Adelic geometry on arithmetic surfaces I: idelic and adelic interpretation of the Deligne pairing
Paolo Dolce

 

University of Udine, Italy
paolo.dolce@uniud.it
Adelic geometry on arithmetic surfaces I: idelic and adelic interpretation of the Deligne pairing
    Paolo Dolce
    December 30, 2023
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For an arithmetic surface X→ B= O_K the Deligne pairing < ,  >(X)×(X)→(B) gives the “schematic contribution” to the Arakelov intersection number. We present an idelic and  adelic interpretation of the Deligne pairing; this is the first crucial step for a full idelic and adelic interpretation of the Arakelov intersection number. 

For the idelic approach we show that the Deligne pairing can be lifted to a pairing < , >_i:(d^1_×)×(d^1_×)→(B), where (d^1_×) is an important subspace of the two dimensional idelic group 𝐀_X^×.  On the other hand, the argument for the adelic interpretation is entirely cohomological.


starttoctoc









§ INTRODUCTION





 §.§ Background

Problems of topological nature on (ℚ, |·|) (where |·| is the usual euclidean absolute value), are commonly solved after an embedding of ℚ in  its completion ℝ with respect to |·|. In this way we can take advantage of the completeness properties of ℝ and the density of ℚ inside ℝ. Any other  absolute value which doesn't give the standard euclidean topology on ℚ is equivalent (in the sense of absolute values, see footnote) to the p-adic absolute value  |·|_p for any prime p, so it makes sense to embed ℚ densely in its completion ℚ_p with respect to |·|_p. The above way of reasoning can be easily generalized for any number field K, and adeles were introduced in 1930s by Chevalley in order to consider simultaneously all the completions of K with respect all possible places[A place 𝔭 is an equivalence class of absolute values on K where two absolute values are declared equivalent if they generate the same topology.]. It is not very useful to study simply the product ∏_𝔭 K_𝔭 of all completions because the resulting space is “too big” and fails to be a locally compact additive group. For any non-archimedean place 𝔭 let 𝒪_𝔭 be the closed unit ball in K_𝔭 under the standard representative of the place 𝔭, then the ring of adeles is defined as a subset of ∏_𝔭 K_𝔭, namely:

    𝐀_K:='∏_𝔭 K_𝔭

where '∏ is the restricted products with respect to the additive  subgroups {𝒪_𝔭𝔭 is non-archimedean}. Note that for archimedean places, the unit ball is not an additive subgroup. The most important features of 𝐀_K were well described in  <cit.> and consist mainly in the fact that 𝐀_K is a locally compact additive group (so it admits a Haar measure), K is discrete in 𝐀_K, and the quotient 𝐀_K/K is compact. Moreover the Pontryagin dual 𝐀_K has a very simple description and 𝐀_K≅𝐀_K. 

The multiplicative version of the adelic theory is the idelic theory, and the group of ideles attached to K is defined as:

    𝐀^×_K='∏_𝔭 K^×_𝔭

where the restricted product is taken with respect to the subgroups 𝒪^×_𝔭:={x∈𝒪_𝔭 |x|_𝔭=1}. 
In this case 𝒪^×_𝔭 has a group structure also in the archimedean case. But a number field K can be seen as the function field of the nonsingular arithmetic curve B= O_K, where O_K is the ring of integers of K,  and we know  that there is a bijection between points of the completed curve B in the sense of Arakelov geometry and places of K. Therefore the ring of adeles attached to K can be described in a more geometric way  related to B:

    𝐀_B:= '∏_b∈BK_b=𝐀_K

where K_b is still the local field attached to the “point” b. So, classical adelic theory can be deduced from 1-dimensional arithmetic geometry. We can adopt a similar approach but starting from 1-dimensional algebraic geometry: fix a nonsingular algebraic projective curve X over a perfect field k with function field denoted by k(X); then to each closed point x∈ X we can associate a non-archimedean local field K_x with its valuation ring denoted by 𝒪_x. The ring of adeles associated to X is then:

    𝐀_X:= '∏_x∈ XK_x ;

In this case 𝐀_X is not a locally compact additive group unless k is a finite field. Each K_x is endowed with a structure of locally linearly compact k-vector space (in the sense of <cit.>), therefore 𝐀_X is again locally linearly compact and one can show similarly to the arithmetic case that: k(X) is discrete in 𝐀_X, the quotient 𝐀_X/k(X) is a linearly compact k-vector space and 𝐀_X is a self dual k-vector space. In other words, from a topological point of view, the passage from arithmetic theory to algebraic theory implies that we substitute the theory of compactness of groups with the theory of linear compactness of vector spaces. In both arithmetic and geometric 1-dimensional case, adelic and idelic theory give a generalization of the intersection theory (i.e. the theory of degree of divisors):


   Ideles can be easily seen as a generalization of line bundles  (resp. Arakelov line bundles), so it is natural to give an extension of the theory of divisors (resp. Arakelov divisors) from an idelic point of view.

   For an algebraic curve X and any divisor D∈(X) we can define a subspace 𝐀_X(D)⊂𝐀_X and a complex 𝒜_X(D). The cohomology of 𝒜_X(D) is equal to the usual Zariski cohomology H^i(X,𝒪_X(D)), therefore we can give an interpretation of (D) in terms of the characteristic of 𝒜_X(D) which will be called the adelic characteristic. For a completed  arithmetic curve B we cannot define a complex 𝒜_B(D) associated to an Arakelov divisor D, since for archimedean points closed unit balls are not additive groups. However, one can recover the Arakelov degree of D as the product of volumes (with respect to  an opportune choice of Haar measures) of certain closed balls in K_b. 
 
The above theory remains valid also in the case of singular curves, because by normalization we can always reduce to the nonsingular case.

The first attempt to construct a 2-dimensional adelic/idelic theory from 2-dimensional geometry was partially made by Parshin in <cit.>, but he treated only rational adeles (i.e. a subset of the actual ring of adeles) for algebraic surfaces. A more structured approach to the adelic theory for algebraic surfaces was given in <cit.>, still with a few mistakes[for example in <cit.> the definition of the object K_x, and consequently of the subspace A_02, is wrong (compare with section <ref> for more details).], and it can be summarized in the following way: let's fix a nonsingular, projective surface (X,𝒪_X) over a perfect field k, then to each “flag”  x∈ y made of a closed point x inside an integral curve y⊂ X we can associate the ring K_x,y which will be a 2-dimensional local field if y is nonsingular at x, or a finite product of 2-dimensional local fields if we have a singularity. Roughly speaking a 2-dimensional local field is a local field whose residue field is again a local field (see subsection <ref>), and in our case K_x,y carries two distinct levels of discrete valuations: there is the discrete valuation associated to the containment x∈ y and the discrete valuation associated to y⊂ X. Formally, K_x,y is obtained through a process of successive localisations and completions starting from 𝒪_X,x. With the symbol 𝒪_x,y we denote the product of valuation rings inside K_x,y. Similarly to the 1-dimensional theory, we perform a “double restricted product”, first over all points ranging on a fixed curve,  and then over all curves in X, in order to obtain the ring of adeles for surfaces:

    𝐀_X:=”∏_x∈ y
     y⊂ X K_x,y⊂∏_x∈ y
     y⊂ X K_x,y .

The topology on K_x,y can be defined  thanks to the construction by completions and localisations, and by starting with the standard 𝔪_x-adic topology on 𝒪_X,x, then the topology on 𝐀_X can be defined canonically. The idelic group attached to X is 𝐀^×_X.

 For 2-dimensional local fields like K_x,y there is a well known theory of differential forms and residues (see for example <cit.>); one can globalise the constructions in order to obtain a k-character  ξ^ω:𝐀_X→ k associated to a rational differential form ω∈Ω^2_k(X)|k and the differential pairing:

    d_ω:𝐀_X×𝐀_X    →    k
    
    (α,β)    ↦   ξ^ω(αβ) .
 
In <cit.> it is shown that: ξ^ω induces the self duality in the category of k-vector spaces,  of  𝐀_X, the subspace 𝐀_X/k(X)^⊥ is linearly compact (orthogonal spaces are calculated with respect to d_ω) and k(X) is discrete in 𝐀_X. 

Both 𝐀_X and 𝐀^×_X carry some important subspaces which in turn lead to the construction of certain complexes 𝒜_X(D), for a divisor D, and 𝒜_X^× (respectively the “adelic complex” and the “idelic complex”). The cohomology of such complexes can be calculated by geometric methods thanks to the following important results:

    H^i(𝒜^×_X)≅ H^i(X,𝒪^×_X) ,


    H^i(𝒜_X(D))≅ H^i(X,𝒪_X(D)) .

For a proof of (<ref>) and (<ref>) see respectively <cit.> and <cit.>.  Again, idelic and adelic theory give an extension of the intersection theory on X:


   
In <cit.> it is shown that the group (X) can be lifted to a subspace if 𝐀^×_X and the intersection pairing on (X) can be extended at the level of ideles.


   In  <cit.> it is shown that the characteristic of the complex 𝒜_X(D) can be used to redefine the intersection pairing between two divisors in terms of adeles, even without using isomorphism (<ref>). Such a theory gives an alternative approach to the Riemann-Roch theorem for algebraic surfaces. 

Most of the 2-dimensional constructions outlined above are true for any 2-dimensional, Noetherian, regular scheme, so in particular for arithmetic surfaces. However,  the adelic theory associated to an arithmetic surface X→ B= O_K (K is a number field) is more complicated and less developed. Locally, the rings K_x,y have a completely  different structure  between each other as 2-dimensional local fields, depending whether y is horizontal or vertical. Moreover, there was the global issue of interpreting the archimedean data of the completed surface X, in the sense of Arakelov geometry, in an adelic and idelic way. A first definition of the “full” (or completed) ring of adeles 𝐀_X has been given only recently in <cit.>.

The purpose of this series of papers is to study the adelic and idelic  theory for a completed arithmetic surface and eventually obtain a 2-dimensional generalisation of the Tate thesis. 




In <cit.> Beilinson shortly described how to attach a n-dimensional adelic theory to any n-dimensional Noetherian scheme  in a very abstract functorial way. Reworks and clarifications of this approach are <cit.> and <cit.>. In particular in <cit.> it is proved  that for dimensions 1 and 2 our explicit theory agrees with Beilinson theory of adeles.





 §.§ Results in this paper

In this first paper we explain the “schematic part” of the idelic and adelic lift of the Arakelov intersection number  (i.e. ignoring the fibres at infinity), whereas a full account of the theory will be published subsequently together with other results. 

Given two Arakelov divisors D=D+∑_σα_σ X_σ and E=E+∑_σβ_σ X_σ, where D,E∈(X), one piece of the Arakelov intersection number D.E is obtained thanks to the Deligne pairing <𝒪_X(D),𝒪_X(E)>∈(B). We define d^1_× to be an arrow of the idelic complex 𝒜^×_ X associated to X, then  we construct an idelic Deligne pairing:

    < , >_i:(d^1_×)×(d^1_×)→(B)

which descends to the Deligne pairing through the composition:

    (d^1_×)×(d^1_×)→(X)×(X)→(X)×(X) .

This will be the arithmetic version  of Parshin idelic lift  given in <cit.>. Our approach will be from local to global:  the main idea consists in globalising Kato's local symbol for 2-dimensional local fields containing a local field (see  <cit.> or <cit.>), which is the generalisation of the usual tame symbol for valuation fields (see appendix <ref>). In the arithmetic framework given by the arithmetic surface φ:X→ B, we have the following constructions: for any point x sitting on a curve y⊂ X we define a ring, which is a finite sum of 2-dimensional local fields, denoted by K_x,y; moreover K_b is the local field associated to the point b∈ B such that φ(x)=b. Then Kato's symbol translates into a skew symmetric, bilinear map:

    ( , )_x,y:K^×_x,y× K^×_x,y→ K^×_b .

Roughly speaking, by composing it with the valuation v_b on K_b and by summing over all  flags x∈ y such that φ(x)=b, we show that we obtain a well defined integer n_b. By repeating the argument for each b∈ B we obtain a divisor ∑_b∈ B n_b[b]. At this point we prove that such a pairing descends to the Deligne pairing.

The adelic theory is very similar to the geometric case and the crucial point consists in  considering the arithmetic analogue of the Euler-Poincare characteristic of coherent sheaves, i.e. the determinant of cohomology. We use the cohomological properties of the adelic complex of the base scheme B in order to give the definition of the adelic determinant of cohomology. Then it is enough to use the fact that the Deligne pairing can be expressed in terms of  the (adelic) determinant of cohomology. 

 



  
Overview of the contents. Section <ref> is a quick review of adelic geometry for arithmetic surfaces, where just by simplicity, we ignore the contribution of fibres at infinity. A more comprehensive introduction to adelic geometry can be found in <cit.>. In section <ref> and <ref> we construct respectively the idelic and adelic Deligne pairing. Finally, appendix <ref> is just a collection of the basic notions of algebraic K-theory needed in this paper and appendix <ref> is a review of the main features of the determinant of cohomology.  



  
Basic notations.
All rings are considered commutative and unitary. When we pick a point x in a scheme X we generally mean a closed point if not otherwise specified, also all sums ∑_x∈ X are meant to be “over all closed points of X”. The cardinality of a set T is denoted as #(T). If F is a field, then F doesn't denote the algebraic closure. For a morphisms of schemes f:X→ S,   the schematic preimage of s∈ S is X_s. Sheaves are denoted with the “mathscr” lateχ font; particular the structure sheaf of a scheme X is 𝒪_X (note the difference with the font 𝒪). For any 𝒪_X-module ℱ and any D∈(X) we put ℱ(D):=ℱ⊗_𝒪_X𝒪_X(D). The notation (·) is used for the notion of “determinant” in the category of free modules over a ring and in the category of free 𝒪_X-modules; the  exact meaning will be clear from the context.  If K is a number field and X→ O_K is an integral scheme over the ring of integers O_K, then the function field of X is denoted by K(X). Finally it is important to point out that the letter K will denote different mathematical objects in this paper (and in different contexts), so the reader should check at the beginning of each section its specific meaning from time to time.



  
Acknowledgements. I would like to thank Ivan Fesenko and Weronika Czerniaswka from the university of Nottingham for the interesting discussion they had with me about the topic and their support. Moreover a special thanks to Fedor Bogomolov and Nikolaos Diamantis for reading my work and to  Dongwen Liu for answering to my questions. 


This work is supported by the EPSRC programme grant EP/M024830/1 (Symmetries and correspondences: intra-disciplinary developments and applications).



§ REVIEW OF 2-DIMENSIONAL ADELIC GEOMETRY



 §.§ Abstract local theory

Let's recall the definition of local field:

A local field (or a 1-dimensional local field) F, is one of the fields listed below:


  (1)F=ℝ endowed with  the usual real absolute value|·|.

  (2) F=ℂ endowed with  the usual complex  absolute value ||·||.

  (3) F is a complete discrete valuation field (the valuation is surjective) such that the residue field F is a perfect field. The valuation ring of F is denoted as 𝒪_F and its maximal ideal is 𝔭_F. Moreover if v is the valuation on F, then the absolute value  is given by |x|_v:=q^-v(x), where q=#(F) if F is a finite field, and q=e:=exp(1) otherwise.

If F is of type (1) or (2), it is an archimedean local field otherwise it is a non-archimedean local field. A local field is topologized with the topology induced by the absolute value. A morphism between local fields is a continuous field homomorphism.


According to our definition, a non-archimedean local field endowed with its natural topology is in general not locally compact.

Remember that if F is a non-archimedean local field there exists only one surjective complete valuation on it (see <cit.>).  A higher local field is a simple generalization of definition <ref>: given a complete discrete valuation field F, it might happen that the residue field F^(1):=F is again a complete discrete valuation field; by taking one more time the residue field we have the field F^(2). In other words, a complete discrete valuation field might originate a potentially infinite sequence of fields  {F^(i)}_i≥ 0 such that F^(0)=F and F^(i+1)=F^(i). Each F^(i) is called the i-th residue field.
 

A n-dimensional local field, for n≥ 2, is a complete discrete valuation field F admitting sequence of residue fields {F^(i)}_i>0  such that  F^(n-1) is a local field. If F^(n-1) is an archimedean local field, then F is called archimedean, otherwise we say that F is non-archimedean. F has mixed characteristic if (F)≠(F).


The simplest n-dimensional local field is the field of iterated Laurent series over a perfect field K:

    F=K((t_1))… ((t_n)) .

 If f=∑ a_jt_n^j∈ F, with a_i∈ K((t_1))…((t_n-1)), we have the complete discrete valuation defined by v(f)=min{j a_j≠ 0}. The valuation ring is 𝒪_F=K((t_1))…((t_n-1))[[t_n]] and the residue field is F^(1)= K((t_1))…((t_n-1)). Clearly F^(n)=K.  When n=2, the elements of K((t_1))((t_2)) are the formal power series ∑_i,j a_i,jt^i_1t_2^j such that a_i,j=0 when the indexes i and j are chosen in the following way: let's plot the couples (j,i) as a lattice on the plane, then we select a semiplane like the one which is not coloured in figure <ref>. The coordinate j is bounded from right, whereas the coordinate i is bounded from above by a descending staircase line. 
 




The above definition of dimension for a n-dimensional local field might seem quite counter-intuitive, indeed a n-dimensional local field can also be a m-dimensional local field for m≠ n. For instance F=K((t_1))… ((t_n))  is m-dimensional for any m=1,…, n. For our purposes it will be clear from the context which dimension we want to take in account. Often it is convenient to consider the maximum amongst all possible dimensions  (when it exists).



Note in the case of archimedean n-dimensional local fields the n-th residue field doesn't exist.




Let's give a less trivial example of higher local field:

Let (K,v_K) be a non-archimedean local field and consider the following set of (double) formal series:

    K{{t}}:={∑_j=-∞^∞ a_jt^j a_j∈ K, inf_j v_K(a_j)>-∞, lim_j→-∞a_j=0}

Addition and multiplication in K{{t}} are defined in the following way:

    ∑_j=-∞^∞ a_jt^j+∑_j=-∞^∞ b_jt^j=∑_j=-∞^∞ (a_j+b_j)t^j


    ∑_j=-∞^∞ a_jt^j·∑_j=-∞^∞ b_jt^j=∑_j=-∞^∞(∑_r=-∞^∞ a_rb_j-r)t^j

and K{{t}}  becomes a field. Note that the series with index r in equation (<ref>) is actually a convergent series in K. We can also define the following discrete valuation v on K{{t}}:

    v(∑_j=-∞^∞ a_jt^j):=inf_j v_K(a_j) .

It is not difficult to verify that v is a well defined valuation and  K{{t}} is complete with respect to v. Let's now analyse the structure of F=K{{t}} as valuation field:

    𝒪_F={∑_j=-∞^∞ a_jt^j∈ K{{t}} a_j∈𝒪_K}


    𝔭_F={∑_j=-∞^∞ a_jt^j∈ K{{t}} a_j∈𝔭_K}

Consider the surjective homomorphism:

    π:  𝒪_F   →   K((t))
    ∑ a_jt^j    ↦   ∑a_j t_j

where clearly a_j is the natural image of a_j in K. Now it is evident that π induces an isomorphism F≅K((t)). In other words F has a structure of 2-dimensional local field such that F^(1)=K((t)) and F^(2)=K.  Clearly such a construction can be iterated several times to get the field:

    K{{t_1}}…{{t_n}} .
 
For example if K=ℚ_p, then K{{t}} is a 2-dimensional local field of mixed characteristic.

Remember that we have the following classical classification theorem for local fields:

Let F be a local field:


  (1) When F is archimedean, then F=ℝ or F=ℂ.

  (2) When F is not archimedean there are two cases:


  (2a) If F=F, then F≅F((t)).

  (2b) If F≠F=p, then F is isomorphic to K_p which denotes a finite extension of ℚ_p.



(1) is true just by definition. For (2) see for example <cit.>.
Such a classification can be extended for higher local fields, in particular any n-dimensional local field can be obtained by “combining” the higher local fields presented in examples <ref> and <ref>:

Let F be a n-dimensional local field with n≥ 2.


  (1) If F= F^(1)=…= F^(n-1), then 

    F≅ F^(n-1)((t_1))…((t_n-1))

and F^(n-1) is isomorphic to one of the four fields listed in theorem <ref>.

  (2) If r∈{2,3,… n} is  the unique number such that F^(n-r)≠ F^(n-r+1)=p, then: 
  

  (2a) When r≠ n,  F is isomorphic to a finite extension of:

    K_p{{t_1}}…{{t_r-1}}((t_r))…((t_n-1)) .


  (2b) When r=n (i.e. in the mixed characteristic case),  F is isomorphic to a finite extension of:

    K_p{{t_1}}…{{t_n-1}} .




See <cit.>.
In this paper we will focus mainly on 2-dimensional local fields, so let's give  a table with all possible 2-dimensional local fields by using the classification theorem:


    4|c|2-dimensional local fields
    1|c|Geometric   2|c|Arithmetic   1|c|Archimedean
    (0,0,0), (p,p,p)   (0,p,p)   (0,0,p)   
    1-3         
    K((t_1))((t_2)) with K perfect      finite extension of K_p{{t}}   K_p((t))   ℂ((t)) or ℝ((t))




For a non-archimedean local field (F,v), we have the notion of local parameter ϖ which is any generator of the maximal ideal 𝔭_F, or equivalently any element such that v(ϖ)=1. Clearly we have the (recursive) generalization for n-dimensional local fields.

Let F be a non-archimedean n-dimensional local field, then a sequence of local parameters for F is a n-tuple (ϖ_1,…,ϖ_n)∈  F×…× F satisfying the following properties:


   ϖ_n is a local parameter for F.

   (ϖ_1,…,ϖ_n-1)∈𝒪_F×…×𝒪_F and the sequence of natural projections (ϖ_1,…,ϖ_n-1) is a sequence of local parameters for the residue field F.


One can obtain a sequence of local parameters, by applying the following algorithm: choose any local parameter for F^(n-1), then pick any of its liftings in F, this will be ϖ_1.  Choose choose any local parameter for F^(n-2), then pick any of its liftings in F, this will be ϖ_2, etc. Let's give another basic  definition: 

Let F be a n-dimensional local field and put  𝒪^(0)_F:=F, then we define recursively the j-th valuation ring (for j≥1):

    𝒪^(j)_F:={x∈𝒪_Fx∈𝒪^(j-1)_F}

It is clear that 𝒪^(1)_F=𝒪_F. For the algebraic properties of 𝒪^(j)_F the reader can check <cit.>.



 From now on in this section we assume that F is a 2-dimensional local field such that:


   F=0 and F^(2)=p.

   F is endowed with a ST-ring[A ST-ring (semi-topological ring) is just a ring endowed with a linear topology as additive group such that the multiplication for any fixed element is continuous. Note that a ST-ring is not necessarily a topological ring. See <cit.> for more details.] topology and there exists a mixed characteristic local field K with a fixed embedding K↪ F of ST-rings (K has the discrete valuation topology).

In this case we say that F is an arithmetic 2-dimensional local field over K. The presence of the local field  K inside F comes from the theory of arithmetic surfaces and it will be explained in section <ref>. 



  
Equal characteristic.
Let F be an arithmetic 2-dimensional local field such that F=0. 

The coefficient field of F (with respect to K) is the algebraic closure of K inside F and it is denoted as k_F.

The coefficient field k_F is a finite extension of K and moreover F=k_F. In particular  F≅ k_F((t)). The valuation field F is naturally endowed with the usual tame symbol ( , )_F: F^×× F^×→ k_F^×, so we can obtain the Kato symbol (or two dimensional tame symbol) by simply composing it with the field norm map:

The Kato symbol for F (with respect to K) is given by:

    ( , )_F|K:N_k_F|K∘ ( , )_F:F^×× F^×→ K^× .



 


  
Mixed characteristic. 
Now we assume that F is an arithmetic 2-dimensional local field of mixed characteristic. By the classification theorem ℚ_p is contained in F and we have the notion of constant field of F which replaces the one of coefficient field:

The constant field of F is the algebraic closure of ℚ_p in F, and it is denoted by k_F.


Note that the definition of the constant field doesn't depend on K so it makes sense for any 2-dimensional local field of mixed characteristic. Of course it might happen that K=ℚ_p. 

Since K is a finite extension of ℚ_p (by the 1-dimensional classification theorem), we know that k_F is an intermediate field between K and F. The constant field k_F is a finite extension of ℚ_p (so also a finite extension of K).

We say that an arithmetic 2-dimensional local field of mixed characteristic F is standard if there is a k_F isomorpshism F≅ k_F{{t}}. When an isomorphism is given, we say that we have fixed a parametrization of F.

We will study standard fields first and extend any result for a generic F thanks to the following result:

There exists a standard field L contained in F such that: [F:L]<∞, k_F=k_L and F=L.

See <cit.>.
So, from now on in this subsection we fix L to be a standard field contained in F with the properties described in proposition <ref>. Clearly we have the following field extensions that need to be kept always in mind (we mark the finite extensions with the superscript 𝔣):

    ℚ_p⊆^𝔣 K⊆^𝔣 k_L=k_F⊆ L≅ k_L{{t}}⊆^𝔣 F .


 
Finally, we want to define the Kato symbol for F and the strategy is the usual one: we start from k_L{{t}} and we extend our arguments to F by checking that everything is independent from parametrizations and from the choice of the standard fields. We will heavily use some K-theoretic notions developed in appendix <ref>.

Fix just for the moment L=k_L{{t}}, then we define:

    ( , )_L|K: L^×× L^×[r, "{ , }"]     K_2(L)[r]    K_2(L)[r, "-^(2)_L"]    K_1(k_L)=k_L^×[r,"N_k_L|K" ]     K^×

where:


   { , } is the natural projection arising from the definition of K_2(L) (see proposition <ref>).

   The morphism  K_2(L)→K_2(L) is the map given by the construction of K_2(L) as projective limit (see equation (<ref>)).

   ^(2)_L is the higher Kato residue map constructed in theorem <ref>. Note that K_1(k_L)=k_L^× because k_L is already complete.
 
Moreover by simplicity we use the following notation:

    ∂_L: K_2(L)[r]    K_2(L)[r, "-^(2)_L"]     k_L^× .



<cit.> gives an explicit description of ^(2)_L which involves winding numbers.
 

Let L be a generic standard field and fix a parametrization: p: L→ k_L{{t}}  then we define:

    ( , )_L|K: L^×× L^×[r, "{ , }"]     K_2(L)[r, "K_2(p)"]     K_2(k_L{{t}})[r, "∂_k_L{{t}}"]     k_L^×[r,"N_k_L|K" ]     K^×

and we put ∂_L:=∂_k_L{{t}}∘ K_2(p).


Let L be a standard field, then the definition of ( , )_L|K doesn't depend on the parametrization of L.

See <cit.>.
At this point we are ready to give the general definition of the Kato symbol:

Let F an arithmetic 2-dimensional local field and let L be a standard field contained in F, then the Kato symbol for F (with respect to K) is given by:

    ( , )_F|K: F^×× F^×[r, "{ , }"]     K_2(F)[r, "K_2(N_F|L)"]     K_2(L)[r, "∂_L"]     k_L^×[r, "N_k_L|K"]     K^×



The definition of ( , )_F|K doesn't depend on the choice of L inside F.

See <cit.>.


 §.§ Adelic geometry

Let's fix B= O_K for a number field K;  φ:X→ B is a B-scheme satisfying the following properties:


   X is  two dimensional, integral, and regular. The generic point of X is η and the function field of X is denoted by K(X).

   φ is proper and flat.

   The generic fibre, denoted by X_K, is a geometrically integral, smooth, projective curve over K. The generic point of B is denoted by ξ.
 
We say that X is an arithmetic surface over B. We consider the set of all possible flags x∈ Y⊂ X where x is a closed point of X contained in an integral curve Y.

From now on a curve Y on X will always be an integral curve and its unique generic point will be denoted with the letter y.  By simplicity we will often identify  Y with its generic point y, which means that by an abuse of language  and notation we will use sentences like “let y⊂ X be a curve on X..." or “let x∈ y⊂ X be a flag on X...”. In other words y  is considered  as a scheme or as a point depending on  the context.

Fix a closed point x∈ X, then:


   𝒪_x:=𝒪_X,x. It is a Noetherian, complete, regular, local, domain of dimension 2 with maximal ideal 𝔪_x.

   K'_x:=𝒪_x.

   K_x:=K(X)𝒪_x⊆ K'_x. Notice that this is not a field.

For a curve y⊂ X we put:


   𝒪_y:=𝒪_X,y. It is a complete DVR  with maximal ideal 𝔪_y.

   K_y:=𝒪_y. It is a complete discrete valuation field with valuation ring 𝒪_y. The valuation is denoted by v_y.

For any point b∈ B we put:


   𝒪_b:=𝒪_B,b. It is a complete DVR.

   K_b:=𝒪_b. It is a local field with finite residue field. The valuation is denoted by v_b.


Fix a flag x∈ y⊂ X, then we have a surjective local homomorphism 𝒪_X,x→𝒪_y,x with kernel 𝔭_y,x induced by the closed embedding y⊂ X (note that 𝔭_y,x is a prime ideal of height 1).

 The inclusion 𝒪_X,x⊂𝒪_x induces a morphism of schemes ϕ:𝒪_x→𝒪_X,x and we define the local branches of y at x as the elements of the set

    y(x):=ϕ^-1(𝔭_y,x)=𝔷∈𝒪_x𝔷∩𝒪_X,x=𝔭_y,x .

If y(x) contains only an element, we say that y is unbranched at x. 



If x is a cusp point on y, one can show that y unbranched at x.




Let 𝔷∈ y(x) be a local branch of a curve y at point x, then let's define the field

    K_x,𝔷:=((𝒪_x)_𝔷) .

in other words: we localise 𝒪_x at the prime ideal 𝔷, then we complete it at its maximal ideal and finally we take the fraction field. By convenience we put 𝒪_x,𝔷:=(𝒪_x)_𝔷.



The proof of the following proposition relies on some basic commutative algebra results:

Let x∈ y⊂ X be a flag and let 𝔷∈ y(x). Then  K_x,𝔷 is a 2-dimensional valuation field  such that 𝒪_K_x,𝔷=𝒪_x,𝔷 and K^(2)_x,𝔷 is a finite extension of k(x).

First of all 𝔷≥𝔭_y,x=1, but if 𝔷=2 then 𝔷 is the maximal ideal of 𝒪_x and we have that 𝔷∩𝒪_X,x=𝔪_x, a contradiction. Therefore 𝔷=1 and (𝒪_x)_𝔷=1. It follows that (𝒪_x)_𝔷 is a Noetherian, complete, local, domain of dimension 1, i.e. a complete DVR which is the valuation ring of the complete discrete valuation field K_x,𝔷. The residue field of K_x,𝔷 is by definition:

    K^(1)_x,𝔷:=(𝒪_x)_𝔷/𝔷(𝒪_x)_𝔷=(𝒪_x/𝔷) .

Note that 𝒪_x/𝔷 is a Noetherian, complete, local domain of dimension 1 (in general we may lose the regularity by passing to the quotient). Consider the normalisation 𝒪_x/𝔷 of 𝒪_x/𝔷; the domain 𝒪_x/𝔷 is obviously normal and again Noetherian and complete. Moreover by Nagata theorem (see <cit.>) 𝒪_x/𝔷 is a Japanese ring, therefore in particular  𝒪_x/𝔷 is a finite 𝒪_x/𝔷-module. Now <cit.> implies that 𝒪_x/𝔷 is also local, and by summing up all the listed property we can conclude that 𝒪_x/𝔷 is a complete DVR with fraction field (𝒪_x/𝔷). This proves that K^(1)_x,𝔷 is a complete valuation field.

It remains to show only that the second residue field K^(2)_x,𝔷 is a finite extension of k(x). By definition K^(2)_x,𝔷 is the residue field of the local ring 𝒪_x/𝔷, but we already know that 𝒪_x/𝔷 is a finite 𝒪_x/𝔷-module, so  K^(2)_x,𝔷 is a finite extension of:


    (𝒪_x/𝔷)/(𝔪_x/𝔷)≅𝒪_x/𝔪_x≅𝒪_X,x/𝔪_x=k(x) .



It is not amongst the purposes of this paper to treat the topology of K_x,𝔷, but it is enough to know that there are several ways to topologise K_x,𝔷, some of them are equivalent, and we end up with  a structure of ST-ring on K_x,𝔷. See for example <cit.> for a survey about topologies on K_x,𝔷.


Let x∈ y⊂ X be a flag and let 𝔷∈ y(x), then we put E_x,𝔷:=K^(1)_x,𝔷 and k_𝔷 (x):=K^(2)_x,𝔷. 

    K_x,𝔷[r, phantom, "⊃"][dr, bend right,dashed, no head
    ]    𝒪_x,𝔷:=𝒪_K_x,𝔷[d][r, phantom, "⊃"]   𝒪^(2)_x,𝔷:=𝒪_K_x,𝔷^(2)[d]
        E_x,𝔷:=K^(1)_x,𝔷[r, phantom, "⊃"][dr, bend right,dashed, no head
    ]    𝒪_E_x,𝔷[d]
           k_𝔷(x):=K^(2)_x,𝔷

The valuation on K_x,𝔷 is v_x,𝔷 and the valuation on E_x,𝔷 is v^(1)_x,𝔷.
Moreover:

    K_x,y:=∏_𝔷∈ y(x)K_x,𝔷 ,  𝒪_x,y:=∏_𝔷∈ y(x)𝒪_x,𝔷 ,  𝒪^(2)_x,y:=∏_𝔷∈ y(x)𝒪^(2)_x,𝔷 ,


    E_x,y:=∏_𝔷∈ y(x)E_x,𝔷 ,   k_y(x):=∏_𝔷∈ y(x)k_𝔷 (x) .



Remember from commutative algebra  the following chain of implications:

    (A regular local)⇒ (A a UFD)⇒(Any prime 𝔭 s.t. (𝔭)=1 is principal) .

So, 𝒪_X,x is a UFD and 𝔭_y,x is principal, but also 𝒪_x is a UFD and 𝔷 is principal.


Let 𝔭_y,x=(ϖ_y) for ϖ_y∈𝒪_X,x, then we can choose the uniformizing parameter for K_x,𝔷 to be ϖ_y. 

We show that ϖ_y generates the maximal ideal of 𝒪_x,𝔷. First of all we notice that the ring 𝒪_X,x/ϖ_y𝒪_X,x≅𝒪_y,x is reduced, and this implies that 𝒪_y,x=𝒪_x/ϖ_y𝒪_x is reduced too. By remark <ref> ϖ_y has a unique factorization ϖ_y=p_1… p_m in 𝒪_x, and all the p_i's are distinct prime elements thanks to the fact that 𝒪_x/ϖ_y𝒪_x is reduced. Again remark <ref> implies that 𝔷=(p_j) for some index j. Any element of 𝔷(𝒪_x)_𝔷 can be written as p_ja/b with b∉𝔷 but:

    p_ja/b=p_1… p_m a/p_1… p_j-1p_j+1… p_m b =ϖ_ya/p_1… p_j-1p_j+1… p_m b

Since p_1… p_j-1p_j+1… p_m b∉𝔷, we can conclude that ϖ_y generates the prime ideal 𝔷(𝒪_x)_𝔷 of (𝒪_x)_𝔷.

If ϖ_y is a uniformizing parameter for the complete valuation field K_y, then it is a uniformizing parameter for K_x,𝔷.

It follows from  proposition <ref> and the fact that (𝒪_X,x)_𝔭_y,x≅𝒪_X,y.




Fix a flag  x∈ y⊂ X. The local homomorphism 𝒪_X,x→𝒪_y,x induces a local homomorphism 𝒪_x→𝒪_y,x which gives a bijective correspondence between the ideals in y(x) and the minimal prime ideals of 𝒪_y,x. Hence the ring of adeles of the curve y is recovered in the following way:

    𝐀_y='∏_x∈ y E_x,y .



Let's denote with v_x,𝔷 the valuation of K_x,𝔷 and with v_y the valuation of K_y. Then the restriction of v_x,𝔷 to K_y is equal to v_y.

By remark <ref> we deduce that E_x,𝔷 contains k(y), which is in turns the residue field of K_y, so the claims follows directly from corollary <ref>.



 The structure of K_x,𝔷 depends on the nature of the curve y and we can distinguish two cases:



  
y is a vertical curve. If φ(y)=b∈ B, then y is a projective curve over the finite field k(b); we assume that k(b) has characteristic p.  K_x,𝔷 has characteristic 0 since we have the embeddings ℚ⊂ K⊂ K(X)⊂ K_x,𝔷 and the residue field E_x,𝔷 has characteristic p since k(b)⊂ k(y)⊂ E_x,𝔷. We conclude that K_x,𝔷 is a two dimensional local field of type (0,p,p) and by the classification theorem we have that K_x,𝔷 is a finite extension of K_p{{t}} where  K_p is a finite extension of ℚ_p.



  
y is a horizontal curve. In this case K_x,𝔷 has still characteristic 0, but we have the embedding K⊆ k(y) given by the surjective map y→ B; therefore E_x,𝔷 has characteristic 0. Moreover, if φ(x)=b, the local homomorphism φ^#_x:𝒪_B,b→𝒪_X,x induces a field embedding k(b)⊆ k(x) and this implies that k_𝔷(x) has characteristic p. We conclude that K_x,𝔷 is a two dimensional local field of type (0,0,p) and by the classification theorem we have that K_x,𝔷≅ K_p((t)).  


If φ(x)=b we have an induced embedding K_b↪ K_x,𝔷, so we can conclude that K_x,𝔷 is an arithmetic 2-dimensional local field over K_b and we can apply the local theory developed in subsection <ref>. 

The ring of adeles 𝐀_X will be the result of a “glueing” of the local data {K_x,y}_x∈ y⊂ X where the couple (x,y) runs amongst all flags in X. The glueing procedure  will be described precisely, but roughly speaking we will define  𝐀_X inside  the big product of rings

    𝐀_X⊂∏_x∈ y,
     y⊂ XK_x,y

as a sort of “double restricted product”.





  
First “restricted product”: the  rings 𝔸_y^(r) and  𝔸_y.
In this paragraph we fix a curve y⊂ X, and denote with 𝔍_x,y the Jacobson radical of 𝒪_x,y. 

Let's put:

    𝔸^(0)_y = {    (α_x,y)_x∈ y∈∏_x∈ y𝒪_x,y∀ s >0, α_x,y∈𝒪_x+𝔍_x,y^s 
       for all but finitely many x∈ y.}⊂∏_x∈ y𝒪_x,y

then for any r∈ℤ

    𝔸^(r)_y:= 𝔪^r_y𝔸_y^(0)⊂∏_x∈ y K_x,y .

Clearly  𝔸^(r)_y⊇𝔸^(r+1)_y and ⋂_r∈ℤ𝔸^(r)_y=0. Moreover we  define 

    𝔸_y:=⋃_r∈ℤ𝔸^(r)_y .



We have the inclusion 𝔸_y⊂∏_x∈ y K_x,y, therefore we can interpret  𝔸_y as a “restricted product” of the rings K_x,y for y fixed and x∈ y. Thus we can write:
 
    𝔸_y='∏_x∈ y K_x,y

 where '∏ here is just a piece of notation without any formal meaning.
 

Each 𝔸_y^(r) can be endowed with a  ind/pro linear topology, and 𝔸_y can be seen as linear direct limit of the topological groups 𝔸_y^(r). More details about such topologies will be given in the second paper of the series.



  
Second “restricted product”: the ring 𝐀_X.
The construction of 𝔸_y can be seen as a  way to take the restricted product of ∏_x∈ y K_x,y. The final step in order to construct the ring of adeles 𝐀_X is to take the restricted product of the groups 𝔸_y over all the curves in X with respect to the subgroups 𝔸_y^(0).


    𝐀_X:={(β_y)_y⊂ X∈∏_y⊂ X𝔸_yβ_y∈𝔸^(0)_y for all but finitely many y}⊂∏_x∈ y,
     y⊂ X K_x,y .

In a more suggestive way, we write by commodity

    𝐀_X=”∏_x∈ y
     y⊂ X K_x,y

where the symbol “”∏” is just a piece of notation which remembers  that we are taking a “double restricted product”.



It is fundamental to recall that 𝐀_X is not the full ring of adeles associated to the completed surface X, bcause we didn't take in account the fibres at infinity.

In order to topologise  𝐀_X we need to recall the description of the restricted product, by means of categorical limits, for linearly topologised groups. Let {G_i}_i∈ I a set of linearly topologised groups and for any i let H_i⊂ G_i be a subgroup endowed with the subspace topology. We denote the family of finite subsets of I as 𝒫_f(I); it forms a directed set with the relation J⊆ J'. For any J∈𝒫_f(I) define

    G_J:=∏_i∈ J G_i×∏_i∉J H_i ,

if  J⊆ J'  the identity in each factor induces an embedding G_J↪ G_J', thus we have a direct system {G_J}_J and it is easy to see that 

    '∏_i G_i=_J G_J ,

where '∏_i G_i is the usual restricted product of the G_i with respect to the subgroups H_i. At this point, on each G_J we put the product topology and '∏_i G_i is endowed with  the linear  direct limit topology. By definition 𝐀_X is the restricted product of the groups 𝔸_y with respect to the subgroups  𝔸_y^(0) for any y⊂ X. Therefore we endow  𝐀_X with the topology described above.

 We now introduce some important subspaces in order to construct the  adelic complexes associated to the surface X. Here the definitions are made “by hands”, but such subspaces can be recovered as a particular case of the general theory of Beilinson adeles (see <cit.>). First of all let's consider the following diagonal embeddings:
 
    K_x⊂∏_y∋ x K_x,y,    K_y⊂∏_x∈ y K_x,y ,

so  we can consider:

    ∏_x∈ XK_x⊂∏_x∈ y
     y⊂ X K_x,y,   ∏_y⊂ XK_y⊂∏_x∈ y
     y⊂ X K_x,y .

Let's define:

    A_012:=𝐀_X ;   A_12:=𝐀_X∩∏_x∈ y
    y⊂ X𝒪_x,y=∏_y⊂ X𝔸^(0)_y;


    A_02:=𝐀_X∩∏_x∈ X K_x ;   A_2:=𝐀_X∩∏_x∈ X𝒪_x ;   A_01:=𝐀_X∩∏_y⊂ X K_y ;


    A_1:=𝐀_X∩∏_y⊂ X𝒪_y ;   A_0:=K(X)

The containment relations are depicted in the following diagram:

    A_0[d,hook'][dl,hook'][dr,hook]      
       A_01[r,hook]    A_012   A_02[l,hook']   
    
    A_1[rr,hook][ur,hook][urr,hook]      A_12[u,hook]      A_2[ll,hook'][ul,hook'][ull,hook']

and we have the adelic complex:

    𝒜_X:     A_0⊕ A_1⊕ A_2 [r, "d^0"]    A_01⊕ A_02⊕ A_12[r, "d^1"]    A_012
        (a_0,a_1,a_2) [r, mapsto]     (a_0-a_1,a_2-a_0,a_1-a_2)    
           (a_01,a_02,a_12)[r, mapsto]    a_01+a_02+a_12 .

If D=∑_y⊂ Xn_y[y] is a divisor of X we can define the subgroups

    𝐀_X(D):=∏_y⊂ X𝔸^(-n_y)_y .

Note that 𝐀_X(D) is a well defined subgroup of 𝐀_X because n_y=0 for all but finitely many y. Let's define the subspaces 

    A_12(D):=A_012∩𝐀_X(D)=𝐀_X(D) .


    A_1(D):=A_01∩𝐀_X(D);   A_2(D):=A_02∩𝐀_X(D);

in order to get the complex

    𝒜_X(D):   A_0⊕ A_1(D)⊕ A_2(D) A_01⊕ A_02⊕ A_12(D) A_012

such that the maps are the same of those in equation (<ref>). Furthermore note that 𝒜_X=𝒜_X(0).
There is also the idelic version of complex (<ref>):

    𝒜^×_X:     A^×_0⊕ A^×_1⊕ A^×_2 [r, "d_×^0"]    A^×_01⊕ A^×_02⊕ A^×_12[r, "d_×^1"]    A^×_012=𝐀_X^×
        (a_0,a_1,a_2) [r, mapsto]     (a_0a^-1_1,a_2a^-1_0,a_1a^-1_2)    
           (a_01,a_02,a_12)[r, mapsto]    a_01a_02a_12

and we have a well defined surjective map:



    p:  (d^1_×)    →   (X)
    
    (α,β,α^-1β^-1)   ↦   ∑_y⊂ X v_y(α_x,y)[y] .




§ IDELIC DELIGNE PAIRING

Let's still consider the arithmetic surface φ: X→ B, and let's denote with 𝐀_B and 𝐀_B^× respectively the nonarchimedean parts of the one dimensional adeles and ideles associated to the base B (in other words we are not considering the archimedean places of K). Remember that for any two divisors D,E on X with no common components we have:

    i_x(D,E):= _𝒪_X,x𝒪_X,x/ (𝒪_X(-D)_x+𝒪_X(-E)_x) .

The Deligne pairing is a bilinear and symmetric map:

    < ,  >(X)×(X)→(B)

which was introduced  for the first time in <cit.>. More details about the construction of < ,  > can be found in <cit.>. First of all let's see how the Deligne pairing can be lifted to a pairing at the level of divisors on X and with target in (B):

There exists a unique pairing

    [[ , ]](X)×(X)→(B)

satisfying the following properties:


  (1) It is bilinear and symmetric.

  (2) It descends to the Deligne pairing 

    < ,  >(X)×(X)→(B) .


  (3) If D,E∈(X) are two  divisors with no common components then [[D,E]] is equal to the class  of the divisor <D, E > in (B) (here the bracket <D,E > denotes the pushforward through φ of the 0-cycle collecting all the local intersection numbers  between D and E). In other words:

    [[D,E]]=∑_x∈ D∩ E[k(x): k(φ (x))]i_x(D,E) [φ(x)]∈(B) .
 

  
For any D,E∈(X) it is enough to put:

    [[D,E]]:=< 𝒪_X(D), 𝒪_X(E)>

where on the right hand side we have the Deligne pairing between invertible sheaves. Uniqueness follows from properties (1)-(3) and what is commonly called “the moving lemma” (<cit.>).
At this point we will try to work in complete analogy to the geometric case and we will use the Kato symbol defined in section <ref> to obtain the map, denoted below with a question mark, which makes the following diagram commutative:

    (d^1_×)×(d^1_×)[rrdd, "?"][d, two heads,"p× p"]       𝐀^×_B[dd ,two heads]
    (X)×(X)[rrd, "[[ , ]]",pos=.2][d, two heads]        
    (X)×(X)[rr, " < ,  >"]       (B)≅^1(B)

As usual, fix a flag x∈ y with 𝔷∈ y(x) and assume that φ(x)=b, then we define

    ( , )_x,𝔷:=( , )_K_x,𝔷|K_b:K_x,𝔷^×× K_x,𝔷^×→ K_b^×

where ( , )_K_x,𝔷|K_b is the Kato symbol defined in section <ref>. Remember that depending on whether y is horizontal or vertical, we have a different expression for ( , )_x,𝔷. Then we put:

    ( , )_x,y:=∏_𝔷∈ y(x)( , )_x,𝔷:K^×_x,y× K^×_x,y→ K^×_b .

It is important to point out that ( , )_x,y maps 𝒪^×_x,y×𝒪^×_x,y to 𝒪^×_b.


The pairing ( , )_x,y is a skew-symmetric bilinear form on K^×_x,y satisfying the following properties:


  (1)Let r,s∈ K^×_x, then for all but finitely many curves y containing x we have that (r,s)_x,y=1 and moreover ∏_y∋ x(r,s)_x,y=1.

  (2) Let y be a vertical curve and let r,s∈ K^×_y, then  ∏_x∈ y(r,s)_x,y=1. In particular (r,s)_x,y∈𝒪^×_b for all but finitely many x∈ y.


Skew symmetry and bilinearity are clear. See <cit.> for (1); note that in <cit.> the proof is made for r,s∈ K(X)^×, but it is easy to see that it actually works also for r,s∈ K^×_x. See <cit.> for (2).



The idelic Deligne pairing 

    < ,  >_i: (d^1_×)×(d^1_×)→^1(B)
 
is given by:


    (r,s)↦ <r,s >_i:=∑_b∈ B n_b(r, s)[b]∈^1(B)

such that:

    n_b(r, s):= ∑_x∈ X_b,
     y∋ xv_b((γ_x,y,β_x,y)_x,y)

for r=(α,β, α^-1β^-1), s=(γ,δ,γ^-1δ^-1)∈(d_1^×) and where v_b is the complete discrete valuation on K_b. It is crucial to emphasize the fact the we consider ∑_b∈ B n_b(r, s)[b] in its linear equivalence class in ^1(B) and not just as a divisor. By simplicity of notation we avoid to mention the canonical map (B)→^1(B).

One can verify that definition <ref> makes sense:

The summations (<ref>) and (<ref>) are finite.

Thanks to the second adelic restricted product over curves it is enough to check it only for a fixed nonsingular vertical curve y⊂ X_b. Let's write β_x,y=fs_x,y∈ K(X)^×𝒪^×_x, then 

    (γ_x,y,β_x,y)_x,y=(γ_x,y,f)_x,y(γ_x,y,s_x,y)_x,y .

<cit.> shows that ∏_x∈ y(γ_x,y,f)_x,y is convergent, and this means that (γ_x,y,f)_x,y∈𝒪^×_b for all but finitely many x∈ y. Moreover if p= k(b), then p is a uniformizing parameter for K_x,y, so γ_x,y=p^r c_x,y with c_x,y∈𝒪^×_x,y. Thus:


    (γ_x,y,s_x,y)_x,y=(p^r,s_x,y)_x,y(c_x,y,s_x,y)_x,y

Obviously (c_x,y,s_x,y)_x,y∈𝒪^×_b, so in order to finish the proof it remains to show that (p^r,s_x,y)_x,y lies in 𝒪^×_b too. Just for simplicity of calculations  let's assume that K_x,y is a standard field and  K_x,y=K_p{{t}} (the argument works easily also for non standard fields). By the explicit expression of the Kato's residue homomorphism (cf. <cit.>) we can calculate that:


    (p^r, s_x,y)_x,y= N_K_p|K_b(p^rw)

where w∈ℤ is the  winding number associated to s_x,y (see <cit.> for details). But we know that s_x,y∈𝒪^×_x=𝒪_K_p^×+t𝒪_K_p[[t]], thus:

    s_x,y=a+t∑_i≥ 0a_it^i=a(1+t∑_i≥ 0a_i/at^i) .

It follows that w=0 and the proof is complete. 




For any b∈ B we have the following decomposition for the big product  (<ref>):

    ∑_x∈ X_b,
     y∋ xv_b(( , )_x,y)=∑_y⊂ X_b,
     x∈ y v_b(( , )_x,y)   +∑_x∈ X_b,
     y∋ x,
     y horiz.v_b(( , )_x,y) .


We put < , >_i as the undetermined function in diagram (<ref>) and we have the following fundamental result:

Consider the notation of diagram (<ref>). The pairing < , >_i satisfies the following properties:


  (1) It is bilinear and symmetric.

  (2) Let r,s,r',s'∈(d^1_×) such that p(r)=p(r') and p(s)=p(s'), then <r,s>_i=<r',s'>_i.

  (3) It descends naturally to a pairing H^1(𝒜_X^×)× H^1(𝒜_X^×)→(B).


Let's fix r=(α,β,α^-1β^-1), s=(γ,δ,γ^-1δ^-1)∈(d^1_×); moreover we can fix b∈ B and work componentwise.


(1) Bilinearity is clear. We will show that as elements of (B) we have <r,s>_i=<s,r>_i+(f) with f∈ K^×. 
 For any flag x∈ y: α_x,y^-1β_x,y^-1∈𝒪^×_x,y and γ_x,y^-1δ_x,y^-1∈𝒪^×_x,y so  we have that:

    0=∑_x∈ X_b,
     y∋ xv_b((α_x,y^-1β_x,y^-1,γ_x,y^-1δ_x,y^-1)_x,y)=
    
    =∑_x∈ X_b,
     y∋ xv_b((α_x,y,γ_x,y)_x,y)_(i)+∑_x∈ X_b,
     y∋ xv_b((α_x,y,δ_x,y)_x,y)_(ii)+
    
    +∑_x∈ X_b,
     y∋ xv_b((β_x,y,γ_x,y)_x,y)_(iii)+∑_x∈ X_b,
     y∋ xv_b((β_x,y,δ_x,y)_x,y)_(iv) .

Now we analyze in detail the underbraced terms in equation (<ref>): for (i) we have the following decomposition thanks to remark <ref>:

    (i)=∑_y⊂ X_b,
     x∈ yv_b((α_x,y,γ_x,y)_x,y)  + ∑_x∈ X_b,
     y∋ x,
     y horiz.v_b((α_x,y,γ_x,y)_x,y)=


    ^(prop.<ref>(2))=0+∑_x∈ X_b∑_y∋ x,
     y horiz.v_b((α_x,y,γ_x,y)_x,y) .

By definition we have that (ii)=n_b(s, r) and (iii)=-n_b(r, s). Finally:

    (iv)=∑_x∈ X_b∑_y∋ x(β_x,y,δ_x,y)_x,y=^(prop.<ref>(1)) 0

By substituting in equation (<ref>) we conclude that  

    n_b(r, s)=n_b(s, r)+∑_x∈ X_b,
     y∋ x,
     y horiz.v_b((α_x,y,γ_x,y)_x,y) .

Let y be an horizontal curve and let x∈ y such that φ(x)=b, then the coefficient field of K_x,𝔷 is k(y)_x. The two dimensional valuation v_x,𝔷 extends the valuation v_y on k(y) and moreover that the norm N_k(y)_x|K_b extends N_k(y)|K. It follows that  ( , )_x,𝔷 extends the one dimensional tame symbol 

    ( , )_y:=N_k(y)|K∘ ( , )_k(y):K^×_y× K^×_y→ k(y)^×→ K^× .

This means that for any two elements u,v∈ K_y, where y is horizontal, we have that:

    (u,v)_x,y=(u,v)_y∈ K

for any x∈ y. Therefore we can rewrite equation (<ref>):

    n_b(r, s)=n_b(s, r)+∑_y horiz.v_b((α_x,y,γ_x,y)_y) .

Let's put f=∏_y horiz.(α_x,y,γ_x,y)_y∈ K^×, then equation (<ref>) implies the following equality:

    <r,s>_i=<s,r>_i+∑_b∈ B v_b(f)[b]=<s,r>_i+(f) .




(2) Let  r'=(α',β', (α')^-1(β')^-1)  and s'=(γ',δ', (γ')^-1(δ')^-1). Since p(r)=p(r') and p(s)=p(s'), then v_y(α_x,y)=v_y(α'_x,y) and v_y(γ_x,y)=v_y(γ'_x,y). This means that γ'_x,y=f_x,yγ_x,y and α'_x,y=g_x,yα_x,y for f_x,y,g_x,y∈𝒪^×_y (for any x∈ y). Then we have the following chain of equalities depending on what we showed in claim (1):


    n_b(r',s')=∑_x∈ X_b,
     y∋ xv_b((f_x,yγ_x,y,β'_x,y)_x,y)=



    =∑_x∈ X_b,
     y∋ x((f_x,y,β'_x,y)_x,y)+∑_x∈ X_b,
     y∋ xv_b((γ_x,y,β'_x,y)_x,y)=



    =∑_x∈ X_b,
     y∋ xv_b((f_x,y,β'_x,y)_x,y)+n_b(r',s)=∑_x∈ X_b,
     y∋ xv_b((f_x,y,β'_x,y)_x,y)+n_b(s,r')+v_b(f)=


    =∑_x∈ X_b,
     y∋ xv_b((f_x,y,β'_x,y)_x,y)+∑_x∈ X_b,
     y∋ xv_b((g_x,yα_x,y,δ_x,y)_x,y)+v_b(f)=(∗)

Where f∈ K^×.

    (∗)=∑_x∈ X_b,
     y∋ xv_b((f_x,y,β'_x,y)_x,y)+ ∑_y⊂ X_b,
     x∈ yv_b((g_x,y,δ_x,y)_x,y)+∑_x∈ X_b,
     y∋ xv_b((α_x,y,δ_x,y)_x,y)+v_b(f)=


    =∑_x∈ X_b,
     y∋ xv_b((f_x,y,β'_x,y)_x,y)_(i)+ ∑_x∈ X_b,
     y∋ xv_b((g_x,y,δ_x,y)_x,y)_(ii)+n_b(r,s)+v_b(fg) .

Note that in the last line we used the fact that n_b(s,r)=n_b(r,s)+v_b(g) for g∈ K^×. We have to show that the terms (i) and (ii) are valuations at b of elements of K^×. Since (α')^-1_x,y(β')^-1_x,y∈𝒪^×_x,y, we have:

    0=∑_x∈ X_b,
     y∋ xv_b((f_x,y,(α')^-1_x,y(β')^-1_x,y)_x,y)=


    =∑_x∈ X_b,
     y∋ xv_b((f_x,y,α'_x,y)_x,y)+∑_x∈ X_b,
     y∋ xv_b((f_x,y,β'_x,y)_x,y)=


    =v_b(h)+∑_y⊂ X_b,
     x∈ yv_b((f_x,y,β'_x,y)_x,y) .

with h=∏_y horiz.(f_x,y,α'_x,y)_y∈ K^×. For (ii) the argument is similar.


(3) Let r,s∈(d^0_×). It means that α=lm^-1, β=tl^-1, γ=uv^-1, δ=zu^-1 for l,u∈ A^×_0=K(X)^×,  m,v∈ A^×_1 and t,z∈ A_2^×. So:


    n_b(r,s)=∑_x∈ X_b,
     y∋ xv_b((uv^-1_x,y,t_x,yl^-1)_x,y)=
    
    =∑_x∈ X_b,
     y∋ xv_b((u,t_x,y)_x,y)+∑_x∈ X_b,
     y∋ xv_b((u,l^-1)_x,y)+
    
    +∑_x∈ X_b,
     y∋ xv_b(v^-1_x,y,t_x,y)_x,y)+∑_x∈ X_b,
     y∋ xv_b((v^-1_x,y, l^-1)_x,y) .

Now it is enough to appeal to one of the arguments previously used to conclude that each summand of equation (<ref>) is either 0 or of the form v_b(f) for f∈ K^×. It means that <r,s>_i=0 in ^1(B).
We want to give an alternative formula for the coefficient n_b(r,s). Notice that:

    -∑_x∈ X_b,
     y∋ xv_b((α_x,y,γ^-1_x,yδ^-1_x,y)_x,y)=
    
    =∑_x∈ X_b,
     y∋ xv_b((α_x,y,γ_x,y)_x,y)+∑_x∈ X_b,
     y∋ xv_b((α_x,y,δ_x,y)_x,y)=
    
    =v_b(f)+∑_x∈ X_b,
     y∋ xv_b((α_x,y,δ_x,y)_x,y)=v_b(f)+n_b(s,r)=v_b(fg)+n_b(r,s)

for f,g∈ K^×. Therefore,  we can also express:


    n_b(r,s)=-∑_x∈ X_b,
     y∋ xv_b((α_x,y,γ^-1_x,yδ^-1_x,y)_x,y) .

In particular if y is a horizontal curve:

    -v_b((α_x,y,γ^-1_x,yδ^-1_x,y)_x,y)=v_y(α_x,y)v_b(N_k(y)_x|K_b(γ^-1_x,yδ^-1_x,y)) .

The following lemmas are fundamental in order to understand the relationship between <r,s>_i and Deligne pairing.

Let X_b⊂ X the fiber over b∈ B and assume that X_b has at least two irreducible components. If D⊂ X_b is an integral curve, then there exists a divisor D'∼ D such that D' doesn't have components contained in X_b.

Consider Γ running amongst all irreducible components of X_b, then put

    S:=⋃_Γ⊂ X_b
    Γ≠ D(Γ∩ D) .

By the moving lemma we can find D'∼ D not passing by S. It is clear by the definition of S that D' cannot have vertical components contained in X_b.



Let D,E two prime divisors on X and let x∈ D∩ E a nonsingular point for both D and E. Moreover let d_x,e_x∈𝒪_X,x be the local equations at x of D and E respectively. Then we have the equality:

    v^(1)_x,D(e_x)=i_x(D,E)

where v^(1)_x,D:E^×_x,D→ℤ is the one dimensional valuation and e_x∈ E^×_x,D is the natural projection through the map 𝒪_x,D→ E^×_x,D.
 
Put  y=D and v=v^(1)_x,D. First of all notice that 𝒪_X,x⊆𝒪_X,y, therefore e_x∈𝒪_y,x  and it is the image on the natural map 𝒪_X,x→𝒪_y,x⊂ k(y). We have to show that v(e_x)=_𝒪_X,x𝒪_X,x/(d_x,e_x), but we know that 𝒪_y,x=𝒪_X,x/d_x𝒪_X,x, thus

    _𝒪_X,x𝒪_X,x/(d_x,e_x)=_𝒪_y,x𝒪_y,x/e_x𝒪_y,x=v(e_x) .


If r,s∈(d^1_×) such that D=p(r) and E=p(s) are two nonsingular prime divisors on X with no common components, then <r,s>_i=[[D,E]].

Fix r=(α,β, α^-1β^-1), s=(γ,δ,γ^-1δ^-1). We want to show that it is enough to restrict to the case when either D or E is horizontal. In any case, by theorem <ref>(2) we always choose δ_x,y in the following way: δ_x,y=1 if x∉ D∩ E and δ_x,y=t_x^-1, where t_x∈𝒪_X,x is the local equation of E at x, if x∈ D∩ E. For any y≠ D, α_x,y∈𝒪^×_y, since p(a)=D, therefore:

    n_b(r,s)=-∑_x∈ D∩ X_bv_b((α_x,D,γ^-1_x,Dδ^-1_x,D)_x,D) .

If D⊆ X_b and E⊆ X_b' with b≠ b', then by proposition <ref>(2) and the choice of δ_x,y we have:
 
    n_b(r,s)=-∑_x∈ D∩ X_bv_b((α_x,D,γ^-1_x,D)_x,D)=0 .

So in such a particular case <r,s>_i=[[D,E]]=0.

If D,E∈ X_b we can apply lemma <ref> and find a divisor D'=∑_j n_jΓ_j∼ D such that Γ_j⊄X_b. Clearly



    [[Γ_j,E]]=∑_j n_j[[Γ_j,E]]
 
therefore from now on we can restrict our calculation to the case where either D or E is horizontal. By symmetry we can fix D to be horizontal and we denote with K(D) its function field. In this case we have an explicit expression given by equation (<ref>):

    n_b(r,s)=∑_x∈ D∩ X_bv_b(N_K(D)_x|K_b(γ^-1_x,Dt_x))=∑_x∈ D∩ X_bv_b(N_K(D)|K(γ^-1_x,Dt_x))=


    =∑_x∈ D∩ X_bv_b(N_K(D)_x|K_b(γ^-1_x,D))+∑_x∈ D∩ E∩ X_bv_b(N_K(D)_x|K_b(t_x)) .

Now by the theory of extensions of valuation fields (see <cit.>), we know that if v_x:=v^(1)_x,D is the valuation on K(D)_x, then:

    v_x=1/[k(x):k(b)]v_b∘ N_K(D)_x|K_b .

Therefore we obtain:

    n_b(r,s)=∑_x∈ D∩ X_b[k(x):k(b)]v_x(γ^-1_x,D)+∑_x∈ D∩ E∩ X_b[k(x):k(b)]v_x(t_x) .

Put by simplicity f=γ^-1_x,D∈ K(D)^×, consider the restricted morphism of arithmetic curves φ:D→ B and the principal divisor (f)∈(D), then:

    φ_∗((f))=∑_b∈ B(∑_x∈ D∩ X_b [k(x):k(b)]v_x(f))[b] .

Moreover v_x(t_x)=i_x(D,E) by lemma <ref>. Equation <ref> implies that in (B) we have the following equality:

    <r,s>_i=φ_∗((f))+[[D,E]] .

But by <cit.> we know that φ_∗((f))=(N_K(D)|K(f))∈(B), so the proof is complete. 
We obtained the idelic representation of Deligne pairing:

Diagram (<ref>) is commutative.

For any two divisors D,E∈(X) define the pairing:

    Θ(D,E):=<r',s' >_i

for a choice  of r',s'∈(d^1_×) such that p(r')=D and p(s')=E. By theorem <ref>(2) Θ is well defined and moreover by <ref>(1), <ref>(3) and <ref> we can conclude that Θ(D,E)=D.E. Thus for any a,b∈(d^1_×) we have that:

    <r,s>_i=Θ(p(r),p(s))=p(r).p(s) .





§ ADELIC DELIGNE PAIRING

A clever and quick adelic interpretation of  intersection theory on algebraic surfaces is given in <cit.> and the strategy is very simple: first of all one defines the adelic Euler-Poincare characteristic χ_a(·) which associates an integer to any divisor on the surface (or more in general to any invertible sheaf) by using just data coming from the adelic complex. Then the adelic intersection pairing is defined accordingly to equation (<ref>) by using χ_a(·) instead of the usual Euler-Poincare characteristic.  Here we try to follow the same approach; so, it is evident that we have to  define the adelic determinant of the cohomology
 
    _a Rφ_∗:(X)→(B)
 
which should be a function involving only adelic data, and then the adelic Deligne pairing according to theorem  <ref>.

For any coherent sheaf ℱ on B and any closed point b∈ B, we define the following  objects:

    K_b(ℱ):=ℱ_ξ⊗_K K_b=(ℱ_b⊗_𝒪_B,b K)⊗_K K_b=ℱ_b⊗_𝒪_B,b K_b ,


    𝒪_b(ℱ):=ℱ_b⊗_𝒪_B,b𝒪_b .


    𝐀_B(ℱ):='∏_b∈ B K_b(ℱ)

where the restricted product is taken with respect to the rings 𝒪_b(ℱ), and

    𝐀_B(ℱ)(0):=∏_b∈ B𝒪_b(ℱ) .

Moreover recall that we have the following one dimensional adelic complex given by:


    𝒜_B(ℱ):   0→ℱ_ξ⊕𝐀_B(ℱ)(0)   →   𝐀_B(ℱ)→ 0 
    
    (f,(α_b)_b)    ↦    (f-α_b)_b

It is important to point out that we want to  consider 𝒜_B(ℱ) as a complex of O_K-modules in the natural way.

Let D be a divisor on X satisfying proposition <ref>. For any invertible sheaf ℒ on B we put by simplicity 𝒢:=φ_∗ℒ(D) and ℋ:=φ_∗(ℒ(D)/ℒ). Then the adelic determinant of cohomology is given by:


    _a Rφ_∗(ℒ):=(H^0(𝒜_B(𝒢)))⊗((H^0(𝒜_B(ℋ))))^∗

Where with ∗ we denote the algebraic dual. Note that _a Rφ_∗(ℒ) is a O_K-module, but by abuse of notation we can consider it as an element in (B) after taking the associated sheaf (_a Rφ_∗(ℒ))^∼ (see <cit.>). In other words we omit the operator ∼ by simplicity of notations. 



The adelic Deligne pairing  between two invertible sheaves ℒ and ℳ on B is defined as:

    <ℒ,ℳ>_ a:=


    _aRφ_∗(𝒪_X)⊗ (_a Rφ_∗(ℒ))^-1⊗(_a Rφ_∗(ℳ))^-1⊗_a Rφ_∗(ℒ⊗ℳ)


It is immediate to verify that the definition of the adelic Deligne pairing coincides with the usual Deligne pairing by using equation (<ref>). Indeed thanks to <cit.> H^0(𝒜_B(𝒢))≅ H^0(B,𝒢) and H^0(B,𝒢)^∼≅𝒢 by affine Serre's theorem (see <cit.>). Obviously the same holds for ℱ.





§ TOPICS IN K-THEORY

Algebraic K-theory is a very wide subject with a long history. It can be approached in many different ways and several links can be build between all approaches (see for example <cit.>). This appendix  is not a short introduction to algebraic K-theory, but just a mere collection of definition and notations needed in this text.


Let G be an abelian group, and fix an integer ≥ 1. A r-Steinberg map is an homomorphism of ℤ-modules f: (F^×)^⊕_r→ G such that f(a_1,…,a_r)=0 whenever there exist two indexes i,j such that i≠ j and a_i+a_j=1.

Let's denote with St(r) the category whose objects are the r-Steinberg maps  f: (F^×)^⊕_r→ G and the morphisms are the commutative diagrams:


    (F^×)^⊕_r[r,"f"][d, "g"']    G 
    
    H[ur, "ϕ"']

where ϕ is a group homomorphism.

The category St(r) has the initial object.

We construct the initial objects by hands. Let's define

    K_r(F):=F^×⊗_ℤ…⊗ _ℤF^×_r times  /  S

where S is the (multiplicative) subgroup generated by the following set:

    {a_1⊗…⊗ a_r a_i+a_j=1  for some  i≠ j} .

The natural image of a pure tensor a_1⊗…⊗ a_r in K_r(F) is denoted by {a_1,…,a_r}.  Clearly we have an induced map:

    { }:  (F^×)^⊕_r   →   K_r(F)
    
    (a_1,…,a_r)   ↦   {a_1,…,a_r}

At this point it is straightforward to see that { }:(F^×)^⊕_r→ K_r(F) is the initial object for St(r).

For r=0 we put K_0(F):=ℤ and in general we call the group K_r(F) constructed in proposition <ref> the r-th K-group of F. Note that K_1(F)=F^×. The map  { }:(F^×)^⊕_r→ K_r(F) is called the r-th symbol map and in the cases r=0,1 it is just the identity.



The groups introduced in definitons <ref> are usually called Milnor K-groups and the standard notation is K_r^M. However in this text we can simplify the notation.

The construction K_r( )  is functorial,  in fact let f:F^×→ L^× be a group homomorphism, then the composition:

    (F^×)^⊕ r (L^×)^⊕ r K_r(L)

is evidently a Steinberg map. By the universal property it induces a morphism K_r(f): K_r(F)→ K_r(L).

When F is a complete discrete valuation field there exists a nice relationship between K-groups of F and K-groups of the residue fields:

Let F be a discrete valuation field (not necessarily complete) then there is a unique group homomorphism:

    ∂_r: K_r(F)→ K_r-1(F)

satisfying the following property:

    ∂_r({x_1,…,x_r-1,ϖ})={x_1,…,x_r-1}

for any local parameter ϖ of F and any x_1,…,x_r-1∈𝒪^×_F .

See <cit.>.



The map ∂_r described in theorem <ref> is called the (Milnor) r-th boundary map.



Consider the tame symbol for a complete discrete valuation field (F,v):

    ( , )_F:F^×× F^×   →F^×
    
    (a,b)   ↦ (a,b)_F=(-1)^v(a)v(b) a^v(b)b^-v(a) .

We have a nice description of the boundary map ∂_2 in relation to the tame symbol.  By the universal property of K_2(F), the tame symbol ( , )_F induces a unique map Ψ:K_2(F)→F^×=K_1(F) such that Ψ({ , })=( , )_F. Let a∈𝒪^×_F and let ϖ be  a local parameter for F, then Ψ({a,ϖ})=(a,ϖ)_F=a; this actually means that ∂_2=Ψ. In other words the 2-nd boundary map for a complete discrete valuation field is exactly the map induced naturally by the tame symbol.

For a discrete valuation field F (not necessarily complete) we have the multiplicative group U_F^(i):=1+𝔭_F^i  for i≥ 1 and we have also the K-theoretic version of it:

    U^iK_r(F):={{a_1… a_r}∈ K_r(F) a_j∈ U_F^(i) ∀ j=1,…,r}

and we put:

    K_r(F):=_i K_r(F)/U^iK_r(F) .

Clearly we have a natural homomorphism K_r(F)→K_r(F) and moreover if F is the completion of F there is an isomorphism K_r(F)≅K_r(F). Now put L=(𝒪_F[[t]] ), for any prime ideal 𝔭 of height 1 in 𝒪_F[[t]] we have that  𝒪_F[[t]]_𝔭 is a discrete valuation ring and in particular F{{t}} is the completion of L at 𝔭= 𝔭_F𝒪_F[[t]]. Consider the set: 

    𝔖:={𝔭∈(𝒪_F[[t]])𝔭=1, 𝔭≠𝔭_F𝒪_F[[t]]} ,

and for any 𝔭∈𝔖 let's denote with ∂^(𝔭)_r:K_r(L)→ K_r-1(k(𝔭)) the r-th boundary map relative to the valuation defined by 𝔭. 

For r≥ 1, the r-th (Kato) residue map on L is given by the following composition:

    _L^(r): K_r(L)[rrr,"(∂^(𝔭)_r)_𝔭∈𝔖"]          ⊕_𝔭∈𝔖 K_r-1 (k(𝔭))[rrr, " ∑_𝔭∈𝔖 K_r(N_k(𝔭)|F)"]            K_r-1(F)



The r-th residue map satisfies:

    _L^(r)(U^(i)K_r(L))⊆ U^(i) K_r-1(F)  ∀ i≥ 1

therefore it induces a homomorphism:

    _F{{t}}^(r):K_r(F{{t}})≅K_r(L)→K_r-1(F) .


See <cit.>.




§ DETERMINANT OF COHOMOLOGY

For an algebraic surface Z over a field k, intersection theory can be introduced by using the Euler-Poincare characteristic χ_k:(Z) →ℤ “restricted” to (Z). In fact, the intersection number between two invertible sheaves ℒ and ℳ on Z can be calculated by the following formula:

    ℒ. ℳ:= χ_k(𝒪_Z)-χ_k(ℒ^-1)-χ_k(ℳ^-1)+χ_k(ℒ^-1⊗ℳ^-1)

 In the Arakelov setting given by the arithmetic surface φ:X→ B, we want to define a map (X) →(B) such that, when we take the “restriction” to (X), we  obtain a formula, similar to (<ref>), relating our map to the Deligne pairing. In other words, we would like to have the arithmetic equivalent notion of the Euler-Poincare characteristic. The answer to our query will be the determinant of the cohomology, denoted by Rφ_∗, and in this section we are going to construct it step by step.
 
The first thing to notice is that χ_k is a cohomological object and in the case of φ: X→ B the “relative cohomology" is captured by the higher direct image functors R^iφ_∗. By keeping in mind that the output of the determinant of the cohomology should be an invertible sheaf on the base B, in analogy with the definition of χ_k, the most reasonable definition should be something like:

    Rφ_∗(ℱ):=⊗_j≥ 0 ( R^jφ_∗ℱ)^(-1)^j=φ_∗(ℱ)⊗ ( R^1φ_∗(ℱ))^-1

Unfortunately equation (<ref>) doesn't make any sense in general, since the higher direct images R^jφ_∗ℱ are not locally free sheaves, so we cannot take the determinant. However, we will cook up a definition of Rφ_∗(ℱ) which agrees  with equation (<ref>) when R^jφ_∗ℱ are locally free. 

The following  proposition is fundamental:

There exists  an effective divisor D on X which doesn't contain any fibre of φ such that for any coherent sheaf ℱ on X we get an exact sequence:

    0→φ_∗ℱ→φ_∗ℱ(D)→φ_∗(ℱ(D)/ℱ)→ R^1φ_∗ℱ→ 0

such that  φ_∗ℱ(D) and φ_∗(ℱ(D)/ℱ) are both locally free sheaves on B.

See  <cit.> or <cit.>.

Let ℱ∈(X) and let D be a divisor as in proposition <ref>; the determinant of the cohomology of ℱ is:

    Rφ_∗(ℱ):=φ_∗(ℱ(D))⊗ (φ_∗(ℱ(D)/ℱ))^-1∈(B)

Moreover Rφ_∗(ℱ) doesn't depend on the choice of D (for the proof of this statement see <cit.> or <cit.>).


Now we want to show that if R^0φ_⋆ℱ=φ_∗ℱ and  R^1φ_⋆ℱ are both locally free, then Rφ_∗(ℱ) is given by equation (<ref>). Consider the exact sequence of equation (<ref>), put 

    f:φ_∗ℱ(D)→φ_∗(ℱ(D)/ℱ) ,
 

    g:φ_∗(ℱ(D)/ℱ)→ R^1φ_∗ℱ ,

and 𝒢=(f)=(g). Then we get the following two short exact sequences of locally free sheaves:

    0→φ_∗ℱ→φ_∗ℱ(D)𝒢→ 0 ;


    0→𝒢→φ_∗ (ℱ(D)/ℱ) R^1φ_∗ℱ→ 0 .

At this point we use the properties of the determinant on short exact sequences and we obtain:

    Rφ_∗(ℱ)=φ_∗(ℱ(D))⊗ (φ_∗(ℱ(D)/ℱ))^-1≅


    ≅φ_∗ℱ⊗𝒢⊗ (𝒢)^-1⊗ ( R^1φ_∗(ℱ))^-1≅


    ≅φ_∗ℱ⊗ R^1φ_∗(ℱ))^-1 .

The relationship between the determinant of cohomology and Deligne pairing is given by the following theorem:

Let ℒ,ℳ  be two invertible sheaves on X, then

    <ℒ,ℳ>≅ Rφ_∗(𝒪_X)⊗ ( Rφ_∗(ℒ))^-1⊗( Rφ_∗(ℳ))^-1⊗ Rφ_∗(ℒ⊗ℳ) .


See <cit.>.




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