Harder-Narasimhan categories

Harder-Narasimhan categories Hua yi Chen No v em ber 11, 2021 Abstract W e prop ose a generalization of Quillen’s ex act category — arithmetic exact category and w e disc uss conditions on s uch categories under whic h one can establish th e notion of Harder-Narasimhan filtrations and H arder-Narsimhan p olygons. F urthermore, w e show the functorialit y of Harder-Narasimhan fi ltrations (indexed by R ), which can n ot b e stated in the classical setting of Harder and Narasimhan’s formalism. 1 In tro duction The notion of Har der-Nar asimhan flag 1 (or c anonic al flag ) of a vector bundle on a s moo th pro jective curve o ver a field was firstly intro duced by Harder and Nar asimhan [10] to study the cohomolog y gro ups of mo duli spaces of vector bundles on curves. Let C b e a smo o th pro jectiv e curve on a field k and E b e a non-zer o lo c a lly free O C -mo dule (i.e. v ector bundle) of finite t yp e. Har der and Nar a simhan pr ov ed that there exists a flag 0 = E 0 ( E 1 ( E 2 ( · · · ( E n = E of E such that 1) each sub-quo tien t E i /E i − 1 ( i = 1 , · · · , n ) is semistable 2 in the s ense of Mumfor d, 2) we have the inequality of succes s ive s lop es µ max ( E ) := µ ( E 1 /E 0 ) > µ ( E 2 /E 1 ) > · · · > µ ( E n /E n − 1 ) =: µ min ( E ) . The Harde r -Narasimha n p olyg o n of E is the concav e function on [0 , rk E ], the graph of which is the co nv ex hull of p oints of co or dinate (rk F , deg( F )), where F r uns ov er all coher en t sub- O C -mo dules of E . Its vertexes ar e of co ordina te (rk E i , de g ( E i )). The av atar of the above constructions in Ara kelo v g eometry was intro duced by Stuhler [19] a nd Grayson [9]. Similar constructions exist also in the theory of filter ed iso cristals [7]. Classically , the canonica l flags hav e no functoriality . Notice tha t alrea dy the length of canonical flag s v ar ies when the vector bundle E changes. How ev er, a s we shall show in this article, if we take in to acco un t the minimal slop es of non- z e ro subbundles E i in the ca nonical flag , which coincide with success ive slop es, i.e. µ min ( E i ) = µ ( E i /E i − 1 ), we obtain a filtration indexed by R which we call Har der-Nar asimhan filtr ation . Such constructio n has the functor iality . 1 In most literature this notion is known as “Harder-N ar asimhan filtration”. How ev er, the so-called “Harder- Narasimhan filtration” is indexed b y a finite set, therefore is i n fact a flag of the ve ctor bundle. Here w e would like to reserve the term “Harder-Narasimhan filtration” for filtration indexed by R , which we shall define later in this article. 2 W e say that a non-zero lo cally free O C -mo dule of finite type F is semistable if f or any non-zero sub-mo dule F 0 of F we hav e µ ( F 0 ) ≤ µ ( F ), where the slop e µ is b y definition the quotien t of the degree by the rank. 1 The ca tegory o f vector bundles on a pro jective v ariety is exa c t in the se ns e of Q uillen [16]. How ev er, it is not the case for the category of Hermitian vector bundles on a pro jective arith- metic v ar iety , o r the c a tegory of vector spaces equipp ed with a filtration. W e shall prop os e a new notio n — arithmetic exact ca tegory — which generalizes sim ultaneously the three cases ab ov e. F urther more we shall discuss the conditions on s uch categor ies under which we ca n establish the notio n of semistability a nd furthermore the existence of Harder -Narasimha n fil- trations. W e a ls o show how to asso cia te to such a filtra tio n a Borel proba bilit y measure on R whic h is a linear combination of Dirac mea sures. This construction is a n imp ortant to ol to study Harde r -Narasimha n p olygons in the author’s forthcoming w ork [5]. W e p oint out that the categorical approach for studying se mistabilit y problems has be en developed in v arious context b y different authors, a mong whom we would like to cite Bridgeland [3], Lafforg ue [12 ] and Rudakov [17]. This article is organized as follows. W e intro duce in the second section the forma lis m o f filtrations in an arbitra ry catego ry . In the third section, we prese nt the arithmetic exact cat- egories which gener a lizes the notion of ex act catego r ies in the sense o f Quillen. W e also g ive several e x amples. The fourth se c tion is devoted to the formalis m of Harder and Narasimhan on an arithmetic exa ct categor y equipped with degr ee and ra nk functions, sub ject to certain conditions which we shall precise (such category will be called Harder-Nar asimhan catego ry in this ar ticle). In the fifth sectio n, w e ass o ciate to each arithmetic ob ject in a Harder-Nar asimhan category a filtration indexed by R , a nd we es ta blish the fonctoriality of this constr uction. W e also ex plain how to apply this cons truction to the study of Harder -Narasimhan p olygons. As an application, we g ive a criterion of Harder-Nar asimhan ca tegories when the underlying exact cat- egory is an Ab elian categ ory . The last section contains se veral examples of Harder -Narasimhan categorie s wher e the a rithmetic ob jects are classica l in p-adic r epresentation theory , a lgebraic geometry and Arakelo v geometr y resp ectively . Ac kno wledgement The r esults in this article is the contin uation o f part of the author’s do ctorial thesis sup ervised by J.-B. Bost to whom the author would lik e to express his gr atitude. The author is also thankful to A. Chamber t-Loir, B. Ke ller a nd C. Mouro ugane for rema r ks. 2 Filtrations in a category In this section we shall intro duce the notion of filtratio ns in a ge neral categ ory and their functorial prop erties. Her e we a re ra ther interested in left contin uo us filtra tions. How ever, for the sake of completeness, a nd for po ssible applica tio ns else where, we s hall also discuss the right contin uous counterpart, which is not dual to the left contin uo us ca se. W e fix throug hout this section a non-empty totally or dered set I . Let I ∗ be the extension of I b y adding a minimal element −∞ . The new totally ordered set I ∗ can b e v iewed as a small categ o ry . Namely , fo r any pair ( i, j ) o f ob jects in I ∗ , Hom( i, j ) is a one po int set { u ij } if i ≥ j , a nd is the empty set otherwis e. The comp osition of morphisms is defined in the obvious wa y . Notice that −∞ is the final ob ject of I ∗ . The subset I o f I ∗ can be viewed as a full sub c ategory of I ∗ . If i ≤ j a re two elements in I ∗ , we s hall use the expression [ i, j ]  resp. ] i, j [, [ i, j [, ] i, j ]  to denote the se t { k ∈ I ∗ | i ≤ k ≤ j }  resp. { k ∈ I ∗ | i < k < j } , { k ∈ I ∗ | i ≤ k < j } , { k ∈ I ∗ | i < k ≤ j }  . Definition 2.1 Let C b e a category and X be an ob ject of C . W e call I -filtr ation of X in C any functor F : I ∗ → C such that F ( −∞ ) = X a nd that, for any morphism ϕ in I ∗ , F ( ϕ ) is a monomorphism. 2 Let F and G b e tw o filtrations in C . W e ca ll morphism of filtr ations fr om F to G any natural transformation from F to G . All filtrations in C and all morphisms of filtrations for m a category , denoted by Fil I ( C ). It’s a full subc a tegory of the catego ry o f functors from I ∗ to C . Let ( X, Y ) be a pa ir of ob jects in C , F b e an I - filtration of X and G b e an I -filtra tion of Y . W e s ay that a mo r phism f : X → Y is c omp atible with the filtrations ( F , G ) if there exists a morphism of filtrations F : F → G such tha t F ( −∞ ) = f . If suc h morphism F exists, it is unique since a ll cano nic a l mor phisms G ( i ) → Y are monomorphic. W e say that a filtration F is exhaustive if lim − → F | I exists a nd if the morphism lim − → F | I → X defined by the system ( F ( u i, −∞ ) : X i → X ) i ∈ I is a n isomorphism. W e say that F is sep ar ate d if lim ← − F exists and is an initial ob ject in C . If i is a n element in I , we denote by I i ) the subset of I co nsisting of all elements strictly smaller (r esp. s tr ictly grea ter ) than i . W e say tha t I is left dense (res p. right dense ) at i if I i ) is non-empty a nd if sup I i = i ). The subsets I i can also b e v ie w ed a s full sub categor ies o f I ∗ . The following tw o easy prop o sitions give cr iteria for I to be dense (left and right resp ec- tively) a t a po in t i in I . Prop ositio n 2.2 L et i b e an element of I . The fol lowi ng c onditions ar e e quivalents: 1) I is left dense at i ; 2) I i is non-empty and the set ] i, j [ is non-empty for any j > i ; 3) I >i is non-empty and the set ] i, j [ is infinite for any i > j . W e say that a filtration F is left c ontinuous at i ∈ I if I is not left dense at i or if the pro jective limit of the restriction of F on I i exists and the mo r phism lim − → F | I >i → F ( i ) defined by the system ( F ( u j i ) : F ( j ) → F ( i )) j >i is an isomo rphism. W e say that a filtra tion F is left c ontinuou s (resp. right c ont inuous ) if it is left co ntin uous (resp. rig h t contin uous) at every element o f I . W e denote by Fil I ,l ( C ) (resp. Fil I ,r ( C )) the full sub catego r y of Fi l I ( C ) formed by all left contin uous (resp. rig h t contin uous) filtrations in C . Given an arbitrary filtratio n F , we w ant to construct a left c ontin uo us filtr ation which is “closest” to the original one. The b est ca ndidate is of course the filtratio n F l such that F l ( i ) = ( lim ← − ki exists, and the canonical morphism lim − → j >i F ( j ) → X defined by the s ystem ( F ( u j, −∞ ) : F ( j ) → X ) j >i is mono morphic, then the filtration F r such that F r ( i ) = ( lim − → j >i F ( j ) , I is right dense a t i, F ( i ) , otherwise , is rig ht contin uous. Ther efore, if the following condition ( M ∗ ) is fulfilled for the category C : any non-empty total ly or der e d system ( X i α i / / X ) i ∈ J of sub obje cts of an obje ct X in C has an induct ive limit, and the c anonic al morphism lim − → X i → X induc e d by ( α i ) i ∈ J is monomorphic, then for an y filtration F in C , the filtra tion F r exists, and F 7− → F r is a functor, which is right adjoint to the for getful functor fro m Fil I ,r ( C ) to Fil I ( C ). Let X be an ob ject in C . All I -filtrations of X and a ll mor phisms of filtr ations equalling to Id X at −∞ form a category , denoted by Fil I X . W e denote by Fil I ,l X (resp. Fil I ,r X ) the full sub c ategory of Fil I X consisting of all left contin uous (resp. right co n tinuous) filtrations of X . The categ ory Fi l I X has a final o b ject C X which sends all i ∈ I ∗ to X and a ll morphisms in I ∗ to Id X . W e call it the trivial filtration of X . If the co ndition ( M ) is v erified for the categor y C , the restriction o f the functor F 7− → F l on Fil I X is a functor fro m Fil I X to Fi l I ,l X , whic h is left adjoint to the forgetful functor Fil I ,l X → Fil I X . Similarly , if the condition ( M ∗ ) is v erified for the ca tegory C , the res triction of the functor F 7− → F r on Fil I X gives a functor from Fi l I X to Fil I ,r X , which is right adjo int to the forg etful functor Fil I ,r X → Fil I X . In the following, we shall discuss functorial constructions of filtrations. Namely , given a morphism f : X → Y in a catego ry C and a filtra tion of X o r Y , w e shall e xplain how to construct a “ natural” filtration o f the other . Suppo se that f : X → Y is a mo r phism in C and G is an I -filtra tion o f Y . If the fiber pro duct in the functor categor y F un ( I ∗ , C ), defined b y f ∗ G := G × C Y C X , exists, where C X (resp. C Y ) is the trivial filtra tion of X (res p. Y ), then the functor f ∗ G is a filtration of X . W e call it the inverse image o f G by the mor phism f . The cano nical pro jection P from f ∗ G to G gives a morphism of filtratio ns in Fil I ( C ) such that P ( −∞ ) = f . In other words, the morphism f is co mpatible with the filtrations ( f ∗ G , G ). Since the fib er pro duct co mm utes to pro jective limits, if G is left contin uo us at a point i ∈ I , then also is f ∗ G . If in the categor y C , a ll fib er pro ducts exist 3 , then for any morphis m f : X → Y in C and any filtration G of Y , the inv erse imag e o f G by f exists, and f ∗ is a functor from Fil I Y to Fil I X which sends Fil I ,l Y to Fil I ,l X . Let C b e a catego ry and f : X → Y be a mor phism in C . W e call admissible de c omp osition of f a ny triplet ( Z , u, v ) such that: 1) Z is an ob ject of C , 2) u : X → Z is a morphism in C and v : Z → Y is a monomor phis m in C such that f = v u . 3 In this case, for any small category D , the category of functors f rom D to C supports fiber pr oducts. In particular, all fiber pro ducts i n the category Fi l I ( C ) exist. 4 If ( Z, u, v ) and ( Z ′ , u ′ , v ′ ) ar e tw o admissible decomp ositio ns of f , we call m orphism of admis- sible de c omp ositions from ( Z, u, v ) to ( Z ′ , u ′ , v ′ ) any morphism ϕ : Z → Z ′ such that ϕu = u ′ and that v = v ′ ϕ . Z ϕ   v A A A A A A A A X u ′ A A A A A A A A u > > } } } } } } } } f / / Y Z ′ v ′ > > } } } } } } } All admiss ible decomp ositions and their mor phisms form a catego ry , denoted by Dec( f ). If the category Dec( f ) has an initial o b ject ( Z 0 , u 0 , v 0 ), we say that f has an image . The mo nomor- phism v 0 : Z 0 → Y is called an image o f f , or an image of X in Y by the morphism f , denoted by Im f . Suppo se that f : X → Y is a mo rphism in C and that F is a filtratio n of X . If for any i ∈ I , the morphism f ◦ F ( u i, −∞ ) : F ( i ) → Y ha s a n image, then w e can define a filtr ation f ♭ F of Y , which asso ciates to each i ∈ I the sub ob ject Im( f ◦ F ( u i, −∞ )) of Y . This filtration is ca lle d the we ak dir e ct image of F b y the mor phism f . If fur ther more the filtra tion f ∗ F := ( f ♭ F ) l is w ell defined, we called it the st r ong dir e ct image b y f . Notice that for any filtration F of X , the morphism f is co mpa tible with filtr ations ( F , f ♭ F ) a nd ( F , f ∗ F ) (if f ♭ F and f ∗ F are well defined). Moreov er, if a ny morphism in C ha s an image, then f ♭ is a functor from Fi l I X to Fil I Y . If in addition the co ndition ( M ) is fulfilled fo r the catego ry C , f ∗ is a functor from Fi l I X to Fil I ,l Y . Prop ositio n 2.4 L et C b e a c ate gory which supp ort s fi b er pr o ducts and such that any morphism in it admits an image. If f : X → Y is a morphism in C , then t he fun ct or f ∗ : Fil I Y → Fil I X is right adjoint to the functor f ♭ . Pr o of. Let F be a filtratio n of Y , G b e a filtration of X a nd τ : G → f ∗ F b e a morphism. F or any i ∈ I let ϕ i : F ( i ) → Y and ψ i : G ( i ) → X be cano nical mo r phisms, and let ( f ♭ G ( i ) , u i , v i ) b e an imag e o f G ( i ) by the mor phism f ψ i . Since the morphis m ϕ i : F ( i ) → Y is monomorphic, ther e exists a unique morphism η i from f ♭ G ( i ) to F ( i ) such that ϕ i η i = v i and that η i u i = pr 1 τ ( i ). f ∗ F ( i ) pr 1 / /   F ( i ) ϕ i   G ( i ) τ ( i ) ; ; w w w w w w w w w u i / / ψ i $ $ H H H H H H H H H f ♭ G ( i ) η i ; ; w w w w w w w w v i # # H H H H H H H H H X f / / Y Hence we have a functor ial bijection Hom Fil I X ( G , f ∗ F ) ∼ − → Ho m Fil I Y ( f ♭ G , F ). ✷ Corollary 2.5 With t he notations of the pr evious pr op osition, if we su pp ose in addition that the c ondition ( M ) is verifie d for the c ate gory C , then for any morphism f : X → Y in C , the functor f ∗ : Fi l I ,l Y → Fil I X is right adjoi nt to the functor f ∗ . 5 Pr o of. F or any filtration F of X and any left co n tinuous filtration G of Y , we ha ve the following functorial bijections Hom Fil I X ( F , f ∗ G ) ∼ − → Ho m Fil I Y ( f ♭ F , G ) ∼ − → Ho m Fil I ,l Y ( f ∗ F , G ) . ✷ In the last part o f the section, we shall discuss a sp ecial t yp e of filtra tions, namely filtrations of finite length, which ar e impo rtant for la ter sections. Let C b e a catego ry . W e say that a filtration F of X ∈ ob j C is of finite length if there exists a finite s ubset I 0 of I suc h that, for any i > j satisfying I 0 ∩ [ j, i ] = ∅ , the mor phism F ( u ij ) is isomorphic. The subset I 0 of I is called a jumping set o f F . W e may hav e different choices of jumping set. In fact, if I 1 is an arbitrary finite subset o f I a nd if I 0 is a jumping set o f F , then I 0 ∪ I 1 is also a jumping set o f F . How ever, the in tersection of all jumping se ts of F is itself a jumping set, called the minimal jumping set of F . Let f : X → Y be a morphism in C . If G is a filtration of finite length o f Y such that f ∗ G is well defined, then the filtration f ∗ G is also of finite length since the fibre pr o duct preser ves isomorphisms. Let C b e a ca tegory , X be an o b ject in C and F b e an I -filtration of X . W e say that F is left lo c al ly c onstant at i ∈ I if I is not left dense a t i or if there exists j < i such that F ( u ij ) is an iso morphism, or equiv alently F ( u ik ) is an isomor phism for an y k ∈ [ j, i [. Similarly , we say that F is right lo c al ly c onstant a t i if I is not right dense at i o r if there exists j > i such that F ( u j i ) is an isomorphism, or equiv a len tly , F ( u ki ) is an isomorphism for any k ∈ ] i, j ]. W e say that the filtr a tion F is left lo c al ly c onstant (resp. right lo c al ly c onstant ) if it is left lo cally constant (resp. right lo cally co ns tant ) at an y p oint i ∈ I . Prop ositio n 2.6 L et C b e a c ate gory, X b e an obje ct in C , F b e a filtr ation of finite length of X , and I 0 b e a jumping set of F . F or any i ∈ I \ I 0 , the filtr ation F is left and right lo c al ly c onst ant at i . Pr o of. Let i ∈ I \ I 0 be an element where I is left dense. Since I 0 is a finite se t, also is I k . Define S =  B ϕ ∗ 0 Id F  : E ⊕ F → E ⊕ F . Since R u =  P ϕ  and since ( B P + ϕ ∗ ϕ )( x i ) = p 1 − λ i B x i + λ i x i = ( (1 − λ i ) x i + λ i x i = x i , 1 ≤ i ≤ k , 0 B x i + x i = x i , k < i ≤ n, 12 the diagr am E ⊕ F S   pr 2 ) ) S S S S S S S S E Ru 5 5 k k k k k k k k τ ) ) S S S S S S S S F E ⊕ F pr 2 5 5 k k k k k k k k is commutativ e, wher e τ =  Id E ϕ  . W e equip E ⊕ F with the Hermitian pro duct h· , ·i 0 such that, for a ny ( α, β ) ∈ ( E ⊕ F ) 2 , we have h α, β i 0 =  S − 1 α, S − 1 β  , where h· , ·i is the or thogonal direct sum o f Hermitia n pro ducts on E and on F . Then for any ( x, y ) ∈ E × E , h τ ( x ) , τ ( y ) i 0 = h S Ru ( x ) , S Ru ( y ) i 0 = h Ru ( x ) , R u ( y ) i = h u ( x ) , u ( y ) i = h x, y i . Finally , the kernel of pr 2 is sta ble by the action of S , so the pr o jections of h· , ·i 0 and of h· , ·i by pr 2 are the sa me. ✷ F r o m the pro of of Pr op osition 3.5, we see that a weak er form (the case where k ϕ k < 1) can be generalized to the family c ase, no matter the family of Hermitian metrics is contin uous o r smo oth. Ultranormed space Let k b e a field equipped with a non-Archimedean absolute v alue k · k under which k is complete. W e denote by V ec k the categ ory of finite dimensional vector spaces ov er k , which is clearly an Abelian categ o ry . Let E be the clas s of shor t e x act seq uence in V ec k . F o r any finite dimension vector spac e E ov er k , we denote b y A ( E ) the set of a ll ultranor ms (see [2] for definition) on E . Supp ose that h is an ultrano rm on E . If i : E 0 → E is a subspac e of E , we denote by i ∗ ( h ) the induce d ultra norm on E 0 . If π : E → F is a quotient space o f E , we denote by π ∗ ( h ) the quotient ultranor m on F . Then ( V ec k , E , A ) is an arithmetic exac t categor y . In particular, the ax iom ( A 7) is justified by the fo llowing prop ositio n, which can b e generaliz e d without any difficult y to Banach space cas e or family case. Prop ositio n 3.6 L et ϕ : E → F b e a line ar map of ve ctor sp ac es over k . Supp ose t hat E and F ar e e quipp e d re sp e ctively with t he ultra norms h E and h F such that k ϕ k ≤ 1 . If we e quip E ⊕ F with the u ltr anorm h such that, for any ( x, y ) ∈ E ⊕ F , h ( x, y ) = max( h E ( x ) , h F ( y )) , then in the de c omp osition E (Id ,ϕ ) / / E ⊕ F pr 2 / / F of ϕ , we have (Id , ϕ ) ∗ ( h ) = h E and pr 2 ∗ ( h ) = h F . Pr o of. In fact, for an y element x ∈ E , h ( x, ϕ ( x )) = ma x( h E ( x ) , h F ( ϕ ( x ))) = h E ( x ) since h F ( ϕ ( x )) ≤ k ϕ k h E ( x ) ≤ h E ( x ). F urther mo re, b y definition it is clear that h F = pr 2 ∗ ( h ). Therefore the pr op osition is true. ✷ Hermitian v ect or bundles Let K b e a num ber field and O K be its integer ring. F or any s cheme X of finite type and fla t over Spec O K such tha t X K is smo oth, we denote by V ec ( X ) the categor y of lo cally 13 free mo dules of finite rank on X . If we denote by E the class of a ll shor t exact seq uence of coherent s heaves in V ec ( X ), then ( V ec ( X ) , E ) is an exact category . Let Σ ∞ be the set of a ll embeddings o f K in C . The spa ce X ( C ) of complex p oints o f X , w hich is a co mplex a nalytic manifold, ca n be written as a disjoint union X ( C ) = a σ ∈ Σ ∞ X σ ( C ) , where X σ ( C ) is the spa ce o f complex po in ts in X × O K ,σ Spec C . Notice that the complex conjugation o f C induces an inv olution F ∞ : X ( C ) → X ( C ) which sends X σ ( C ) onto X σ ( C ). W e call Hermitian ve ctor bund le on X any pair ( E , h ) where E is an ob ject in V ec ( X ) and h = ( h σ ) σ ∈ Σ ∞ is a collec tion such that, for a n y σ ∈ Σ ∞ , h σ is a contin uous Hermitian metric on E σ ( C ), E σ being E ⊗ O K ,σ C , sub ject to the condition that the collection h = ( h σ ) σ ∈ Σ ∞ should b e inv ariant under the action of F ∞ . The collection of Hermitia n metrics h is calle d a Hermitian structur e on E . One ca n cons ult for example [1] and [4 ] for details. If i : E 0 → E is an injective homomor phism of O X -mo dules in V ec ( X ), we denote b y i ∗ ( h ) the collection of induced metr ic s on ( E 0 ,σ ( C )) σ ∈ Σ ∞ ; if π : E → F is a surjective homomorphism o f O X - mo dules in V ec ( X ), w e denote by π ∗ ( h ) the collection of quo tient metric on ( F σ ( C )) σ ∈ Σ ∞ . F or any ob ject E in V ec ( X ), let A ( E ) b e the set of all Hermitian structures o n E . The family version of Pro p os ition 3.4 implies that ( V ec ( X ) , E , A ) is a n arithmetic exact category . The family version of Pr op osition 3.5 implies that, if ( E , h E ) and ( F , h F ) ar e tw o Hermitian vector bundles over X and if ϕ : E → F is a homomorphism o f O X -mo dules in V ec ( X ) such that, for any x ∈ X ( C ), k ϕ x k < 1, then ϕ is compatible with ar ithmetic structures . W e say that a Hermitia n s tructure h = ( h σ ) σ ∈ Σ ∞ on a vector bundle E o n X is smo oth if for any σ ∈ Σ ∞ , h σ is a smoo th Hermitian metric. F or a n y vector bundle E on X , let A 0 ( E ) b e the set of all smo o th Hermitian structures on E . Then ( V ec ( X ) , E , A 0 ) is a lso an arithmetic exact category . If ( E , h E ) and ( F , h F ) ar e t wo s mo o th Hermitian vector bundles ov er X , then any homomorphism ϕ : E → F which has nor m < 1 at every complex p oint of X is compatible with arithmetic structures. Filtrations in an Ab elian category Let C b e a n essentially sma ll Abelia n catego ry and E be the clas s of sho rt exact sequence s in C . It is well known that any finite pro jective limit (in par ticular any fib er pro duct) exists in C . F urthermor e, an y morphism in C has an image, which is is omorphic to the cokernel of its k ernel, or the k ernel of its co kernel. F or any ob ject X in C , w e deno te by A ( X ) the set 5 of iso morphism clas ses of left contin uous I -filtra tio ns of X , where I is a totally ordered s et, as explained in the beg inning of the second sectio n. F or any le ft co nt inuous I -filtra tio n F of X , we denote by [ F ] the iso morphism clas s of F . If u : X 0 → X is a monomorphism, w e define u ∗ [ F ] to be the cla ss o f the inv erse imag e u ∗ F . If π : X → Y is an epimorphism, we define π ∗ [ F ] to be the cla ss o f the str ong direct imag e π ∗ F . W e assert that ( C , E , A ) is an ar ithmetic exact catego ry . In fact, the ax ioms ( A 1 ) — ( A 5) are clea rly satisfied. W e now verify the ax iom ( A 6). Consider the diagram (1) in Definition 3.3, which is the right sagittal square of the following diag ram (2). Supp ose given an I - filtration F of Y . F or any i ∈ I , w e no te Y i = F ( i ) a nd we denote b y b i : Y i → Y the canonical 5 This is a set b ecause C is essent ially small. 14 monomorphism. Z i / / c i / /   v i   Z   v   X i p i > >   u i   / / a i / / X p > > > > ~ ~ ~ ~ ~ ~ ~ ~   u   W i / / d i / / W Y i q i > > > > | | | | | | | | / / b i / / Y q > > > > ~ ~ ~ ~ ~ ~ ~ ~ (2) Let d i : W i → W be the imag e of q b i in W and q i : Y i → W i be the ca nonical epimorphism. Let ( Z i , c i , v i ) b e the fiber pr o duct of v and d i , and ( X i , a i , u i ) b e the fib er pro duct o f u and b i . Therefor e, in the diagram (2), the tw o coronal square and the right sagittal square are cartesian, the inferior square is co mmutative. As v pa i = q ua i = q b i u i = d i q i u i , there exists a unique morphism p i : X i → Z i such that c i p i = p a i and that v i p i = q i u i . It is then not har d to verify that the left sa gittal squar e is ca rtesian, ther efore p i is an epimorphism, so Z i is the image of pa i . The axiom ( A 6) is therefore verified. Fina lly , the verification of the ax iom ( A 7) follows fro m the following prop os itio n. Prop ositio n 3.7 L et X and Y b e two obje cts in C and let F (r esp. G ) b e an I -filtr ation of X (r esp. Y ). If f : X → Y is a morphism which is c omp atible with the filtr ations ( F , G ) , then ther e ex ists a filtr ation H on X ⊕ Y such that Γ ∗ f H = F and pr 2 ∗ H = G , wher e Γ f = (Id , f ) : X → X ⊕ Y is the gr aph of f and pr 2 : X ⊕ Y → Y is the pr oje ction onto the se c ond factor. Pr o of. Let H be the filtration such that H ( i ) = F ( i ) ⊕ G ( i ). Clea rly it is left co nt inuous, a nd pr 2 ♭ H = G . Ther efore pr 2 ∗ H = G l = G . Mor eov er, for any i ∈ I , consider the s quare F ( i ) φ i / / (Id ,f i )   X (Id ,f )   F ( i ) ⊕ G ( i ) Φ i / / X ⊕ Y (3) where φ i : F ( i ) → X and Φ i = φ i ⊕ ψ i : F ( i ) ⊕ G ( i ) → X ⊕ Y are cano nical inclusions, f i : F ( i ) → G ( i ) is the morphism through which the restrictio n o f f on F ( i ) (i.e., f φ i ) factorizes. Then the squa re (3) is c o mm utative. Supp ose that α : Z → X and β = ( β 1 , β 2 ) : Z → F ( i ) ⊕ G ( i ) are t wo mor phisms such that (Id , f ) α = Φ i β . Z β 1 $ $ α $ $ β $ $ F ( i ) φ i / / (Id ,f i )   X (Id ,f )   F ( i ) ⊕ G ( i ) Φ i / / X ⊕ Y (4) Then we have α = φ i β 1 and f α = ψ i β 2 . So ψ i β 2 = f α = f φ i β 1 = ψ i f i β 1 . 15 As ψ i is a monomorphism, we o btain that f i β 1 = β 2 . So β 1 : Z → F ( i ) is the only mor phism such that the diagram (4) comm utes. Hence w e get F = (Id , f ) ∗ H . ✷ Notice that the categ o ry of arithmetic ob jects C A is equiv alent to the category Fil I ,l ( C ) of left co n tinuous filtrations. Mor eov er, there exist some v a riants of ( C , E , A ). F or exa mple, if for any ob ject X in C , we denote b y A 0 ( X ) the set of iso morphism classes of I -filtra tions which are sepa rated, exhaustive, left contin uo us and o f finite length. Then ( C , E , A 0 ) is also an arithmetic ex act categ ory . F urthermore, the categ ory C A 0 is equiv alent to Fil I , self ( C ), the full sub c ategory of Fil I ,l ( C ) consisting of filtrations which are s epara ted, e xhaustive, l eft continous and of f inite length. 4 Harder-Narasimhan categories In this section we intro duce the for malism o f Harder -Narasimhan filtrations (indexed by R ) on arithmetic exa ct categ o ries. Let ( C , E , A ) b e an arithmetic exac t c a tegory . W e s ay that an arithmetic ob ject ( X , h ) is non-zer o if X is non-zero in C . Since C is essentially small, the isomorphism classe s of ob jects in C A form a set. W e denote by E A the class of diagrams of the fo r m 0 / / ( X ′ , h ′ ) i / / ( X, h ) p / / ( X ′′ , h ′′ ) / / 0 where ( X ′ , h ′ ), ( X , h ) and ( X ′′ , h ′′ ) are arithmetic ob jects and 0 / / X ′ i / / X p / / X ′′ / / 0 is a dia gram in E such tha t h ′ = i ∗ ( h ) and h ′′ = p ∗ ( h ). Let K 0 ( C , E , A ) be the free Ab elian gro up generated by isomor phism classes in C A , mo dulo the subgr o up gene r ated by elements of the fo rm [( X , h )] − [( X ′ , h ′ )] − [( X ′′ , h ′′ )], where 0 / / ( X ′ , h ′ ) i / / ( X, h ) p / / ( X ′′ , h ′′ ) / / 0 is a diagra m in E A , in other words, 0 / / X ′ i / / X p / / X ′′ / / 0 , and i ∗ ( h ) = h ′ , p ∗ ( h ) = h ′′ . The group K 0 ( C , E , A ) is ca lled the Gr othendie ck gr oup o f the arithmetic exa c t category ( C , E , A ). W e have a “ for getful ” ho momorphism from K 0 ( C , E , A ) to K 0 ( C , E ), the Grothendieck g r oup 6 of the exa ct category ( C , E ), which s ends [( X , h )] to [ X ]. In order to establish the semi-stability o f arithmetic o b jects and furthermore the Harder- Narasimhan for malism, we need tw o auxiliary homomorphisms of gro ups. The first one, fro m K 0 ( C , E , A ) to R , is called a de gr e e fun ction on ( C , E , A ); and the second one, from K 0 ( C , E ) to Z , which takes stric tly p ositive v alues on elements of the form [ X ] with X non-zer o, is called a r ank function on ( C , E ). Now let d deg : K 0 ( C , E , A ) → R b e a degree function on ( C , E , A ) and rk : K 0 ( C , E ) → Z be a rank function o n ( C , E ). F or any arithmetic ob ject ( X, h ) in ( C , E , A ), we s ha ll use the expr e s sions d deg( X , h ) and r k( X ) to denote d deg([( X , h )]) and rk([ X ]), and call them the arithmetic de gr e e and the r ank o f ( X , h ) res pectively . If ( X , h ) is non-zer o, the quotient b µ ( X, h ) = d deg( X , h ) / rk( X ) is called the arithmetic slop e of ( X , h ). W e say that a non- z e ro 6 Whic h is, b y definition, the free Ab elian group generate d by is omorphism classes in C , modulo the s ub- group generate d b y elemen ts of the form [ X ] − [ X ′ ] − [ X ′′ ], where 0 / / X ′ / / X / / X ′′ / / 0 is a diagram in E . 16 arithmetic ob ject ( X , h ) is semistable if for a n y non-z ero arithmetic subo b ject ( X ′ , h ′ ) of ( X , h ), we hav e b µ ( X ′ , h ′ ) ≤ b µ ( X , h ). The following pro po sition pr ovides so me basic prop erties of ar ithmetic deg rees and of a rith- metic slop es. Prop ositio n 4.1 L et us ke ep the notations ab ove. 1) If 0 / / ( X ′ , h ′ ) / / ( X, h ) / / ( X ′′ , h ′′ ) / / 0 is a diagr am in E A , then d deg( X , h ) = d deg( X ′ , h ′ ) + d deg( X ′′ , h ′′ ) . 2) If ( X , h ) is an arithmetic obje ct of r ank 1 , then it is semistable. 3) Any non-zer o arithmetic obje ct ( X , h ) is semistable if and only if for any non-t rivial arith- metic quotient ( X ′′ , h ′′ ) (i.e., X ′′ do es not r e duc e to zer o and is not c anonic al ly isomorphic to X ), we have b µ ( X , h ) ≤ b µ ( X ′′ , h ′′ ) . Pr o of. Since d deg is a homomo r phism from K 0 ( C , E , A ) to R , 1) is clear. 2) If ( X ′ , h ′ ) is an arithmetic sub ob ject of ( X , h ), then it fits into a diagra m 0 / / ( X ′ , h ′ ) f / / ( X, h ) / / ( X ′′ , h ′′ ) / / 0 in C A . Since X ′ is non- zero, rk( X ′ ) ≥ 1. Therefore rk( X ′′ ) = 0 and hence X ′′ = 0. In other words, f is an isomor phis m. So we have b µ ( X ′ , h ′ ) = b µ ( X , h ). 3) F o r any dia gram 0 / / ( X ′ , h ′ ) / / ( X, h ) / / ( X ′′ , h ′′ ) / / 0 in E A , ( X ′′ , h ′′ ) is non- trivial if and only if ( X ′ , h ′ ) is non- trivial. If ( X ′ , h ′ ) a nd ( X ′′ , h ′′ ) a re bo th non-triv ia l, we hav e the following equality b µ ( X, h ) = rk( X ′ ) rk( X ) b µ ( X ′ , h ′ ) + rk( X ′′ ) rk( X ) b µ ( X ′′ , h ′′ ) . Therefore b µ ( X ′ , h ′ ) ≤ b µ ( X, h ) ⇐ ⇒ b µ ( X ′′ , h ′′ ) ≥ b µ ( X , h ). ✷ W e are now able to in tro duce conditio ns ensuring the e x istence and the uniquenes s of Harder-Nar asimhan “ fla g”. The conditions will b e prop osed a s a x ioms in the coming definition, and in the theorem w hich follows, we shall prov e the existence and the uniq ue ne s s of Harder- Narasimhan “flag ”. Definition 4.2 Let ( C , E , A ) ba an ar ithmetic exact catego ry , d deg : K 0 ( C , E , A ) → R b e a degree function and rk : K 0 ( C , E ) → Z be a rank function. W e say that ( C , E , A, d deg , rk) is a Har der-Nar asimhan c ate gory if the following t wo a xioms are verified: ( HN 1) F o r any non- zero arithmetic ob ject ( X , h ), ther e exists an ar ithmetic sub ob ject ( X des , h des ) of ( X , h ) such that b µ ( X des , h des ) = sup { b µ ( X ′ , h ′ ) | ( X ′ , h ′ ) is a non-zero arithmetic subob ject o f ( X, h ) } . 17 F urther more, for a n y no n- zero a rithmetic s ub ob ject ( X 0 , h 0 ) of ( X , h ) such that b µ ( X 0 , h 0 ) = b µ ( X des , h des ), there ex ists an admissible mono mo rphism f : X 0 → X des such that the diagram X 0 f / / j " " E E E E E E E E X des i   X is commutativ e and that f ∗ ( h des ) = h 0 , where i and j are canonica l a dmissible monomorphisms. ( HN 2) If ( X 1 , h 1 ) and ( X 2 , h 2 ) ar e tw o semistable ar ithmetic ob jects such that b µ ( X 1 , h 1 ) > b µ ( X 2 , h 2 ), there exis ts no non-zero morphism from X 1 to X 2 which is compatible with arithmetic str uctures. With the notations of Definition 4.2, if ( X , h ) is a non- z e ro ar ithmetic ob ject, then ( X des , h des ) is a s emistable arithmetic o b ject. If in a ddition ( X , h ) is not semistable, we say that ( X des , h des ) is the ar ithmetic sub ob ject which destabilizes ( X , h ). Theorem 4 .3 L et ( C , E , A, d deg , rk) b e a H ar der-Nar asimhan c ate gory. If ( X, h ) is a non-zer o arithmetic obje ct, then ther e exists a se quenc e of admissible monomorphisms in C : 0 = X 0 / / X 1 / / · · · / / X n − 1 / / X n = X , (5) unique up to a unique isomorphi sm, such that, if for any inte ger 0 ≤ i ≤ n , we denote by h i the induc e d arithmetic structur e (fr om h ) on X i and if we e quip, for any inte ger 1 ≤ j ≤ n , X j /X j − 1 with the quotient arithmetic structu r e (of h j ), then 1) for any inte ger 1 ≤ j ≤ n , the arithmetic obje ct X j /X j − 1 define d ab ove is semistable; 2) we have the ine qualities b µ ( X 1 /X 0 ) > b µ ( X 2 /X 1 ) > · · · > b µ ( X n /X n − 1 ) . Pr o of. First we prove the existence b y inductio n o n the ra nk r o f X . The case wher e ( X , h ) is semistable is trivial, and a fortiori the existence is true for r = 1. Now we consider the case where ( X , h ) isn’t semistable. Let ( X 1 , h 1 ) = ( X des , h des ). It’s a semistable ar ithmetic ob ject, and X ′ = X/X 1 is non- zero. The rank o f X ′ being strictly smaller than r , we can therefore apply the inductio n hypothes is on ( X ′ , h ′ ), where h ′ is the quo tien t arithmetic str ucture. W e then obtain a sequence o f a dmissible monomorphis ms 0 = X ′ 1 f ′ 1 / / X ′ 2 / / · · · / / X ′ n − 1 f ′ n − 1 / / X ′ n = X ′ verifying the desired condition. Since the canonical mor phism from X to X ′ is an a dmissible epimor phis m, for any 1 ≤ i ≤ n , if we note X i = X × X ′ X ′ i , then b y the axiom ( Ex 6), the pro jection π i : X i → X ′ i is an admissible epimor phism. F o r any integer 1 ≤ i < n , we have a canonic a l mo r phism from X i to X i +1 and the square X i f i / / π i     X i +1 π i +1     X ′ i f ′ i / / X ′ i +1 (6) 18 is car tesian. Since f ′ i is an mo nomorphism, also is f i (cf. [1 3] V. 7). On the other hand, since the squar e (6) is ca rtesian, f i is the kernel of the co mpo sed morphism X i +1 π i +1 / / / / X ′ i +1 p i / / / / X ′ i +1 /X ′ i , where p i is the canonical morphism. Since π i +1 and p i are admissible epimorphisms, also is p i π i +1 (see a xiom ( Ex 4)). The r efore f i is an admiss ible monomorphism. Hence we obtain a commutativ e diagram 0 = X 0 / / X 1 f 1 / / π 1   X 2 / / π 2   · · · / / X n − 1 f n − 1 / / π n − 1   X n = X π   0 = X ′ 1 f ′ 1 / / X ′ 2 / / · · · / / X ′ n − 1 f ′ n − 1 / / X ′ n = X ′ where the hor izontal mo rphisms in the lines are a dmissible mono morphisms and the vertical morphisms ar e admiss ible epimorphisms. F urthermo re, fo r any integer 1 ≤ i ≤ n − 1, we hav e a natural isomor phism ϕ i from X i +1 /X i to X ′ i +1 /X ′ i . W e denote b y g i (resp. g ′ i ) the ca nonical morphism from X i (resp. X ′ i ) to X (res p. X ′ ). Let h i = g ∗ i ( h ) (resp. h ′ i = g ′ i ∗ ( h ′ )) be the induced ar ithmetic structure on X i (resp. X ′ i ). After the axiom ( A 6), π i ∗ ( h i ) = π i ∗ f ∗ i ( h ) = f ′∗ i π ∗ ( h ) = h ′ i . Ther efore ϕ i ∗ sends the q uotient arithmetic s tructure o n X i +1 /X i to that on X ′ i +1 /X ′ i . Hence the a rithmetic ob ject X i +1 /X i is semistable a nd w e hav e the equality b µ ( X i +1 /X i ) = b µ ( X ′ i +1 /X ′ i ). Finally , since X 1 = X des , we have b µ ( X 2 /X 1 ) = rk( X 2 ) b µ ( X 2 ) − rk( X 1 ) b µ ( X 1 ) rk( X 2 ) − rk( X 1 ) < b µ ( X 1 ) . Therefore the se q uence 0 = X 0 / / X 1 / / · · · / / X n − 1 / / X n = X satisfies the de- sired conditions. W e then prov e the uniqueness o f the sequence (5). By induction we only need to prov e that X 1 ∼ = X des . Let i b e the first index such that the cano nical morphism X des → X factor- izes through X i +1 . The comp osed morphism X des → X i +1 → X i +1 /X i is then non-zero. Since X des and X i +1 /X i are semistable, w e have b µ ( X des ) ≤ b µ ( X i +1 /X i ). This implies i = 0 and b µ ( X des ) = b µ ( X 1 ). Therefor e the morphism X 1 → X factorizes through X des . So we hav e X des ∼ = X 1 . ✷ F r o m the pro of above we see that the axio m ( HN 1) suffices for the existence. It is the axiom ( HN 2) which ensure s the uniqueness. Definition 4.4 With the notatio ns of Theorem 4.3, the sequence (5) is called the Har der- Nar asimhan se quenc e of the (non-zero) arithmetic ob ject ( X , h ). Sometimes w e write instea d 0 = X 0 / / X 1 / / · · · / / X n − 1 / / X n = X for underlining the arithmetic str uctures. The rea l num ber s b µ ( X 1 ) and b µ ( X/ X n − 1 ) are called resp ectively the maximal slop e and the minimal slop e o f X , denoted by b µ max ( X ) and b µ min ( X ). W e po in t o ut tha t for any in teger 1 ≤ i ≤ n , 0 = X 0 / / X 1 / / · · · / / X i − 1 / / X i is the Harder -Narasimha n sequence of X i . Therefore we hav e b µ min ( X i ) = b µ ( X i /X i − 1 ). Finally , we define by conv ent ion b µ max (0) = − ∞ and b µ min (0) = + ∞ . 19 Corollary 4.5 L et ( C , E , A, d deg , rk) b e a H ar der-Nar asimha n c ate gory and X b e a non-zer o arithmetic obje ct. 1) F or any non-zer o arithmetic su b obje ct Y of X , we have b µ max ( Y ) ≤ b µ max ( X ) . 2) F or any non-zer o arithmetic qu otient Z of X , we have b µ min ( Z ) ≥ b µ min ( X ) . 3) We have the ine qualities b µ min ( X ) ≤ b µ ( X ) ≤ b µ max ( X ) . Pr o of. Let 0 = X 0 − → X 1 − → · · · − → X n − 1 − → X n = X be the Harder-Nar asimhan sequence of X . 1) After replacing Y by Y des we may suppo se tha t Y is semistable. Let i b e the first index such that the canonical morphis m Y → X fa c to rizes throug h X i +1 . The co mp os e d morphism Y → X i +1 → X i +1 /X i is non-zero and c ompatible with arithmetic structures. T her efore b µ ( Y ) ≤ b µ ( X i +1 /X i ) ≤ b µ max ( X ) . 2) After replac ing Z by a s e mistable quotient we may supp ose that Z is itse lf semistable. Let f : X → Z b e the ca nonical mor phis m. It is an a dmissible epimo r phism. Let i b e the smallest index such that the co mp os e d morphism X i +1 → X f → Z is non-zer o. Since the comp osed morphism X i → X f → Z is zero, w e obtain a non-zero morphism from X i +1 /X i to Z which is compatible with arithmetic structur e s after Axiom ( A 6). X i +1 / / / /     X     X i +1 /X i / / / / X/X i / / / / Z Therefore b µ ( Z ) ≥ b µ ( X i +1 /X i ) ≥ b µ min ( X ). 3) W e hav e d deg( X ) = n X i =1 d deg( X i /X i − 1 ). Therefor e b µ ( X ) = n X i =1 rk( X i /X i − 1 ) rk( X ) b µ ( X i /X i − 1 ) ∈  b µ min ( X ) , b µ max ( X )  . ✷ It is well known that if E and F ar e tw o vector bundles on a smo oth pro jective cur ve C such that µ min ( E ) > µ max ( F ), then there isn’t any non-zero ho momorphism from E to F . The following r e sult (Prop osition 4.7) g eneralizes this fac t to Harder -Narasimha n categor ies. Lemma 4.6 L et ( C , E , A ) b e an arithmetic exact c ate gory. Supp ose that any epimorphism in C has a kernel. L et ( X, h X ) and ( Z , h Z ) b e t wo arithmetic obje cts , ( Y , h Y ) b e an arithmetic quotient of ( X , h X ) , and f : Y → Z b e a morphism in C . Denote by π : X → Y the c anonic al admissible epimorph ism. The morphism f is c omp atible with arithmetic stru ctur es if and only if it is the c ase for f π . Pr o of. Since π is compatible w ith arithmetic str uctures, the compatibility o f f with arithmetic structures implies that of f π . I t then suffices to verify the converse assertion. By definition there exists an arithmetic ob ject ( W, h W ) and a decomp osition X / / i / / W p / / / / Z of f π 20 such that i ∗ h W = h X and p ∗ h W = h Z . Let T b e the fiber copro duct of i and π and let j : Y → T a nd q : W → T b e canonical morphisms. After Axiom ( Ex 5), j is an a dmis s ible monomorphism. Let τ : U ֌ X b e the kernel o f π . W e assert that q = Coker( iτ ). On one hand, w e hav e q iτ = j π τ = 0. On the other hand, if α : W → V is a morphism in C such that αiτ = 0 , then there exists a unique morphism β : Y → V such that β π = αi since π is a cokernel of τ . Ther efore, there exits a unique mo rphism γ : T → V such that γ q = α . So q is a cokernel of i τ , hence an a dmissible epimor phism. The morphisms p : W ։ Z and f : Y → Z induce a mo r phism g : T → Z : U   τ   X / / i / / π     W q     p A A A A A A A A Y / / j / / T g / / Z Since g is an epimorphism, by hypothesis it ha s a kernel. After Axiom ( E x 7 ), it is an admis- sible epimo r phism. Fina lly if we deno te by h T the ar ithmetic structure q ∗ h W on T , we have g ∗ ( h T ) = p ∗ ( h W ) = h Z and j ∗ ( h T ) = π ∗ ( i ∗ h W ) = π ∗ ( h X ) = h Y . ✷ Prop ositio n 4.7 L et ( C , E , A, d deg , rk) b e a Har der-Nar asimhan c ate gory. Supp ose t hat any epimorphi sm in C has a kernel. If X and Y ar e two arithmetic obje cts and if f : X → Y is a non-zer o morphism c omp atible with arithmetic structu r es, then b µ min ( X ) ≤ b µ max ( Y ) . Pr o of. Let 0 = X 0 / / X 1 / / · · · / / X n − 1 / / X n = X be the Harder-Nar asimhan sequence of X . F or any integer 0 ≤ i ≤ n , let h i : X i → X b e the canonica l monomo rphism. Let 1 ≤ j ≤ n b e the first index suc h that f h j is non- zero. Since f h j − 1 = 0, the mo rphism f h j factorizes through X j /X j − 1 , so we get a non-zer o mo rphism g from X j /X j − 1 to Y . After Lemma 4.6, g is compatible with arithmetic s tr uctures. Let 0 = Y 0 / / Y 1 / / · · · / / Y m − 1 / / Y m be the Ha rder-Nara simhan sequenc e of Y . Let 1 ≤ k ≤ n b e the firs t index such tha t g factorizes through Y k . If π : Y k → Y k / Y k − 1 is the cano nical morphis m, then π g is non-zero since g do esn’t factorize thro ugh Y k − 1 . F urthermore, it is co mpatible with arithmetic structures. Therefore, we hav e b µ min ( X ) ≤ b µ ( X j /X j − 1 ) ≤ b µ ( Y k / Y k − 1 ) ≤ b µ max ( Y ) . ✷ Corollary 4.8 Ke ep the notations and t he hyp othesis of Pr op osition 4.7. 1) If in addition f is monomorphic, then b µ max ( X ) ≤ b µ max ( Y ) . 2) If in addition f is epimorphic, then b µ min ( X ) ≤ b µ min ( Y ) . Pr o of. Suppose that f is monomor phic. Let i : X des → X b e the ca nonical morphism. Then the comp osed morphism f i : X des → Y is non- zero a nd compa tible with a r ithmetic structures. 21 Therefore b µ max ( X ) = b µ min ( X des ) ≤ b µ max ( Y ). The pro of of the o ther asse r tion is similar . ✷ If the a rithmetic str ucture A is trivia l, then any mor phism in C is compatible with arithmetic structures. Therefore in this case we may r emov e the hypothes is o n the existence of kernels in Pr op osition 4.7 and in Corollar y 4.8. Ho wev er, we don’t know whether in gener al case we can remove the hypothesis that any e pimo rphism in C has a kernel, altho ugh this conditio n is fulfilled for all examples that we hav e dis cussed in the previous sec tio n. In the following, we give an ex ample of Harder-Nar asimhan category , which will play an impo rtant role in the next section. Let C b e an Abe lia n ca tegory and E be the cla ss of all s hort exact sequences in C . W e supp ose given a ra nk function r k : K 0 ( C ) → Z . In this exa mple we take the totally or dered set I as a subset of R (with the induced order ). F or any ob ject X in C , let A 0 ( X ) b e the set of isomor phism class e s in Fil I , self X . W e hav e shown in the previo us section tha t ( C , E , A 0 ) is an arithmetic exact category . An y arithmetic ob ject X = ( X , h ) of this a rithmetic exact category ma y be consider ed, after c ho osing a r epresentativ e in h , a s an ob ject X in C eq uipped w ith an R -filtration ( X λ ) λ ∈ I which is separated, exhaustive, left contin uous and of finite length. W e define a real num ber 7 d deg( X ) = X λ ∈ I λ  rk( X λ ) − sup j >λ,j ∈ I rk( X j )  . The summation ab ove turns out to b e finite since the filtration is of finite length and its v alue do esn’t dep end on the choice of the repr esentativ e in h . If X = ( X , ( X λ ) λ ∈ I ) and Y = ( Y , ( Y λ ) λ ∈ I ) are tw o ar ithmetic o b jects and if f : X → Y is an isomo rphism which is compatible with arithmetic structures, then for any λ ∈ I , we hav e rk( X λ ) ≤ r k( Y λ ). Therefore we hav e d deg( X ) ≤ d deg( Y ) by Abel’s summation formula. W e now show that the function d deg defined above extends naturally to a ho momorphism from K 0 ( C , E , A 0 ) to R . Let 0 / / X ′ u / / X p / / X ′′ / / 0 be a short exa ct sequence in C . Supp ose that F ′ = ( X ′ λ ) λ ∈ I (resp. F = ( X λ ) λ ∈ I , F ′′ = ( X ′′ λ ) λ ∈ I ) is an R -filtration o f X ′ (resp. X , X ′′ ) which is s eparated, exhaustive, left co nt inuous and of finite length, and such that F ′ = u ∗ ( F ), F ′′ = p ∗ ( F ). Then for any real num ber λ ∈ I we hav e a cano nical ex act sequence 0 / / X ′ λ / / X λ / / X ′′ λ / / 0 . Therefore, d deg( X , [ F ]) = d deg( X ′ , [ F ′ ]) + d deg( X ′′ , [ F ′′ ]). Notice that an non-zero a rithmetic ob ject X = ( X , [ F ]) is semistable if and only if the filtration F has a jumping set which reduces to a one point set. If X is se mistable and if { λ } is a jumping set of F , then the arithmetic slop e of X is just λ . Therefor e, if X = ( X , [ F ]) and Y = ( Y , [ G ]) ar e tw o semis ta ble arithmetic ob jects such tha t λ := b µ ( X ) > b µ ( Y ), then any morphism f : X → Y which is compa tible with filtrations sends F ( λ ) = X int o G ( λ ) = 0, therefore is the ze r o mo rphism. If X = ( X , [ F ]) is a non-zer o arithmetic ob ject, w e denote by X des the non-zero ob ject in the filtr ation F ha ving the maximal index. The existence of X des is justified by the finiteness and the left cont inuit y o f F . The arithmetic subo b ject X des of X is semistable. F urthermor e, 7 Here sup ∅ = 0 b y con v ent ion. 22 for any no n-zero arithmetic sub ob ject Y = ( Y , [ G ]) of X , we hav e b µ ( Y ) = 1 rk( Y ) X λ ∈ I λ  rk( G ( λ )) − sup j >λ,j ∈ I rk( G ( j ))  ≤ 1 rk( Y ) X λ ∈ I b µ ( X des )  rk( G ( λ )) − sup j >λ,j ∈ I rk( G ( j ))  = b µ ( X des ) . The equa lit y holds if and only if Y is semistable and of slop e b µ ( X des ), in this case, the canonical morphism from Y to X facto r izes through X des since it is compatible with filtrations . Hence we hav e prov ed that ( C , E , A 0 , d deg , rk) is a Harder-Nara simhan catego ry . Suppo se that X = ( X , [ F ]) is a no n-zero a rithmetic ob ject, where F = ( X λ ) λ ∈ R . If E = { λ 1 > λ 2 > · · · > λ n } is the minimal jumping set of F (i.e. the intersection of a ll jumping sets of F , which is itself a jumping set of F ), then 0 / / X λ 1 / / X λ 2 / / · · · / / X λ n = X is the Har der-Nara simhan se q uence of X . F urther more, b µ ( X λ 1 ) = λ 1 , and for a ny 2 ≤ i ≤ n , b µ ( X λ i /X λ i − 1 ) = λ i . 5 Harder-Narasimhan filtrations and p olygons W e fix in this sec tion a Harder- Na rasimhan category ( C , E , A, d deg , rk). W e s hall intro- duce the notions o f Harder -Narasimhan filtrations and Harder-Na rasimhan mea sures for an arithmetic o b ject in ( C , E , A, d deg , rk). W e shall als o explain that if D is an Abelia n category equipp e d with a rank function and if there exists an exa c t functor F : C → D which pre- serves ra nk functions, then for any non- zero a rithmetic ob ject X in C , the Harde r -Narasimha n filtration of X induces a filtr a tion of F ( X ), whic h defines an arithmetic o b ject F ( X ) of the Harder-Nar asimhan categor y defined by R -filtra tions in D which are separa ted, exhaustive, left contin uous and of finite le ng th. F urther more, the Harder-Nar asimhan p olyg on (resp. measure) of F ( X ) coincides with that o f X . Therefore, filter e d ob jects in Abe lia n categ ories equippe d with rank functions can be considered in some sense as mo dels to study Harder-Na rasimhan po lygons. Prop ositio n 5.1 L et X b e a non-zer o arithmetic obje ct and 0 = X HN 0 / / X HN 1 / / · · · / / X HN n − 1 / / X HN n = X b e its Har der-Nar asimhan se quenc e. If for any r e al num b er λ we denote by 8 i X ( λ ) = max { 1 ≤ i ≤ n | b µ ( X HN i / X HN i − 1 ) ≥ λ } and X λ = X HN i X ( λ ) , then ( X λ ) λ ∈ R is an R -filt r ation of the obje ct X in C . F urthermor e, this filtr ation is sep ar ate d, ex haustive, left c ont inu ous and of fi n ite length. Pr o of. If λ > λ ′ , then i X ( λ ) ≤ i X ( λ ′ ), hence ( X λ ) λ ∈ R is an R -filtration of X . Moreover, for any λ ∈ R , X λ ∈ { X HN 0 , · · · , X HN n } , ther e fore this filtration is of finite length. When λ > b µ max ( X ), we ha ve i X ( λ ) = 0, which implies that X λ = X HN 0 = 0 is the zero ob ject, so 8 By conv en tion max ∅ = 0. 23 the filtration is separa ted. When λ < b µ min ( X ), i X ( λ ) = n , so X λ = X , i.e., the filtration is exhaustive. T o pr ov e the left contin uit y of this filtra tion, it suffices to verify that the function λ 7→ i X ( λ ) is left c o nt inuous. Actually , this function is left lo cally constant: if i X ( λ ) = 0, then for an y integer 1 ≤ i ≤ n , we hav e b µ ( X HN i /X HN i − 1 ) < λ , so there exists ε 0 > 0 such that for any 0 ≤ ε < ε 0 , we have b µ ( X HN i /X HN i − 1 ) < λ − ε , i.e., i X ( λ − ε ) = 0; if i X ( λ ) = n , then for any int eger 1 ≤ i ≤ n and an y real n umber ε ≥ 0, we have b µ ( X HN i /X HN i − 1 ) ≥ λ ≥ λ − ε , so i X ( λ − ε ) = n ; finally if 1 ≤ i X ( λ ) ≤ n − 1, then we ha ve b µ ( X HN i X ( λ ) / X HN i X ( λ ) − 1 ) ≥ λ and b µ ( X HN i X ( λ )+1 / X HN i X ( λ ) ) < λ , hence there exis ts ε 0 > 0 such that, for any 0 ≤ ε < ε 0 , w e hav e b µ ( X HN i X ( λ ) / X HN i X ( λ ) − 1 ) ≥ λ − ε and b µ ( X HN i X ( λ )+1 / X HN i X ( λ ) ) < λ − ε , i.e., i X ( λ − ε ) = i X ( λ ). ✷ Definition 5.2 With the notatio ns of Pro po sition 5 .1, the filtration ( X λ ) λ ∈ R is called the Har der-Nar asimhan filtr ation (or c anonic al fi ltr ation ) of X , denoted by HN( X ). Clea rly , b µ min ( X λ ) ≥ λ fo r any λ ∈ R . W e define the Har der-Nar asimhan filt ra tion (or c anonic al filtr ation ) of the zer o ob ject to b e its only R -filtration which a s so ciates to each λ ∈ R the zero ob ject itself. Theorem 5 .3 Ke ep the notations of Pr op osition 5.1. Su pp ose in addition t hat any epimor- phism in C has a kernel in the c ase wher e A is non-trivial. Then any morphism in C A is c omp atible with Har der-Nar asimhan filtr ations. Pr o of. Let f : X → Y b e a mo r phism which is co mpatible with arithmetic struc tur es. The case where X or Y is zero is trivial. W e now supp ose that X and Y are non- z ero. Let 0 = X HN 0 / / X HN 1 / / · · · / / X HN n − 1 / / X HN n = X be the Har der-Naras imhan sequence of X a nd 0 = Y HN 0 / / Y HN 1 / / · · · / / Y HN m − 1 / / Y HN m = Y be the Harder-Nara simhan sequence of Y . F or all integers 0 ≤ i < j ≤ m , let P j,i be the canonical mo rphism fro m Y HN j to Y HN j / Y HN i . F or any in teger 0 ≤ i ≤ n , let U i be the canonical monomo r phism fro m X HN i to X . Suppose that λ is a r eal num b er. If i X ( λ ) = 0 or if i Y ( λ ) = 0 , we define F λ as the zero mor phism from X λ to Y λ ; if i Y ( λ ) = m , we hav e Y λ = Y and we define F λ as the comp osition f U i X ( λ ) ; other wise we have b µ ( X HN i X ( λ ) / X HN i X ( λ ) − 1 ) ≥ λ and b µ ( Y HN i Y ( λ ) / Y HN i Y ( λ ) − 1 ) ≥ λ , but b µ ( Y HN j /Y HN j − 1 ) < λ for any j > i Y ( λ ). W e will prove by induction that the morphism f U i X ( λ ) factorizes through Y HN i Y ( λ ) . First it is obvious that the mo rphism f U i X ( λ ) factorizes throug h Y HN m = Y . If it factorizes through certain ϕ j : X HN i X ( λ ) → Y HN j , where j > i Y ( λ ), then the c o mpo sition P j,j − 1 ϕ j m ust be zero since (see Pro po sition 4.7 and the remar k a fter its pro o f ) b µ ( Y HN j /Y HN j − 1 ) < λ ≤ b µ ( X HN i X ( λ ) / X HN i X ( λ ) − 1 ) = b µ min ( X HN i X ( λ ) ) . So the mo rphism f U i X ( λ ) factorizes thr ough Y HN j − 1 . B y induction we obtain that f U i X ( λ ) fac- torizes (in unique way) through a mor phism F λ : X i X ( λ ) → Y i Y ( λ ) . The family of mor phisms F = ( F λ ) λ ∈ R defines a natural tr ansformation such that ( F, f ) is a morphism of filtrations. Therefore the mo r phism f is compatible with Harder-Nar a simhan filtrations. ✷ 24 Remark 5.4 Theo rem 5.3 implies that HN defines a ctually a functor from the catego ry C A to the full sub-categor y Fil R , self ( C ) of Fi l R ( C ) consisting of R -filtrations whic h are separa ted, exhaustive, left con tinuous and o f finite length, which sends an arithmetic ob ject X to its Harder-Nar asimhan filtra tion. Corollary 5.5 Supp ose in the c ase wher e A is non-trivial that any epimorph ism in C has a kernel. L et X and Y b e two arithmetic obje cts and f : Y → X b e a morphism which is c omp atible with arithmetic s t ructur es. If b µ min ( Y ) ≥ λ , then t he morphism f factorizes thr ough X λ . Pr o of. Since f is compatible with ar ithmetic structures, it is co mpatible with Ha rder-Nara simhan filtrations. So the r estriction of f o n Y λ factorizes through X λ . As b µ min ( Y ) ≥ λ , w e hav e Y λ = Y , therefore f factorizes through X λ . ✷ Let X b e a non-zero ar ithmetic ob ject and 0 = X HN 0 / / X HN 1 / / · · · / / X HN n − 1 / / X HN n = X be its Harder -Narasimhan sequenc e . F or any integer 0 ≤ i ≤ n , we no te t i = rk X HN i / rk X . F or any integer 1 ≤ i ≤ n , w e note λ i = b µ ( X HN i /X HN i − 1 ). Then the function P X ( t ) = n X i =1 d deg( X HN i − 1 ) rk X + λ i ( t − t i − 1 ) ! 1 1 [ t i − 1 ,t i ] ( t ) is a poly gon 9 on [0 , 1], ca lled the normalize d Har der-Nar asimhan p olygon of X . T he function P X takes v alue 0 at the orig in, and its first o rder deriv a tiv e is given by P ′ X ( t ) = n X i =1 λ i 1 1 [ t i − 1 ,t i [ ( t ) . The pro babilit y measure ν X := n X i =1 rk( X HN i ) − r k( X HN i − 1 ) rk X δ λ i = n X i =1 ( t i − t i − 1 ) δ λ i is ca lled the Har der-Nar asimhan me asu r e o f X . W e define the Harder-Nara simhan mea s ure of the zero arithmetic o b ject to be the zero meas ure on R . After Prop ositio n 5.1, if X is a non-zer o a rithmetic ob ject and if ( X λ ) λ ∈ R is the Harder -Narasimhan filtration of X , then the Harder-Na rasimhan mea s ure ν X of X is the first order deriv ative (in distribution s e nse) of the function t 7− → − rk( X t ). Finally we point out that the Harder-Nar asimhan p olyg on of a non-ze r o arithmetic ob ject X can be uniquely determined in an explicit way from its Harder-Nar asimhan meas ure. Prop ositio n 5.6 Supp ose in the c ase wher e A is non-trivial that any epimorph ism in C has a kernel. If X and Y ar e two non-zer o arithmetic obje cts and if f : X → Y is an isomorphi sm which is c omp atible to arithmetic st ructur es, then b µ ( X ) ≤ b µ ( Y ) , and ther efor e d deg( X ) ≤ d deg( Y ) . 9 Namely a conca ve function hav ing v alue 0 at the origin and which is piecewise linear. 25 Pr o of. Let ( X λ ) λ ∈ R and ( Y λ ) λ ∈ R be the Harder- Narasimhan filtratio ns of X and o f Y resp ec- tively . Theorem 5.3 implies that f is compa tible with filtrations. Hence rk( X λ ) ≤ r k( Y λ ) for any λ ∈ R . Therefo r e, b y taking an interv al [ − M , M ] containing supp( ν X ) ∪ s upp( ν Y ), we obtain b µ ( X ) = Z M − M t d ν X ( t ) = − Z M − M t d rk( X t ) = h − t rk( X t ) i M − M + Z M − M rk( X t )d t ≤ M rk ( X M ) + Z M − M rk( Y t )d t = M r k( Y M ) + Z M − M rk( Y t )d t = b µ ( Y ) . ✷ Let D b e a n Abelia n categor y and r k b e a r ank function on D . I t is interesting to calcula te explicitly the Harder -Narasimhan filtration o f an o b ject Y in D , equipp ed with an R -filtratio n F = ( Y λ ) λ ∈ R which is separated, exhaustive, left contin uous and of finite length. Let U = { λ 1 > · · · > λ n } b e the minimal jumping s e t of the filtra tion F , then 0 / / Y λ 1 / / Y λ 2 / / · · · / / Y λ n = Y is the Harder-Nar a simhan sequence of Y = ( Y , [ F ]). Therefore, the Har der -Narasimha n filtra- tion of Y is just the filtration F its e lf. So w e hav e P ′ Y ( t ) = n X i =1 λ i 1 1 [ t i − 1 ,t i [ where t 0 = 0, a nd for a n y 1 ≤ i ≤ n , t i = rk( Y λ i ) / rk( Y ). F urthermore, ν Y = n X i =1 ( t i − t i − 1 ) δ λ i . Let F : C → D b e an exact functor from C to an Ab elian categ o ry D . The functor F induces a functor b F : C A → D which sends an a rithmetic ob ject X to F ( X ), it als o induces a homo- morphism of g roups K 0 ( F ) : K 0 ( C , E , A ) → K 0 ( D ). Since F is exact, it sends mono mo rphisms to monomorphisms, therefor e it induces a functor e F : Fi l R , self ( C ) → Fi l R , self ( D ). If X is an arithmetic ob ject of ( C , E , A ), then e F (HN( X )) is an R -filtra tion of F ( X ). The following prop o- sition s hows that we ca n recov er the Harder -Narasimhan p olygon a nd the Harder- Na rasimhan measure of X fro m the filtration e F (HN( X )). Prop ositio n 5.7 Supp ose given a r ank fun ction rk on K 0 ( D ) (which defines a Har der-Nar asimhan c ate gory s tructur e on D ) such that the fun ctor F pr eserves r ank functions (i.e. rk( F ( X )) = rk( X ) for any X ∈ o b j C ). Then for any arithmetic obje ct X in C A , t he normalize d Har der- Nar asimhan p olygon of the filtr ation F ( X ) = ( F ( X ) , [ e F (HN( X ))]) c oincides with t hat of X , and the Har der-Nar asimhan me asur e of F ( X ) c oincides with that of X . Pr o of. Since the Harder -Narasimha n filtration of F ( X ) coincides with e F (HN( X )), the func- tion t 7− → − rk( HN( X )( t )) iden tifies with t 7− → − rk( e F (HN( X ))( t )). There fo re ν F ( X ) = ν X and hence P F ( X ) = P X . ✷ Let ( C , E , A ) be an arithmetic exac t category , d deg b e a degree function on ( C , E , A ) and rk b e a rank function on ( C , E ). If ( C , E ) is a n Ab elian ca tegory , then the a xioms for ( C , E , A, d deg , rk) to b e a Har der-Naras imhan categ ory can be considerably simplified. W e shall show this fact in Pr o po sition 5 .8. 26 Prop ositio n 5.8 Sup ose t hat ( C , E ) is an Ab elian c ate gory. Then ( C , E , A, d deg , rk) is a Har der- Nar asimhan c ate gory if the fol lowing c onditions ar e satisfie d: 1) for any non-zer o arithmetic obje ct X , ther e exists a non-zer o arithmetic sub obje ct Z of X such that b µ ( Z ) = sup { b µ ( Y ) | Y is a non-zer o arithmetic sub obje ct of X } ; (7) 2) for any non-zer o obje ct X in C and for any two arithmetic structu r es h X and h ′ X on X , if Id X : ( X , h X ) → ( X, h ′ X ) is c omp atible with arithmetic struct u r es, then b µ ( X, h X ) ≤ b µ ( X, h ′ X ) . Note that the condition 1) is verified once { b µ ( Y ) | Y is a non-zer o a rithmetic sub ob ject of X } is a finite set, or equiv a lent ly { d deg( Y ) | Y is a non-z e r o arithmetic sub ob ject of X } is a finite set for an y non-ze r o ar ithmetic ob ject X . The following technical lemma, whic h is dual to Lemma 4.6, is useful for the pro of of Prop ositio n 5.8. Lemma 5.9 L et ( C , E , A ) b e an arithmetic exact c ate gory. Supp ose t hat any monomorphism in C has a c okernel. L et ( X , h X ) and ( Y , h Y ) b e two arithmetic obje cts and f : X → Y b e a morphism in C . Supp ose that ( Y , h Y ) is an arithmetic sub obje ct of an arithmetic obje ct ( Z , h Z ) and u : Y → Z is the inclusion morphism. Then the morphism f is c omp atible with arithmetic structur es if and only if it is the c ase for uf . Pr o of of Pr op osition 5.8. Supp ose that X des is a non-zero a r ithmetic sub ob ject o f X verifying (7), whose ra nk r is ma x imal. Supp o se that Z is another non-zer o arithmetic subo b ject of X verifying (7). Co nsider the short e x act sequence 0 / / Z ∩ X des / / Z ⊕ X des / / Z + X des / / 0 , where Z ∩ X des is the fib er pro duct Z × X X des and Z + X des is the canonical image o f Z ⊕ X des in X . Therefore, d deg( Z ∩ X des ) + d deg( Z + X des ) = d deg( Z ) + d deg( X des ) = α (rk ( Z ) + r k( X des )) , so d deg( Z + X des ) = α (rk ( Z ) + r k( X des )) − d deg( Z ∩ X des ) ≥ α (rk( Z ) + rk ( X des ) − rk( Z ∩ X des )) = α rk( Z + X des ) , which means that b µ ( Z + X des ) = α , and hence rk( Z + X des ) = rk( X des ) s ince rk( X des ) is maximal. As r k is a rank function, we obtain Z = X des . Therefore , the axiom ( HN 1 ) is fulfilled. W e now verify the axiom ( HN 2). Let X = ( X, h X ) and Y = ( Y , h Y ) b e tw o semistable arithmetic ob jects. Supp o se that there exists a non-zero mor phism f : X → Y which is compatible with arithmetic ob jects. Let Z b e the image of f in Y , u : Z → Y b e the canonical inclusion and π : X → Z be the canonical pro jection. The fac t that f is compatible with a rithmetic structur es implies that the identit y mo rphism Id Z : ( Z , π ∗ h X ) → ( Z , u ∗ h Y ) is compatible with arithmetic structures (after Lemmas 4 .6 and 5.9). Therefore, the semistability of X and of Y , co m bining the condition 2), implies that b µ ( X ) ≤ b µ ( Z, π ∗ h X ) ≤ b µ ( Z, u ∗ h Y ) ≤ b µ ( Y ).  27 Corollary 5.10 L et ( C , E ) b e an Ab elian c ate gory e quipp e d with a r ank fun ction rk , n ≥ 2 b e an inte ger, ( A i ) 1 ≤ i ≤ n b e a family of arithmetic structu r es on ( C , E ) and A = A 1 × · · · × A n . Supp ose given for any 1 ≤ i ≤ n a de gr e e function d deg i on ( C , E , A i ) such that 1) { d deg i ( Y ) | Y is a non- zer o arithmetic sub obje ct of X } is a finite set for any non-zer o arith- metic obje c X ; 2) ( C , E , A i , d deg i , r k) is a Har der-Nar asimhan c ate gory. Then for any α = ( a i ) 1 ≤ i ≤ n ∈ R n ≥ 0 , if we denote by d deg α = P n i =1 a i d deg i , then ( C , E , A, d deg α , r k) is a Har der-Nar asimhan c ate gory. 6 Examples of Harder-Narasimhan categories In this sectio n, we sha ll give some exa mple of Harder- Narasimhan c a tegories. Filtrations in an extension of Ab elian categories Let C and C ′ be tw o Abelian catego ries and F : C → C ′ be an exa c t functor which sends a non-zer o o b ject o f C to a non-zero ob ject of C ′ . Let E (res p. E ′ ) b e the class of all exact sequences in C (resp. C ′ ). Supp ose given a rank function rk ′ : K 0 ( C ′ , E ′ ) → R . Let I b e a non- empt y s ubset of R , eq uipped with the induced order. F or an y ob ject X in C , let A ( X ) be the set o f is omorphism cla sses of ob jects in Fi l I , self F ( X ) . Suppo se that h = [ F ] is an element in A ( X ). F or any monomorphism u : X 0 → X , we define u ∗ ( h ) to b e the class [ F ( u ) ∗ F ] ∈ A ( X 0 ). F or any epimor phism p : X → Y , we define p ∗ ( h ) to b e [ F ( p ) ∗ F ] ∈ A ( Y ). Similarly to the the case of filtrations in an Abelia n categ ory , ( C , E , A ) is an arithmetic exact categ ory . By definition w e know that if X i = ( X i , [ F i ]) ( i = 1 , 2 ) are tw o arithmetic o b jects, then a morphism f : X 1 → X 2 in C is compatible with arithmetic s tr uctures if and only if F ( f ) is compatible with filtrations ( F 1 , F 2 ). F or any ar ithmetic o b ject X of ( C , E , A ), we define the arithmetic deg ree of X = ( X , [ F ]) to b e the r e a l num b er d deg( X ) = X λ ∈ I λ  rk ′ ( F ( λ )) − sup j >λ,j ∈ I rk ′ ( F ( j ))  . Since F is an ex act functor , d deg extends naturally to a homomo rphism from K 0 ( C , E , A ) to R . In the previous section we hav e shown that if we define, for a n y ob ject X ′ in C ′ , A ′ ( X ′ ) as the set of all iso morphism classe s of ob jects in Fil I , self X ′ , then ( C ′ , E ′ , A ′ ) is an ar ithmetic categor y . F urther more, if fo r any ar ithmetic o b ject X ′ = ( X ′ , [ F ′ ]), we define d deg ′ ( X ′ ) = X λ ∈ I λ  rk ′ ( F ′ ( λ )) − sup j >λ,j ∈ I rk ′ ( F ′ ( j ))  , then d deg ′ extends na turally to a homomorphism K 0 ( C ′ , E ′ , A ′ ) → R , and ( C ′ , E ′ , A ′ , d deg ′ , r k ′ ) is a Ha rder-Nar a simhan category . Notice tha t for any ob ject ( X , [ F ]) in C A , we have d deg( X , [ F ]) = d deg ′ ( F ( X ) , [ F ]) . Prop ositio n 6.1 Denote by rk the c omp osition rk ′ ◦ K 0 ( F ) . Then ( C , E , A, d deg , rk) is a Har der- Nar asimhan c ate gory. 28 Pr o of. Since F is an ex a ct functor which sends non-zero ob jects to non- zero ob jects, the homomorphism r k is a rank function. Let X = ( X, [ F ]) b e a non-zer o arithmetic ob ject in C A . First we s how that S :=  d deg( Y ) | Y is an a rithmetic sub ob ject of X  is a finite set. Let U = { λ 1 , · · · , λ n } be a jumping set of F . If u : Y → X is a monomorphism, then U is a lso a jumping set of F ( u ) ∗ F , therefo re, d deg( Y , [ F ( u ) ∗ F ]) ∈ n n X i =1 a i λ i    ∀ 1 ≤ i ≤ n, a i ∈ N , 0 ≤ a 1 + · · · + a n ≤ rk( X ) o . The la tter is cle a rly a finite set. Therefore, the c ondition 1) of P rop osition 5.8 is sa tisfied. If X is a n ob ject in C and if F and G are tw o filtra tions of F ( X ) such that Id F ( X ) = F (Id X ) is com- patible to filtrations ( F , G ), then after Prop ositio n 5 .6, d deg ′ ( F ( X ) , [ F ]) ≤ d deg ′ ( F ( X ) , [ G ]) and therefor e b µ ( X , [ F ]) ≤ b µ ( X , [ G ]). After Pr op osition 5.8, ( C , E , A, d deg , rk) is a Har der- Narasimhan catego ry . ✷ Remark 6.2 B y Coro llary 5 .10, we ca n easily generaliz e the formalism of Harder and Nar asimhan to the cas e of b o jects in C equipp ed with several filtr a tions of their images by F in C ′ . Filtered ( ϕ , N ) -mo dules Let K b e a field of characteristic 0, equipp ed with a discr e te v aluation v such that K is complete fo r the top ology defined by v . Supp o se that the residue field k of K is of characteristic p > 0. Le t K 0 be the fraction field of Witt vector r ing W ( k ) and σ : K 0 → K 0 be the absolute F r o be nius endomo rphism. W e call ( ϕ, N ) -mo dule (see [8], [20], a nd [6] for details) any finite dimensional vector s pace D over K 0 , equipp ed with 1) a bijective σ -linear endomo rphism ϕ : D → D , 2) a K 0 -linear endomo rphism N : D → D such that N ϕ = pϕN . Let C be the catego r y o f all ( ϕ, N )-modules. It’s an Abelian category . W e denote by E the class o f all s hort exact sequences in C . There exis ts a natural r ank function rk on the category C defined b y the rank of vector space over K 0 . F urthermore, we hav e an exa ct functor F from C to the categor y V ec K of all finite dimensional vector spaces over K , which s ends a ( ϕ, N )-mo dule D to D ⊗ K 0 K . Co nsider the arithmetic structur e A on ( C , E ) such that, for any ( ϕ, N )-mo dule D , A ( D ) is the s e t of isomorphis m classes of Z -filtra tions o f F ( D ) = D ⊗ K 0 K . Then ( C , E , A ) b ecomes a n ar ithmetic exact c a tegory . The ob jects in C A are calle d filter e d ( ϕ, N ) -mo dules . T o each ( ϕ, N )-module D we as s o ciate a n integer deg ϕ ( D ) = − v (det ϕ ). If D = ( D , [ F ]) is a filtered ( ϕ, N )-mo dule, w e define deg F ( D ) := X i ∈ Z i  rk K ( F ( i )) − rk K ( F ( i + 1))  and d deg( D ) = deg F ( D ) + deg ϕ ( D ) . It is clea r that d deg is a degr e e function on ( C , E , A ). Prop ositio n 6.3 ( C , E , A, d deg , rk) is a Har der-Nar asimhan c ate gory. 29 Pr o of. Let X = ( X , [ F ]) b e a non- zero filtered ( ϕ, N )-mo dule. W e hav e shown in the previous example that S F = { deg F ( Y ) | Y is an arithmetic sub ob ject of X } is a finite set. By the iso clinic dec ompo sition we obtain that S ϕ = { deg ϕ ( Y ) | Y is a sub ob ject of X } is also a finite set. Ther e fore, e S = { b µ ( Y ) | Y is an a rithmetic sub ob ject of X } is a finite set, and he nc e the co ndition 1) of P rop osition 5.8 is verified. Suppo se that X is a ( ϕ, N )-mo dule and F a nd G ar e t wo Z -filtrations of X such that Id X is compatible with filtrations ( F , G ). W e have shown in the previous example that deg F ( X, F ) ≤ deg F ( X, G ). Hence d deg( X , F ) ≤ d deg( X , G ). Therefore, the condition 2) of Prop ositio n 5.8 is verified, and hence ( C , E , A, d deg , rk) is a Harder -Narasimha n categ ory . ✷ Note that semis table filtered ( ϕ, N )-modules having slo pe 0 are nothing but admissible filtered ( ϕ, N )-mo dules. In classical literature, such filtered ( ϕ, N )-modules are said to be weakly admis s ible. In fa c t, Co lmez and F ontaine [6] hav e prov ed that all weakly a dmiss ible ( ϕ, N )-mo dules are admissible, which had b een a conjecture of F o n taine. T orsion free shea ves on a p olarized pro jective v ariet y Let X b e a g eometrically nor mal pro jective v ariety of dimension d ≥ 1 ov er a field K a nd L be an ample inv ertible O X -mo dule. W e denote by TF ( X ) the catego ry of tor s ion free coherent sheav es on X . Notice that if 0 / / E ′ / / E / / E ′′ / / 0 is an exact seq uenc e of coherent O X -mo dules s uc h that E ′ and E ′′ are tor sion free, then also is E . Therefore, TF ( X ) is an exact sub-ca tegory o f the Abelia n categor y of a ll coherent O X -mo dules on X . Let E be the class of all exact s equences in TF ( X ) and let A be the tr ivial arithmetic structure on it. If E is a torsion free coherent O K -mo dule, we denote b y rk( E ) its rank a nd by deg ( E ) the intersection num b er c 1 ( L ) d − 1 c 1 ( E ). The mapping deg (resp. r k) extends natura lly to a homomorphism from K 0 ( TF ( X )) to R (resp. Z ). A classical result [1 4] (see also [18]) shows that ( TF ( X ) , E , A, deg , r k) is in fact a Ha rder-Nara simhan catego ry . Hermitian v ector bundles on the sp ectrum of an algebraic integer ring Let K be a num ber field and O K be its integer ring. W e denote by Pro ( O K ) the category of all pro jective O K -mo dules of finite type. Let E be the family of short exac t sequence s of pro jective O K -mo dules o f finite type. Then ( Pro ( O K ) , E ) is an exact ca tegory . W e denote b y Σ f the set of a ll finite places of K whic h ide ntifies with the set of closed p oints of Sp e c O K . If p is an element in Σ f , we de no te by v p : K × → Z the v aluation asso ciated to p which sends a no n-zero element a ∈ O K to the leng th of the Artinian lo c a l ring O K, p /a O K, p . Let F p := O K, p / p O K, p be the residue field a nd N p be its cardina l. W e denote by | · | p the absolute v alue on K such that | x | p = N − v p ( x ) p for any x ∈ K × . Let Σ ∞ be the set of all embeddings of K in C , whose car dinal is [ K : Q ]. F or any σ ∈ Σ ∞ , let | · | σ : K → R ≥ 0 be the Archimedian a bsolute v alue suc h that | x | σ = | σ ( x ) | . The complex conjugation defines an inv o lution σ 7→ σ on Σ ∞ . The pro duct formula asser ts that for any x ∈ K × , | x | p = 1 for almost all finite places p , and we hav e Y p ∈ Σ f | x | p Y σ ∈ Σ ∞ | x | σ = 1 . Notice that a Hermitian vector bundle over Sp ec O K is nothing other than a pair E = ( E , ( k · k σ ) σ ∈ Σ ∞ ), where E is a pro jective O K -mo dule of finite t yp e E , and for any σ ∈ Σ ∞ , 30 k · k σ is a Hermitian metric on E ⊗ O K ,σ C suc h tha t k x ⊗ z k σ = k x ⊗ z k σ . The rank of the Hermitian vector bundle E is just defined to b e that of E . The rank function on Pro ( O K ) extends naturally to a homomo rphism from K 0 ( Pro ( O K )) to Z . If E is a Hermitian v ector bundle of rank r , the (nor malized) Arakelov degr ee of E is b y definition d deg n E = 1 [ K : Q ]  log #( E / O K s 1 + · · · + O K s r ) − 1 2 X σ ∈ Σ ∞ log det( h s i , s j i σ )  , where ( s 1 , · · · , s r ) ∈ E r is a n ar bitr ary element in E r which defines a basis of E K ov er K . This definition do esn’t dep e nd o n the choice of ( s 1 , · · · , s r ). F or more details, see [1] and [4]. If for any pro jective O K -mo dule of finite t yp e E , we denote by A ( E ) the set of all Her- mitian structures on E , then ( Pro ( O K ) , E , A ) is an arithmetic exact category , as we hav e shown in the previous sectio n. The category Pro ( O K ) A is the category of all Hermitian vec- tor bundles ov er Sp ec O K and all homomor phism of O K -mo dules having norm ≤ 1 at e very σ ∈ Σ ∞ . F urther more, if 0 / / E ′ / / E / / E ′′ / / 0 is a sequence in E A , then we hav e the equality d deg n ( E ) = d deg n ( E ′ ) + d deg n ( E ′′ ). 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