REGULATORS SYNTOMIC AND p-ADIC INTEGRATION I: RIGID SYNTOMIC REGULATORS
AMNON BESSER
1. Introduction The syntomic cohomology, more precisely the cohomology of the sheaves s(n) on the syntomic site of a scheme, where introduced in [FM87] in order to prove comparison isomorphisms between crystalline and p-adic tale cohomology. It can e be seen as an analogue of the Deligne-Beilinson cohomology in the p-adic world (for an excellent discussion see [Nek98]). In particular, when X is a smooth scheme over the ring of integers V of a nite extension K of Qp there should exist higher Chern classes from algebraic K-theory into the syntomic cohomology of X. Such classes have been constructed, sometimes under certain additional assumptions, by Gros [Gro90] and by Niziol [Niz97]. Syntomic cohomology comes in dierent avors (much like Deligne-Beilinson cohomology). The versions discussed above are well behaved only for proper schemes. In particular, they do not have the homotopy property for ane spaces. This makes computations dicult because most constructions in K-theory go through non proper schemes. In [Gro94], Gros introduced, using the rigid cohomology of Berthelot [Ber96, Ber97], rigid syntomic cohomology for a scheme X which is smooth over an unramied base. When the scheme X is ane he constructs rigid syntomic regulators, ci,j : Kj (X) H 2ij (X, s(i)X/K,rig ) , from K-theory into his rigid syntomic cohomology. Using these regulators Gros is able to show that the value of the syntomic regulator on certain cyclotomic elements in the higher K-theory of number elds is, when properly normalized, given by the values of p-adic polylogarithms at roots of unity. It should be mentioned here that there is another method of controlling syntomic cohomology, due to Somekawa [Som92]. In this method one assumes X has a compactication where the complement is a relative normal crossings divisor. Somekawa is able to prove the result of Gros for all cyclotomic elements. We should note however that loc. sit. is not yet published to the best of our knowledge. The following philosophy exists: Philosophy 1. There should be a p-adic Beilinson conjecture that relates special values of p-adic L-functions to syntomic regulators. Special cases of this are the results of [Gro90] and [KNQD98]. One should be able to derive some general conjecture from [PR95]. For results about CM elliptic curves see the discussion below. The main result of this work is an extension of the constructions of Gros to an arbitrary smooth V-scheme X. For such a scheme X we dene in section 3
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i syntomic cohomology Hsyn (X, n) and in section 4 we construct Chern classes from K theory to it. Our denition takes into account more growth conditions than that of Gros: we also consider log singularities. The result is that Hsyn is always nite dimensional (proposition 3.5). Our cohomology maps when possible to the version of Gros (proposition 6.4) and to the version of Niziol (proposition 6.7). Another objective of this work is to begin to develop tools for computations in syntomic cohomology. Our main result here is the construction of a modied syn tomic cohomology, denoted Hms (X, ), in section 5. This cohomology is related to syntomic cohomology by a natural map (proposition 5.6.2) which is an isomorphism in most cases of interest (proposition 5.6.3). It is signicantly easier to compute when the base V is ramied. We have also found that the original rigid syntomic cohomology of Gros (without log singularities), extended to the case of ramied base, is also useful in some computations (see for example [BdJ98]). It again can come with an original or modied avor, the latter being most useful. Let us discuss a bit of applications. In a sequel to this paper [Bes98] we com2 pute the syntomic regulator K2 (X) Hsyn (X, 2) when X is smooth and proper of relative dimension 1 over V. We show that there is a precise relation between this regulator and the p-adic regulator constructed by Coleman and de Shalit [CdS88]. In particular, for elliptic curves with complex multiplication their results in conjunction with ours relate the syntomic regulator with special values of a p-adic L-function of E, in line with the philosophy 1. In [Bes97] we will build on the results of this paper and embed syntomic cohomology in some other cohomology theory which has Poincar duality. This is e very useful for computations involving cycles. We will show how to relate p-adic Abel-Jacobi maps to a generalization of Colemans p-adic integration theory [Col85]. Finally, in [BdJ98] we intend to show how to compute syntomic regulators on the wedge complexes introduced in [DJ95] using p-adic polylogarithms. This is a generalization of the results of Gros on cyclotomic elements described above. Throughout this work V is a complete valuation ring with maximal ideal p, quotient eld K and residue eld of characteristic p. When is perfect we let V0 V be the Witt ring of and K0 its quotient eld. All schemes will be of nite type over V. We would like to thank Gros, Berthelot, de Jeu and Scholl for helpful conversations.
2. Rigid and de Rham complexes In this section we do the preparation to the construction of syntomic cohomology in the next section by constructing certain complexes computing rigid and (ltered parts of) de Rham cohomology. For the purpose of constructing Chern classes, it is very useful to lift cohomology to the level of the derived category, and even to the level of complexes. In the constructions below we will habitually write R but we will actually mean a particular complex representing this object in the derived category and maps between these objects will be represented by maps of complexes commuting on the nose. We will explain below (proposition 2.17 how such a choice of complexes and maps can be achieved. The rst step is to construct the complexes computing rigid cohomology. Here we assume that V is a discrete valuation ring. We will consider schemes X which are of nite type over .
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Denition 2.1. A rigid datum for X over V consists of an open immersion j : X X together with a closed immersion X P into a formal V-scheme P which is smooth in a neighborhood of X. We will write this datum as (X, j, P). A morphism between data (X, j, P) and (X , j , P ) is a commutative diagram X X P = u
j j
X X P , where is proper and u is smooth in a neighborhood of X. The collection of all rigid data becomes a category under this notion of morphisms. We denote this category by RD(X, V). Remark 2.2. Without further mention we will always assume that rigid data for X exist. This is certainly the case if X is quasi-projective. If this condition is not satised, one can carry out all the constructions using simplicial formal schemes, but we will not discuss this in this work. Lemma 2.3. The category RD(X, V) is ltered. Proof. Given Dr = (X r , jr , Pr ), r = 0, 1, 2 in RD(X, V), with (resp. without) maps Dr D0 for r = 1, 2, we can consider D3 = (X 3 , j3 , P3 ) RD(X, V) where P3 = P1 P0 P2 (resp. P3 = P1 V P2 ), X 3 is the closure in X 1 X 0 X 2 (resp. X 1 X 2 ) of the image of X and j3 is the obvious map. Then there clearly exists a commutative diamond (2.1) D3 ??? ?? D1 ? D2 ?? ?? D0 ,
(resp. without the maps to D0 ). For a p-adic formal V-scheme P there is an associated rigid analytic K-space, the generic ber of P, denoted PK . There is a canonical specialization map sp : PK P, which is continuous when PK is given its strong Grothendieck topology and P its Zariski topology. Berthelot introduces the notion of a tube. If Y is a locally closed subset of the special ber of a formal V-scheme P, the tube of Y in P, denoted ]Y [P , is a rigid analytic K-subspace of PK whose underlying set is the set sp1 (Y ) of points whose specialization is in Y . Now let (X, j, P) RD(X, V) and let Z = X X. Berthelot introduces the notion of a strict neighborhood of ]X[P inside ]X[P . By denition this is a subset U ]X[P , open in the strong Grothendieck topology, such that {U, ]Z[P } is a covering of ]X[P in the same topology. Berthelot denes a functor j from the category of sheaves on ]X[P to itself by j (F ) = --- jU F, lim --
U
where the direct limit is over all U which are strict neighborhoods of ]X[P in ]X[P and jU is the canonical embedding.
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Denition 2.4 (Berthelot). Let D = (X, j, P) RD(X, V). The rigid complex Rrig (X/K)D is dened by Rrig (X/K)D := R(]X[P , j P ). ]X[ As remarked above, we will take this as dening an actual complex rather than merely an object of a derived category. It is easy to see that the association D Rrig (X/K)D is a contravariant functor from RD(X, V) to the category of bounded below complexes of K-vector spaces. A fundamental theorem of Berthelot ([Ber97] Theorem 1.4 and Corrolaire 1.7) asserts that maps of rigid data induce quasiisomorphisms of rigid complexes. This motivates the following denition. Denition 2.5. The rigid complex of X over K is denes as Rrig (X/K) :=
DRD(X,V)
lim ----
Rrig (X/K)D .
The complex Rrig (X/K) is quasi-isomorphic to each of the Rrig (X/K)D . We next discuss functoriality. Let X and Y be -schemes as above and let f : X Y be a -morphism. We dene a rigid datum for f as a tuple consisting of rigid data for X and Y , (X, jX , PX ) and (Y , jY , PY ) respectively, and maps f : X Y , f : PX PY compatible with f in the obvious sense. We denote by RD(f, V) the category of all rigid data for f . As before, it is easily seen that the category RD(f, V) is ltered. Let P1 and P2 be the two projection functors from RD(f, V) to RD(X, V) and RD(Y, V) respectively. Lemma 2.6. The functor P2 is surjective. Proof. Following [Ber97] bottom of page 338 and page 340, Let (X, jX , PX ) and (Y , jY , PY ) be rigid data for X and Y respectively. We than constructs a new rigid datum for X in the following way. We take P = PX PY , X to be the closure of the graph of f in X Y and j : X X to be the obvious map. Then (X , j, P) together with (Y , jY , PY ) and the projections on Y and PY dene an object of RD(f, V) mapping to (Y , jY , PY ). Corollary 2.7. The association X Rrig (X/K) extends to a contravariant functor from -schemes to complexes of K-vector spaces. Proof. It is easy to see that to any D RD(f, V) corresponds a map Rrig (Y /K)P2 (D) Rrig (X/K)P1 (D) and that this map is natural in D. We therefore obtain a diagram
D RD(Y,V)
lim ----
Rrig (Y /K)D lim ----
DRD(f,V)
lim ----
Rrig (Y /K)P2 (D) lim ---- Rrig (X/K)D .
Rrig (X/K)P1 (D)
DRD(f,V)
D RD(X,V)
The left pointing arrow is an isomorphism (not just a quasi-isomorphism) by lemma 2.6. This gives the map associated to f . It is easy to check that we get a functor. Functoriality allows us to extend the denition of the rigid complex to simplicial schemes in the standard fashion. Denition 2.8. Let X be a simplicial -scheme. Applying the functor Rrig (?/K) we obtain a cosimplicial object in the category of complexes of K-vector spaces. We dene Rrig (X /K) to be the total complex of the associated double complex.
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This construction is functorial on the category of simplicial -schemes. We have the usual spectral sequence: Proposition 2.9. Let X = (Xn )nZ0 be a simplicial -scheme. Then there exists a spectral sequence
i,j i+j i E2 = Hrig (Xj /K) Hrig (X /K).
Proposition 2.10. Let X be a -scheme and let U X be the covering associated to a nite Cech covering of X (we view X as a simplicial scheme which is X in each degree). Then the canonical map Rrig (X/K) Rrig (U /K) is a quasiisomorphism. Proof. Let the Cech covering be {U1 , . . . , Un }. Then Un =
|I|=n+1
UI ,
UI :=
iI
Ui .
We choose a compactication j : X X and an embedding X P. This then denes a compactication UI X X, denoted jI , for each UI and we thus get a rigid datum DI = (X, jI , P) for each UI . The identity maps on X and P dene rigid data for all the morphisms between the UI that appear in the denition of U . It follows that Rrig (U /K) is quasi-isomorphic to the total complex of the double complex
R(]X[P , jI P ). ]X[ |I|=n+1
It follows from [Ber96, Prop. 2.1.8] or [Ber97, 1.2.ii] that this last complex is quasi-isomorphic to R(]X[P , j ) and hence to Rrig (X/K). ]X[
P
We state 2.11, 2.12 and 2.13 below for schemes but they immediately extend to simplicial schemes as well. Proposition 2.11. Let V V be a nite map of discrete valuation rings where V has residue eld and fraction eld K and let X be a -scheme. Then there is a canonical base change map K K Rrig (X/K) Rrig (X /K ), which is a quasi-isomorphism. The base change map is functorial in the obvious sense with respect to diagrams V V V and commutes with the maps induced by morphisms of -schemes. Proof. Let D = (X, j, P) be a rigid datum for X over V. One obtains a rigid datum D = (X , j , P V V ) for X over V . In this situation the proof of [Ber97, Proposition 1.8] shows the existence of a map K K Rrig (X/K)D Rrig (X /K )D . Taking direct limits give the required map and the functoriality statements are straightforward. Corollary 2.12. Suppose is perfect and recall that V0 is the Witt ring of . Let : V0 V0 be the map induced by the p-power map on . Then there exists a canonical and natural -semilinear map : Rrig (X/K0 ) Rrig (X/K0 ).
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Proof. Let be the projection X and let X X ,Fr be the relative X Frobenius map. Here the map in the last tensor product is the Frobenius map of , i.e., the p-power map. The map is obtained as the composition Rrig (X/K0 ) K0 Rrig (X/K0 ) Rrig (X ,Fr /K0 ) Rrig (X/K0 ), where the base change map is with respect to the map . Naturality is easily veried. The following lemma will be needed for the comparison between syntomic cohomology and modied syntomic cohomology. Its truth is obtained by a careful application of the functoriality properties of the base change. Lemma 2.13. Suppose, under the assumptions of corollary 2.12 that is a nite eld with q = pr elements, which implies that Frr : X X is -linear. Then r = (Frr ) as endomorphisms of Rrig (X/K). It is convenient to use a dierent model for the rigid complex which takes into account more data. Denition 2.14. An extended rigid datum for X over V consists of a rigid datum D = (X, j, P) RD(X, V) together with a strict neighborhood U of ]X[P in ]X[P . A map from (D, U ) to (D , U ) is a map of rigid data D D such that the induced map on tubes takes U into U . The category of extended rigid data is denoted ER(X, V). Given a morphism f : X Y over , an extended rigid datum for f consists of rigid data, (DX , UX ) and (DY , UY ), for X and Y over V respectively, a map f : X Y extending f and a rigid map UX UY commuting with the specialization maps to X and Y . The collection of extended rigid data for f forms a category denoted ER(X, V). The categories ER(X, V) and ER(f, V) are again ltered: In the situation of the proof of lemma 2.3, suppose we were given in addition corresponding strict neighborhoods Ui for i = 0, 1, 2. Then one can take U3 = (U1 U0 U2 ) ]X 3 [P3 . There are obvious functors RD(?, ?) ER(?, ?) obtained by taking U = ]X[P . Denition 2.15. Let D = (X, j, P, U ) ER(X, V). The rigid complex Rrig (X/K)D is dened by Rrig (X/K)D := R(U, j ) U The rigid complex Rrig (X/K) is given by Rrig (X/K) :=
DER(X,V) base change (FrX ) 1id
lim ----
Rrig (X/K)D .
The complex Rrig (X/K) is clearly functorial in X by the same argument that proved the functoriality of Rrig (X/K). It follows from [Ber97, 1.2.iv] that for an extended rigid datum (D, U ), with D = (X, j, P), the map induced on rigid complexes by the map (D, U ) (D, ]X[P ) is a quasi-isomorphism. This implies that all maps of extended rigid data induce quasi-isomorphisms on the associated rigid complexes, hence that Rrig (X/K) Rrig (X/K)D for any datum D. There = is an obvious natural transformation Rrig Rrig which is a quasi-isomorphism by the result above.
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The next step is to dene a de Rham complex. This was already done by Huber [Hub95, Chapter 7] so we do not go into all the details. We need to know not only a complex computing de Rham cohomology, but also complexes computing all the ltered parts. Here K can be any eld of characteristic 0. Let X be a smooth K-scheme. A de Rham datum for X is an injection i : X Y where Y is a smooth and proper K-scheme and D := Y X is a divisor with normal crossings. Denition 2.16. To a de Rham datum D = (Y, i) and to every k Z0 we associate a complex, called the k-th ltered part of the de Rham complex of X with respect to the datum D, dened by Filk RdR (X/K)D := R(Y, k log D ). Y The k-th ltered part of the de Rham complex of X is dened by Filk RdR (X/K) := --- Filk RdR (X/K)D , lim --
D
where the limit is over all de Rham data D. We will write RdR (X/K) for Fil0 RdR (X/K). Note that the Filk , in spite of their name, are not subcomplexes of RdR (X/K) but there are natural maps Filk RdR (X/K) RdR (X/K). The nal ingredient needed for the construction of syntomic cohomology is a comparison between de Rham and rigid cohomology. Let X be a smooth Vscheme with generic ber XK and closed ber X . We will dene a functorial map RdR (XK /K) Rrig (X /K). We stress that this map is not a quasi-isomorphism in general. The datum required for the denition is a compactication X X together with a de Rham datum for XK , i : XK Y . The compactication j gives an rise to an extended rigid datum (X , j , X, XK ), where X is the p-adic complean tion of X and XK is the rigid analytic K-space associated with XK [Ber96, 0.3.3]. an It is not so hard to see that indeed XK is a strict neighborhood of ]X [X inside ]X [X = X K . We obtain a map RdR (XK /K)(i,Y ) = R(Y, log(Y XK ) ) R(Y, i K ) Y X (2.2)
an R(XK , K ) R(XK , K ) X X an an R(XK j K ) = Rrig (X /K)(X X an an ,j ,X,XK ) an j
.
By taking the limit over all X and Y we obtain the required map. Functoriality is evident. Now it remains to explain why all the constructions can be made on the level of complexes Proposition 2.17. There exists a way to choose the complexes below, representing their name sakes in the derived categories of complexes of K or K 0 -vector spaces, in such a way that all morphisms in the derived categories sense between them we have used above are in fact represented by maps between these complexes. The complexes are: R(U, j ) and R(U, ) when U is a strict neighborhood of a tube (the U U an latter required for the case U = XK in (2.2)), R(X, ) when X is a smooth X k K-scheme, and R(Y, Y log(Y X) ) and R(Y, i ) when i : X Y is a de X Rham datum for X.
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Proof. The method for constructing these complexes is standard. It uses a construction in [SD72] and was used by Beilinson to construct Zariski sheaves computing Deligne cohomology in [Bei85, 1.6.5]. We only explain how to construct the complexes R(U, j ) in a functorial way with respect to morphisms of extended rigid U data and leave the other cases as an exercise. Consider the category A whose object are 4-tuples D = (X, j : X X, P, U ) where X and X are -schemes and (X, j, P, U ) is an extended rigid datum for X. Maps are the obvious commuting diagrams. Let B be the category whose objects are pairs (D, F ) where D A and F is a sheaf on U = UD in the Grothendieck topology of U . A map (D, F ) (D , F ) consists of a map f : D D in A together with a map of sheaves F fU F , where fU is the map UD UD which is part of f . Then B is, in the terminology of [SD72, 1.2.2], biltered by toposes over A (loc. sit., 4.1.0). We can consider the section category (B) of loc. sit., 1.2.8. Explicitly, an object of F (B) is given by a collection of sheaves FD on UD , for every D A, together with morphisms of sheaves f : FD fU FD for every morphism f : D D in A such that one has (2.3) (f g) = g f , id = id .
By loc. sit., 1.2.12, (B) is a topos. By loc. sit., 1.3.10 there is a collection I B of abelian objects of (B) such that the following two properties hold: Any abelian F (B) injects into I IB . For I IB and for any D A, the sheaf ID on UD is asque.
The association D jD D , together with the natural maps jD D fU jD D U U U denes a complex of abelian objects of (B). We choose a resolution of this complex by a complex of objects of IB . This gives us for each D A a complex of asque sheaves ID on UD together with a morphisms of complexes of sheaves f : ID fU ID for every morphism f : D D in A satisfying (2.3). Taking global sections on UD now gives a functor D (UD , ID ) into complexes of vector spaces. This is enough to construct functorially Rrig .
3. Syntomic cohomology and product structures In this section we will dene syntomic cohomology and state some of its fundamental properties, including the product structure. We begin with a bit of homological algebra. Suppose we are given complexes X , Y and Z with maps f : X Z and g : Y Z . Then one can form the naive bered product X Z Y whose n-th component is X n Z n Y n . It is of course equal to the kernel of f g : X Y Z . Therefore, one should prefer to use instead the slightly dierent construction, called the quasi-bered product, X Z Y := Cone(f g)[1]. We have the well known Lemma 3.1. In the situation above, if the map f g is surjective, then the two construction are quasi-isomorphic via the map (3.1) (x, y) (x y, 0).
It will be convenient to use both construction in what follows. p i Notice that we have canonical maps Z [1] X Z Y X Y coming from the cone construction. Let us write pA and pB for the composition of p with the rst and second projection respectively. The following construction of the cup
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product is a variant of one of Niziol [Niz93], which is itself a variant of a construction of Beilinson. Alternatively, it is a special case of the construction of [Bei86, 1.11]:
Lemma 3.2. Suppose We are given complexes Xi , Yi , Zi and maps fi , gi as above for i = 1, 2, 3, and that we are given maps of complexes : X1 X2 X3 , and similarly for Y and Z, which are (strictly) compatible with the maps f i and gi in the obvious sense. Then, 1. There exist a map (bottom horizontal), making the following diagram commute, where the top horizontal map is induced by the maps . (X1 Z1 Y1 ) (X2 Z2 Y2 ) X3 Z3 Y3 (X1 Z1 Y1 ) (X2 Z2 Y2 ) X3 Z3 Y3 . 2. On homology one has the following projection formula for z H (Z1 ) and w H (X2 Z2 Y2 ):
((i1 ) (z)) w = (i3 ) [x (g2 ) (pB2 ) w] . Proof. (Compare with [Niz93, Prop. 3.1] or [Bei86, Lemma 1.11]) One chooses a parameter and denes the cup product by the formula (x1 , y1 , z1 ) (x2 , y2 , z2 ) = x1 x2 , y1 y2 , (3.2) +(1)
deg x1
z1 (f2 (x2 ) + (1 )g2 (y2 )) ((1 )f1 (x1 ) + g1 (y1 )) z2 .
All of these products are known to be homotopic for dierent values of . Checking the required properties is straightforward from this formula. For the second part one species = 0. We are now ready to dene syntomic cohomology. Let X be a smooth V-scheme. By the constructions of section 2 we have, for any n Z0 , the following diagram of complexes and maps between them (3.3) Rrig (X /K0 ) Rrig (X /K) Rrig (X /K) RdR (XK /K) Filn RdR (XK /K). We also have a -linear map : Rrig (X /K0 ) Rrig (X /K0 ) and both diagram and map are functorial in X. Denition 3.3. The syntomic complex of X twisted by n is dened to be Rsyn (X, n) := Cone 1 pn [1]Rrig (X /K) Filn RdR (XK /K),
where the two maps dening the bered product are Cone 1 : Rrig (X /K0 ) Rrig (X /K0 ) [1] pn Rrig (X /K0 ) Rrig (X /K) Rrig (X /K)
and the map induced by the left pointing arrows of (3.3). The i-th homology of i Rsyn (X, n) will be denoted Hsyn (X, n).
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The above construction is evidently functorial in X. We can therefore dene Rsyn for simplicial schemes as in denition 2.8. We have the analogue of proposition 2.10: Proposition 3.4. Let X be a smooth V-scheme and let U X be the covering associated to a nite Cech covering of X. Then the canonical map Rsyn (X, n) Rsyn (U , n) is a quasi-isomorphism for any n Z0 . Proof. Because Rsyn is dened as an iterated cone, it is enough to check the statement of the proposition on each of the components of the cone. But for the de Rham components it is well known and for the rigid components it was proved in proposition 2.10. We proceed to show some of the fundamental properties of syntomic cohomology. Proposition 3.5. There is a long exact sequence, (3.4) 1 i1 i1 i1 i1 Hrig (X /K0 ) Filn HdR (XK /K) Hrig (X /K0 ) Hrig (XK /K)
i Hsyn (X, n)
2 i i i i Hrig (X /K0 ) Filn HdR (XK /K) Hrig (X /K0 ) Hrig (XK /K) , where the maps 1 and 2 are given in the appropriate degrees by (3.5) (x, y) 1 pn x, x y .
Here, for the second component we have identied both x and y with their images i i in Hrig (XK /K). In particular, if K is nite over K0 , then Hsyn (X, n) is a nite dimensional K0 -vector space for every i and n. Proof. By writing explicitly the quasi-bered product in term of cones, one nds Rsyn (X, n) Cone(Rrig (X /K0 ) Filn RdR (XK /K) = Rrig (X /K0 ) Rrig (X /K))[1], where the map dening the cone is given by (3.5) (the reader should compare at this point the construction of [Niz97, 2.1]). This immediately gives the result. Remark 3.6. Let us consider the special case where X is a smooth K-scheme considered as a V-scheme. In this case we have X = , so Rrig (X /?) = 0 with ? = K or K0 and the same is true with Rrig . The long exact sequence (3.4) shows that
i i Hsyn (X, n) Filn HdR (X/K). =
This is perhaps to be expected since this is the absolute cohomology for varieties over a eld. Denition 3.7. The cup product map on syntomic cohomology,
i j i+j : Hsyn (X, n) Hsyn (X, m) Hsyn (X, n + m),
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is constructed as follows: By lemma 3.2 it is enough to construct a product Cone(1 n ) Cone(1 m ) Cone(1 n+m ), with n = /pn . This is achieved by the formula, similar to (3.2), (x1 , z1 ) (x2 , z2 ) = x1 x2 , (3.6) z1 (x2 + (1 )m (x2 )) +(1)deg x1 ((1 )x1 + n (x1 )) z2 .
This denition is compatible with the denitions given by Niziol, Kato, Gros and many others. 4. Construction of syntomic regulators In this section we construct syntomic Chern classes,
2jp cp : Kp (X) Hsyn (X, j). j
The method follows mostly Huber [Hub95, Chapter 18] with some input from Gros [Gro90] and Deligne [Del74]. The main step in the construction is to repeat the computation of the de Rham cohomology of B GLn by Deligne [Gro90, Chapter II] for rigid cohomology. We briey recall the setup from loc. sit., but using the notation of Deligne in [Del74, 6]. We will work simultaneously over any of the bases , V, V0 K or K0 , making the needed adjustments. If G is an algebraic group (over any of the bases above) acting on a scheme X we let [X/G] be the simplicial scheme such that [X/G]n = (Gn X)/G where G acts by g (g0 , . . . , gn , x) = (g0 g 1 , . . . , gn g 1 , gx) and the face and degeneracy maps are the obvious ones [Del74, 6.1.2]. Note that the quotients are well dened and in fact there is an isomorphism Gn X [X/G]n given by (for example) (g1 , . . . gn , x) (1, g1 , . . . gn , x). Lemma 4.1. Let X be a principal G-bundle over S = X/G. Then the map [X/G] S induces an isomorphism on rigid cohomology. Proof. (sketch). If we knew how to write rigid cohomology as a sheaf cohomology this would follow from [Del74, 6.1.2.2]. We need to check that what we know about rigid cohomology is sucient for a proof. Let X = cosq(X S). There is a canonical isomorphism of simplicial schemes over S, X [X/G] = [Del74, 6.1.2.a]. If there is a section S X, then it extends to a section s : S X to the canonical map : X S. It is well known that the map s is homotopic to the identity map of X and this homotopy induces a homotopy on the rigid complexes showing the result in this case. In the general case we have a nite covering, U = Ui , of S such that the restriction of X to each Ui has a section. Let U = cosq(U S). An application of the spectral sequence 2.9, proposition 2.10, and the special case of a map with a section discussed above now shows that the cohomology of the bisimplicial set (cosq(X S Un Un ))?,n = (cosq(Xm S U Xm ))m,? is isomorphic to the cohomology of U , hence of S, on the one hand, and to the cohomology of X on the other hand.
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AMNON BESSER
Fix N n in Z0 . Let E = Gn and F = GN be two vector group schemes a a and let Hom(E, F ) be the corresponding scheme of homomorphisms. There is a ltration of Hom(E, F ) by open subschemes Hom(E, F ) = Un Un1 U0 , where Ul is dened by the invertability of at least one n l minor. Lemma 4.2. The scheme Ul Ul1 is a smooth subscheme of Hom(E, F ) of codimension l(l n + N ). Proof. This is proved in [Gro90, II.2.4] for schemes over V0 but the proof is the same in any of the other cases. The group G = GLn acts on Hom(E, F ) in the obvious manner, preserving the ltration by the Ui . The scheme U0 is the so called Stiefel variety of n-frames on F and is denoted by Stief(E, F ). We have (4.1) Stief(E, F )/G Grassn (F ) = ,
where Grassn (F ) is the grassmanian of n-dimensional subspaces of F . Proposition 4.3. The canonical map
Hrig ([Hom(E, F )/G] /K) Hrig ([Stief(E, F )/G] /K)
is an isomorphism in degrees 2(N n). Proof. (Compare [Gro90, Corollaire II.2.8]). It is enough to show the same for the map induced on rigid cohomology by each of the inclusions [Ul1 /G] [Ul /G] . By lemma 4.2 we see that on the n-th component, [Ul /G]n [Ul1 /G]n is a closed subscheme of [Ul /G]n of codimension l(ln+N ) N n+1. By purity for rigid coi i homology [Ber97, Corollaire 5.7] the map Hrig ([Ul /G]n /K) Hrig ([Ul1 /G]n /K) is an isomorphism if i 2(N n). The result now follows from the spectral sequence 2.9.
2i Proposition 4.4. There are canonical classes xi Fili HdR (B GLn /K) such that we have isomorphisms
(4.2)
K[x1 , . . . , xn ] HdR (B GLn /K) Hrig (B GLn /K).
If K = K0 and we identify the classes xi with their images in Hrig , then we have (xi ) = pi xi . Proof. Let be the one point space. Then B GLn = [/G] [Del74, 6.1.3]. We have a G-equivariant diagram, Stief(E, F ) JJ JJ JJ JJ JJ $ / Hom(E, F ) t G ttt tt tt ztt 0
SYNTOMIC REGULATORS
13
where 0 denotes the 0 section to . It induces a corresponding diagram of cohomologies,
i i Hrig ([Stief(E, F )/G] /K) o Hrig ([Hom(E, F )/G] /K) iTTTT jjj5 TTTT jjjj TTTT jj jj TTT jjjj r i Hrig (B GLn /K)
An easy diagram chase using proposition 4.3 shows that the left diagonal map is injective for i 2(N n). A similar argument shows the same for de Rham cohomology. By lemma 4.1 and (4.1) it now follows that the two horizontal maps in the commutative diagram
i i Hrig (B GLn /K) Hrig (Grassn (F )/K)
(4.3)
i i HdR (B GLn /K) HdR (Grassn (F )/K)
are injective for i 2(N n). The map on the right is an isomorphism since Grassn (F ) is proper. In de Rham cohomology we have a good theory of characteristic classes. Let 2i xi Fili HdR (B GLn /K) be the i-th Chern class of the universal bundle. Then (xi ) are the Chern classes of the universal vector bundle over Grassn (F ) and it is known that these generate the cohomology ring of Grassn (F ). It follows that is surjective, hence that if i 2(N n) all maps in diagram (4.3) are isomorphisms. Varying N we nd the isomorphisms (4.2). It now follows that the properties of the classes xi can be tested in the cohomology of Grassn ), (F where they are well known: As Grassn (F ) is proper we have an isomorphism H 2i (Grass (F )/K0 ) H 2i (Grass (F )/V0 ) K0 , =
dR n cr n
under which xi correspond to the crystalline Chern classes of the universal bundle and therefore have the right behavior under Frobenius. We can now dene Chern classes in syntomic cohomology. From proposition 4.4 it follows that B GLn has cohomology only in even dimensions. Using the long exact sequence (3.4) we easily obtain an isomorphism H 2i (B GLn V, i) {x Fili H 2i (B GLn /K0 ), (x) = pi x}. =
syn dR
In particular, we see that the classes xi of proposition 4.4 dene classes, denoted n 2i Ci , in Hsyn (B GLn V, i). Considering the usual inductive system of B GLn -s, obtained by the inclusions in the upper left corner GLn GLn+1 , we see that n the Ci are compatible under the induced maps on cohomology because the de Rham universal classes are known to do so. We thus obtained cohomology classes Ci in the cohomology of the ind-scheme B GL which we call the universal syntomic Chern classes. Theorem 4.5. Let X be a smooth V-scheme. There exist functorial Chern classes
2jp cp : Kp (X) Hsyn (X, j), j 2jp 2jp such that their composition with the map Hsyn (X, j) Filj HdR (XK /K) obtained from the sequence(3.4) gives the usual Chern classes in de Rham cohomology.
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AMNON BESSER
Proof. We follow Hubers treatment in [Hub95, Chapter 18]. By [Hub95, Proposition 18.1.5] (whose proof is also valid in our case) we have an isomorphism lim --- p Tot(Z Z B GL(U )) Kp (X), --
U
where the direct limit is over all nite ane Cech coverings U of X. By [Hub95, Proposition 18.1.7 b] there are induced maps (4.4) Kp (X) --- p Tot(Z B GL(U )), lim --
U
K0 (X) Z.
For simplicial schemes U and Y , let B(U , Y ) be the simplicial cosimplicial abelian group which is the Q-vector space generated by Hom(Un , Ym ) in degree (m, n) and let A(U , Y ) be the associated complex [Hub95, Denition 18.2.1]. Summation of pullback maps give a map of simplicial cosimplicial groups, B(U , Y ) [Hom(Rsyn (Ym , j), Rsyn (Un , j)]m,n ,
where Hom here means in the category of complexes (this is why we insisted on dening the syntomic cohomology on the level of complexes). By taking the associated complexes and then the total complexes we obtain a map (4.5) A(U , Y ) R Hom(Rsyn (Y , j), Rsyn (U , j)) .
In the special case that Y = B GL, we have by [Hub95, Lemma 18.2.4] a map p Tot(Z B GL(U )) H p (A(U , B GL)) . Composing this with the map induced on homology by (4.5) and applying to the universal class Cj H 2j (Rsyn (B GL, j)) we get a map
2jp p Tot(Z B GL(U )) Hsyn (U , j). 2jp If U is as in (4.4), then Hsyn (U , j) Hsyn (X, j) by proposition 3.4. This = 2jp completes the construction. The result about the composition with the projection to de Rham cohomology follows from the universal case and functoriality.
Remark 4.6. As in [Hub95, Denition 18.2.6], the same construction yields Chern classes for split simplicial smooth V-schemes of nite combinatorial dimension. In particular, we obtain Chern classes in relative cohomology using simplicial cones. Denition 4.7. The Chern character ch : Ki (X) ch =
j1 j 2ji Hsyn (X, j) is given by
(1)j1 i c (j 1)! j
(+ Rank
if
i = j = 0).
Proposition 4.8. The Chern character is multiplicative. Proof. This reduces as usual to properties of the universal Chern classes. Because the syntomic cohomology is the same as de Rham cohomology for B GLn , there is nothing to prove.
SYNTOMIC REGULATORS
15
5. Modified syntomic cohomology In this section we dene a certain modication of the rigid syntomic cohomology of section 3. The dierence is that we replace the semi-linear Frobenius by a linear Frobenius. This makes the theory easier to compute. For the purpose of computing regulators in higher K-theory the modied theory is as good as the original one. In this section we need the additional assumption that Fp . The following notion is due to Coleman. Denition 5.1. Let X be a -scheme. A Frobenius endomorphism, : X X, of degree q = pr is any -endomorphism of X obtained in the following way: Let X be an Fq -scheme and let : X X Fq be a -isomorphism. Then r = 1 (Fr id ) . It is clear that if is a Frobenius endomorphism of degree q then k is a Frobenius endomorphism of degree q k . Denition 5.2. The category of Frobenius endomorphisms of X is the category whose objects are Frobenius endomorphisms : X X. There is a unique morphism between and k for any k 1. Lemma 5.3. The category of Frobenius endomorphisms of X is ltered. Proof. It is not hard to see that suciently high powers of any two Frobenius endomorphisms become identical. Fix an integer n. We associate to each Frobenius endomorphism a certain complex, in such a way that we get a functor on the category of all Frobenius endomorphisms. To of degree q we associate the complex Cone 1 : Rrig (X/K) Rrig (X/K) [1]. qn
To the morphism m we associate the map of cones induced by the commutative diagram Rrig (X/K) Rrig (X/K) Pm1 =
s=0
1( /q n )
(5.1)
( /q n )s
Rrig (X/K) Rrig (X/K). Denition 5.4. The modied syntomic complex associated with a Frobenius endomorphism is the complex Rms (X, n) := Cone 1 qn [1]Rrig (X /K) Filn RdR (XK /K),
1( /q n )n
where q is the degree of and the cone is the one discussed above. The modied syntomic complex of X is Rms (X, n) = --- Rms (X, n) , lim --
where the direct limit is over the category of all Frobenius endomorphisms and the connecting maps are the ones dened above. The homology of the modied syni tomic complex is called modied syntomic cohomology and denoted by Hms (X, n).
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AMNON BESSER
Lemma 5.5. The modied syntomic complexes, and hence the modied syntomic cohomologies, are functorial. Proof. One need only observe that any morphism of varieties over is already dened over some nite eld, which implies that for any morphism f : X Y and for a conal collection of Frobenius endomorphisms : Y Y there is a Frobenius endomorphism : X X making the obvious diagram commute. Most of the basic properties of the modied syntomic cohomology are concentrated in the following proposition. Proposition 5.6. (5.2) Rms (X, n) --- Cone 1 n : Filn RdR (XK /K) Rrig (X/K) [1], = lim -- q where the limit is over all Frobenius endomorphisms , the notation 1 /q n stands for this map composed with the map RdR Rrig and the transition maps are constructed using a diagram analogous to (5.1). Furthermore, if is any xed Frobenius endomorphism of degree q, then we also have the quasi-isomorphism (5.3) Rms (X, n) = lim --- Cone 1 --
k
1. There is a canonical quasi-isomorphism,
qn
k
: Filn RdR (XK /K) Rrig (X/K) [1].
2. If is a nite eld, then there is a canonical and functorial map : Rsyn (X, n) Rms (X, n) 3. There are canonical and functorial maps (5.4) Rrig (X /K)[1] Rsyn (X, n), Rrig (X /K)[1] Rms (X, n). When is a nite eld these maps are compatible with the map . These maps induce isomorphisms, i1 i Hms (X, n) Hrig (X /K)/ Filn HdR (XK /K) , (5.5) = i1 and, if is nite, (5.6)
i1 i Hsyn (X, n) Hrig (X /K)/ Filn HdR (XK /K) = i1
.
,
(at least) in the following two cases: X is proper over V and 2n = i, i 1, i 2, X is ane and n i > reldim X. In particular, if in either of these cases is a nite eld, then induces an isomorphism on degree i cohomology. 4. Suppose V is a nite extension of V with eld of fractions K and let X = XV V . Then there exists a canonical base change quasi-isomorphism R ms (X, n)K K Rms (X , n). 5. There are cup products in modied syntomic cohomology compatible with the products in syntomic cohomology under the map and also compatible with base change.
SYNTOMIC REGULATORS
17
Proof. 1. Let be a Frobenius endomorphism of X . In the denition 5.4 of Rms (X, n) we may replace the quasi-bered product by an ordinary product because the cone on the left hand side surjects on Rrig (X /K). The resulting complex is easily seen to be isomorphic to the level complex of (5.2). The second part of the assertion follows because for a xed the collection of powers k is conal in the category of Frobenius endomorphisms. 2. We rst construct a map Cone(1 /pn : Rrig (X /K0 ) ) Cone(1 /q n : Rrig (X /K0 ) ) for some Frobenius endomorphism . Suppose is a nite eld with q = pr elements. Then = Frr is a Frobenius endomorphism of X and by lemma 2.13 we have r = on Rrig (X /K0 ). It follows that we can dene the required map by using a diagram similar to (5.1). This map can then be composed with the extension of scalars map and the canonical map between Rrig and Rrig to give a map Cone(1 /pn : Rrig (X /K0 ) ) Cone(1 /q n : Rrig (X /K) ) .
By the construction of the (modied) syntomic complexes we now obtain a map Rsyn (X, n) Rms (X, n) by taking the identity maps on the other components of the quasi-bered product. This map we may compose with the map to the limit on all Frobenius endomorphisms to complete the construction. For schemes over our particular commutes with all maps and this easily gives functoriality. 3. The maps (5.4) are evident from the denition of the (modied) syntomic complexes, as is the compatibility with the map because we have taken the identity map on Rrig (X /K) when dening it. We show that these maps induce isomorphisms on cohomology in the stated cases for syntomic cohomology, the proof for modied cohomology being essentially the same. We abbreviate Cone for Cone(1 /pn )[1]. From the construction of syntomic cohomology as a quasibered product, which is again a cone, we get the following long exact sequence.
i1 i1 i H i1 (Cone) Filn HdR (XK /K) Hrig (X /K) Hsyn (X, n) i i H i (Cone) Filn HdR (XK /K) Hrig (X /K) . i1 i1 i It follows that the map Hrig (X /K)/ Filn HdR (XK /K) Hsyn (X, n) is an isoi1 i i morphism if H (Cone) = H (Cone) = 0 and the map Filn HdR (XK /K) i Hrig (X /K) is an injection. This last requirement holds in the cases considered i because in the proper case HdR (XK /K) Hrig (X /K) and when n > reldim X we = i n i have Fil HdR (XK /K) = 0. The long exact sequence for the cohomology of Cone, i2 i2 Hrig (X /K0 ) Hrig (X /K0 ) H i1 (Cone) i1 Hrig (X /K0 ) Hrig (X /K0 ) H i1 (Cone) i1 i Hrig (X /K0 ) Hrig (X /K0 ) i 1/pn 1/pn 1/pn
,
shows that the i-th and i 1-th cohomologies of Cone vanish when 1 /pn is an isomorphism on Hrig (X /K0 ) in degrees i, i 1 and i 2. This now follows from the theory of weights. By [ClS98] and [Chi97] the K0 -linear Frobenius, which j is a power of , has weight j when acting on Hrig (X /K0 ) when X is proper and has mixed weights between j and 2j in general. In the proper case it follows that if 2n = j for j = i 2, i 1 and i, then the operator /pn has no xed vector j on Hrig (X /K0 ) because some power of it does not. It follows that 1 /pn is
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AMNON BESSER
injective, hence bijective, on the degree i 2, i 1 and i cohomologies. In the i second case this is no longer true a-priori for j = i but Hrig = 0 because X is ane and i > reldim X. 4. Apply base change (proposition 2.11 for rigid cohomology) in each component. 5. The construction of the cup product is almost identical to the one we did for syntomic cohomology. The cup product on Cone(1 /q n : Rrig (X /K) ) is given by the formula (3.6) with m = /q m . One then needs to check that these products are compatible up to homotopy under the transition maps. This can be done by a direct laborious computation. A much more conceptual and general of understanding this is given in [Bes97]. This type of compatibility also implies that the product is compatible with the map . Compatibility with base change is clear. Remark 5.7. 1. We expect the base change isomorphism of proposition 5.6.4 to exist for innite extensions as well, at least on the level of cohomology. 2. Using the model (5.3) for modied syntomic cohomology it is easy to see that the cup product is given in level k by the formula (3.6) with m being ( /q n )k composed with Filn RdR (XK /K) Rrig (X /K). 3. Suppose K = K0 . One can extend the map to Rrig (X /K0 ). An argument similar to the proof of proposition 5.6.1 shows that Rsyn (X, n) Cone(Filn RdR (XK /K) Rrig (X /K)) . = This gives rise to a long exact sequence
i1 i Filn HdR (XK /K) Hrig (X /K) Hsyn (X, n) i1 i Filn HdR (XK /K) Hrig (X /K) i 1/pn 1/pn 1/pn
.
In the cases discussed in proposition 5.6.1 this reduces to a short exact sequence
i1 i 0 Filn HdR (XK /K) Hrig (X /K) Hsyn (X, n) 0. i1 i1 The isomorphism (5.6) is induced by the map sending x Hrig (X /K) to i the image in Hsyn (X, n) of (1 /pn )x. A similar analysis applies to modied syntomic cohomology. See proposition 5.9.3 for a special case. 1/pn
When we compose the syntomic Chern classes with the canonical map : Hsyn Hms we obtain modied syntomic Chern classes and Chern characters. Alternatively, one can construct these directly using the same techniques as before and universal Chern classes which are the images of the syntomic ones under the map in the cohomology of B GLn . This makes the following lemma evident Lemma 5.8. The modied syntomic Chern classes commute with base change, i.e., when X is an V-scheme, V is a nite extension of V and X /V is the scheme obtained by base change to V , there is a commutative diagram
2jp Kp (X) Hms (X, j) cp j cp j
2jp Kp (X ) Hms (X , j).
SYNTOMIC REGULATORS
19
i From here until the end of the section we consider the cohomology Hms (X, i) (so the degree equals the twist) of a smooth ane V-scheme X = Spec(A). Fix a compactication j : X X over V. Let P = X. We get a rigid datum D = (X , j , P). By the proof of proposition 1.10 in [Ber97] we see that the complex Rrig (X /K)D is quasi-isomorphic to the complex
,K := --- , lim U A --
U
where the limit is over all strict neighborhoods of ]X [X K . We remark that this complex is in fact the complex of dierentials of the dagger algebra A used in the Monsky-Washnitzer cohomology [MW68, vdP86], but we will not need this fact here. Now x a Frobenius endomorphism of degree q of X . It follows from lifting theorems for dagger algebras ([Col85, Thm A-1] or [vdP86, Thm 2.4.4.ii]) that there is a lifting of to the dagger algebra A . This implies that there are strict neighborhoods U U and a map : U U whose reduction is , so the collection, (X , j , P, U ), (X , j , P, U ) ER(X , V), :U U ,
belongs to ER(, V). We therefore obtain a commutative diagram ,K A
Rrig (X /K) Rrig (X /K), where the vertical maps are quasi-isomorphisms. Now choose any de Rham datum (Y, i) for X. By Hodge theory [Del71, Corollaire 3.2.13.ii] we see that the space i (XK )log := H 0 (Y, i log(Y XK ) ) Y
i is independent of the choice of (Y, i) and is isomorphic to Fil i HdR (XK /K) = i i H (Y, Y log(Y XK ) ).
,K A
Proposition 5.9. Let X and be as above. 1. There is a canonical isomorphism (5.7)
i Hms (X, i) --= lim (, h) : -- k
i (XK )log , h i1 /di2 , A ,K A ,K dh = 1 qi
k
,
where we abusively identied with its image in i ,K . The connecting map A between level k and level km is given by
m1
(, h)
,
s=0
( /q n )sk h .
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AMNON BESSER
2. The cup product H i (X, i) H j (X, j) H i+j (X, i + j) is given in level k by the formula 1 , h1 ) (2 , h2 ) = 1 2 , (5.8) +(1)i h1 + (1 ) qi qj
k k
2 1 h2 .
(1 ) +
i1 i 3. When i > reldim X the isomorphism Hrig (X /K) Hms (X, i) of (5.5) is given by the formula i1 u Hrig (X /K) i1 /di2 (0, (1 ( /q i )k )u) A ,K A ,K k>0
.
Proof. The rst part follows immediately from the discussion above and (5.3). The second part follows easily from remark 5.7.2. The last part is straightforward (compare remark 5.7.3). Proposition 5.10. For X = Spec(A), and as above, The composed map A K1 (X) 1 Hms (X, 1), 1 is given as follows: Let f A and let f be its reduction. As f is dened over some k k k nite eld, there is some power of , say , of degree q , such that f k = f q . k It follows that f k f q (mod p) and therefore that the rigid function f0 := fq f k
k
c1
satises log f0 0 ,K . With all that, under the isomorphism (5.7) the cohomology A class c1 (f ) is given in degree k by 1 dlog f, log f0 qk .
Proof. By replacing by k we may always assume k = 1. We will abuse the notation to write c1 (f ) for the rst component of the regulator, which is dened 1 under our assumption. We start with the case X = Gm (so A = V[T, T 1]), f = T and is dened by (T ) = T q . Since the modied syntomic Chern class lifts the de Rham Chern class, which for T is just dlog T , We see that 1 c1 (T ) = (h, dlog T ), where dh = dlog T dlog (T ) = 0. 1 q The proposition in this case amounts to the statement that h = 0. To see this we use the involution : A A dened by (T ) = T 1 . As the Chern class is functorial, and as commutes with , we see that c1 (T 1 ) = (h, dlog T ) = ( h, dlog T ) = (h, dlog T ). 1 As c1 is a group homomorphism, we have (0, 0) = c1 (T 1 T ) = (2h, 0), which 1 1 proves what we wanted in this case. The next step is to show that the formula of the proposition is consistent with changing the map . For this we will need a lemma.
SYNTOMIC REGULATORS
21
Lemma 5.11. Let Z = Gm /V Z = P1 and P = Z. Let U be a strict neighborhood of ]Z [P in PK . Let Z , resp. Z be the diagonals in Z Z , resp. Z Z , and let = (U U ) ]Z [PP . Finally, let z be the standard parameter on ZK and let x and y be its two pullbacks to . Then, the image of log(x/y) A() in the 0-th component of Rrig (Z /K) is 0. Proof. First of all we notice that indeed log(x/y) is a rigid function on . This is because x y (mod p). Let be the image of log(x/y) in the 0-th component. Then d is the image of dlog(x/y) = dlog x dlog y. But both dlog x and dlog y are pullbacks of dlog z from U , so by the construction of Rrig (Z /K) as a ltered direct 0 limit we have d = 0. It follows that denes a class [] Hrig (Z /K) K. To show that the image is 0, we now use the diagonal map : U . This map, together with all the associated data, denes an object of ER(id, V). It follows that the image of [] in K is the same as that of log(x/y) = 0. Remark 5.12. Notice that in the above proof the map does not dene a morphism of extended rigid data. It is therefore essential to verify rst that d = 0. Corollary 5.13. Let X = Spec(A) as above, let f A and let , be two morphisms U U whose reduction is . Then the functions log(f q / (f )), log [f q /(( ) (f ))] A(U ) have the same image in Rrig (X , V). Proof. We need to show that log [ (f )/(( ) (f ))] has image 0 in Rrig (X , V). This follows because this function is the pullback from (in the notation of the lemma) of log(x/y) by the map U given by (f , f ). We can now complete the proof of proposition 5.10. Corollary 5.13 tells us that when the result is proved for one endomorphism , it is true for all of them. In p...