − → H 0 ( U , K 2 , U ) T − − − − → L C ∈V (Γ( D )) H 0 ( C − S ( D ) , O ∗ C ) − − − − → L e ∈ S ( D ) Z e which is ex act at the term H 0 ( U , K 2 , U ), whe r e the second map is the direct sum of tame s ym b ols along the irreducible comp onents a nd the third map is the sum o f the maps which fo r every C ∈ V (Γ( D )) assigns every element of H 0 ( C − S ( D ) , O ∗ C ) to its diviso r co nsidered a s a zero cycle supp orted o n S ( D ). The kernel of the latter is a finitely generated a belian group of r ank H (Γ( D ) , Z ) hence claim ( i ) is clear. By Corolla ry 6.9 the kernel Ker( {·} ) of the ma p: {·} : H 0 ( U , K 2 , U ) → H (Γ( X ∞ ) , F ∗ ) is p -divisible. Therefore its image with res pect to the map T ab ov e is finite b e- cause the maximal p -divisible subgr o up of a finitely gener ated ab elian group is finite. Hence the kernel of T in Ker( {·} ) is a subgro up of finite index which lie in H 0 ( X , K 2 , X ). Therefore cla im ( ii ) holds . According to Parshin’s co njectur e the group H 0 ( X , K 2 , X ) should b e torsion. Then the same is tr ue for Ker( {·} ) ∩ H 0 ( X , K 2 , X ) and therefor e Ker( {·} ) is tor- sion as well by cla im ( ii ). The Bass c onjecture states that H 0 ( X , K 2 , X ) should be a finitely g enerated a belian group. Hence the sa me is tr ue for its subgr oup Ker( {·} ) ∩ H 0 ( X , K 2 , X ). Note that this gro up is also p -divis ible: every ele ment of Ker( {·} ) ∩ H 0 ( X , K 2 , X ) has a p - th ro ot in Ker( {·} ) ⊆ H 0 ( U , K 2 , U ). On the other hand if px ∈ H 0 ( X , K 2 , X ) for some x ∈ H 0 ( U , K 2 , U ) then x ∈ H 0 ( X , K 2 , X ) using the lo calization sequenc e the same way we did in the pro of o f Coro llary 6.12 alre ady . Hence Ker( {·} ) ∩ H 0 ( X , K 2 , X ) is a finite group w ho se order is not divisible by p so Ker( {·} ) is finite a s w ell by cla im ( ii ). Cla im ( ii i ) is now proved. ON THE KERNEL AND THE IMAGE OF THE RIGID ANA L YTIC REGULA TOR 27 Because the maximal p -divisible subgr oup of H (Γ( X ∞ ) , F ∗ ) is finite the image of H 0 ( X , K 2 , X ) with resp ect to the rigid ana lytic regulator is torsion by Pro positio n 6.5. But the torsio n of H (Γ( X ∞ ) , F ∗ ) is finite so Im( {·} ) is finitely genera ted and claim ( v ) is true by claim ( i ). O n the o ther hand note that a finitely g enerated subgroup Λ ⊂ H (Γ( X ∞ ) , F ∗ ) is p -satur ated if and only if p n Λ = Λ ∩ p n H (Γ( X ∞ ) , F ∗ ) for every n ∈ N . The la tter ho lds for Im( { ·} ) by Co rollary 6 .12 so c la im ( iv ) is true. Let Reg : H 0 ( U , K 2 , U ) → H (Γ( X ∞ ) , Z ) denote the tame regulator which is defined as follows. F or every k ∈ H 0 ( U , K 2 , U ) a nd for every edge e ∈ E (Γ( X ∞ )) we define Reg( k )( e ) as the v alua tion of the tame symbo l of k a lo ng the ir reducible comp onen t o ( e ) of X ∞ with resp ect to the v aluation corresp onding to the clo sed po in t whic h is the image of e with re spect to the norma liz ation map. By Theor em 5.6 of [17] the dia gram: H 0 ( U , K 2 , U ) {·} − − − − → H (Γ( X ∞ ) , F ∗ )   y Reg   y v H (Γ( X ∞ ) , Z ) H (Γ( X ∞ ) , Z ) commutes where v is the ma p induced by the normalize d v aluation on F . If D = X ∞ then the kernel of Re g contains H 0 ( X , K 2 , X ) as a subg r oup of finite index acco rding to the complex we wro te down ab ov e. Since H 0 ( X , K 2 , X ) is p -divisible its ima g e with resp ect to the reg ulator {·} is finite. Hence the kernel of the map v in Im( {·} ) is finite, to o. Therefor e Im( {·} ) must be discrete as cla im ( v i ) says.  References 1. M. 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