Finite polynomial cohomology for general varieties

Nekovář and Nizioł (Syntomic cohomology and p-adic regulators for varieties over p-adic fields, 2013) have introduced in a version of syntomic cohomology valid for arbitrary varieties over p-adic fields. This uses a mapping cone construction similar to the rigid syntomic cohomology of (Besser, Israel J Math 120(1):291–334, 2000) in the good-reduction case, but with Hyodo–Kato (log-crystalline) cohomology in place of rigid cohomology. In this short note, we describe a cohomology theory which is a modification of the theory of Nekovář–Nizioł, modified by replacing 1-φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 - \varphi $$\end{document} with other polynomials in φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document}. This is the analogue for bad-reduction varieties of the finite-polynomial cohomology of (Besser, Invent Math 142(2):397–434, 2000); and we use this cohomology theory to give formulae for p-adic regulator maps, extending the results of (Besser, Invent Math 142(2):397–434, 2000; Besser, Israel J Math 120(1):335–360, 2000; Besser, Israel J Math 190(1):29–66, 2012) to varieties over p-adic fields, without assuming any good reduction hypotheses.


Preliminaries from p-adic Hodge theory 1.Filtered (ϕ, N)-modules and their cohomology
We recall some standard constructions for (φ, N )-modules, following §2.4 of [10]. Let K be a p-adic field (i.e. the field of fractions of a complete DVR V of mixed characteristic (0, p), with perfect residue field k). Definition 1.1. 1 We define a filtered (ϕ, N , G K )-module over K to be the data of a finitedimensional K nr 0 -vector space D, where K 0 is the maximal unramified subfield of K and K nr 0 its maximal unramified extension, equipped with the following structures: • a K nr 0 -semilinear, bijective Frobenius ϕ; • a K nr 0 -linear monodromy operator N satisfying N ϕ = pϕ N ; • a K nr 0 -semilinear action of G K ; • and a decreasing, separated, exhaustive filtration of the K -vector space D K = (D ⊗ K nr 0 K ) G K by K -vector subspaces Fil i D K .
These are the objects of a pre-abelian category MF (φ,N ,G K ) . (It is not an abelian category, since morphisms may not be strictly compatible with the filtration.)

Variants
We now construct a variant of the complex C st (D) in which the semilinear Frobenius is replaced by a "partially linearized" one, and 1 − ϕ by a more general polynomial.
We choose a finite extension L/Q p contained in K , and we write f = [L 0 : Q p ] and q = p f . We can then define D st,L = D st ⊗ L 0 L, which we equip with an L-linear operator given by extending scalars from the L 0 -linear operator ϕ f on D st .
with the maps given by u, v → (P( )u, N u, u − v) and (w, x, y) → N w − P(q )x.
for the cohomology of the complex C • st,L ,P (D).

Remark 1.2.2
In applications of the theory, we are almost always interested in the case when K is finite over Q p and L = K ; but in order to set up the theory we need a base-change compatibility which seems to be easier to prove for varying K but fixed L, which is why we have set up the theory for L = K .
We have natural compatibilities when we change the polynomial P or the field L: Note that the map of Definition 1.2.4 is an isomorphism if L /L is an unramified extension.
Remark 1.2.5 One can define, in the obvious fashion, a pre-abelian category MF ( ,N ,G K ,L) of "filtered ( , N , G K )-modules with coefficients in L", whose objects are finite-dimensional L K nr 0 -vector spaces D L with an L-linear q-th power Frobenius , a monodromy operator N , a G K -action, and a filtration on There is a natural extensionof-coefficients functor and the functors C • st,L ,P (−) naturally factor through this. For each polynomial P ∈ 1 + T L[T ], one can define an object 1 P of MF ( ,N ,G K ,L) by taking 1 P = L[X ]/P(X ), with acting as multiplication by X (and N and G K acting trivially, and filtration concentrated in degree 0). Then the complex C • st,L ,P (D) computes the Ext groups Ext i (1 P , D) in this category, and the change-of-P map of Definition 1.2.3 is then obtained by contravariant functoriality from the obvious map 1 P Q → 1 P . Note, however, that 1 P is not weakly admissible in general.  Proof This is a straightforward exercise in homological algebra. Let λ ∈ K and choose polynomials a(T 1 , T 2 ) and b(T 1 , T 2 ) such that a(T 1 , T 2 )P 1 (T 1 ) + b(T 1 , T 2 )P 2 (T 2 ) = (P 1 P 2 )(T 1 T 2 ). Then the cup-products in the various degrees are given in Table 1. One verifies easily that changing the value of λ, or the polynomials a and b, changes the product by a chain homotopy.

Convenient modules
Definition 1.4.1 Let D be a filtered (ϕ, N , G K )-module. We say D is convenient (for some choice of L and P) if D is crystalline and P( ) and P(q ) are bijective as endomorphisms of D st,L .
The inverse of this isomorphism is given by where ι is the natural inclusion D st,L → D K ; note that we must have N w = P(q )x since (w, x, y) is a cocycle, but by assumption N = 0 and P(q ) is bijective, so in fact we have x = 0. Note that this map commutes with the change-of-P and change-of-L maps (where defined).
The following special case will be of importance below: Proposition 1.4.2 If P(1) = 0 and P(q −1 ) = 0, then there is an isomorphism The key to our description of syntomic regulators is the following. We now take L = K .

Proposition 1.4.3 Suppose that D is convenient
where the right-hand vertical map is the cup-product of the previous section.
Remark 1.4.4 The utility of this proposition is that if D is convenient, it allows us to completely describe an element of H 1 st,L ,P (D) via its cup-products with elements of H 0 st,L ,Q (D * (1)) for suitable polynomials Q. This is the key to our computations of regulator maps below. Note that even if P is some very simple polynomial (such as 1 − T ), we still need to be able to choose Q freely for this to work.

Finite polynomial cohomology for general varieties 2.1 Summary of the theory of Nekovář-Nizioł
We briefly summarize some of the main results of the paper [10]. Let Var(K ) be the category of varieties over K (i.e. reduced separated K -schemes of finite type).
Theorem 2.1.1 There exists a functor X → R syn (X h , * ) from Var(K ) to the category of graded-commutative differential graded E ∞ -algebras over Q p , equipped with period morphisms The cohomology theory X → R syn (X h , r ) has pushforward maps for projective morphisms, and has a functorial map from Voevodsky's motivic cohomology, compatible with the étale realization map via ρ syn .
For X a variety over K , let H j HK (X h ) and H j dR (X h ) be the extensions of Hyodo-Kato and de Rham cohomologies defined by Beilinson [1], and ι B dR : The main result of [1] shows that we have where K varies over finite extensions of K , and that there is a canonical isomorphism of (ϕ, N , Theorem 2.1.2 There is a "syntomic descent spectral sequence" , compatible with the period morphism ρ syn on the abutment.

Definition of P-syntomic cohomology
We now develop a theory which very closely imitates that of [10], but modified to use general polynomials in Frobenius in the place of 1 − ϕ, and replacing the p-power Frobenius ϕ with a "partially linearized" Frobenius. We choose a finite extension L/Q p is a finite extension contained in K , with residue class degree f = [L 0 : Q p ], and P ∈ 1 Let (U, U ) be an "arithmetic pair" (in the sense of op.cit.), log-smooth over V × (i.e. Spec(V ) with the log structure associated to the closed point).
Let R cr (U, U ) Q be the rational crystalline cohomology (defined as . This is a complex of K 0 -vector spaces, hence of L 0 -vector spaces, and we write R cr (U, There is a K 0 -semilinear Frobenius ϕ on R cr (U, U ) Q , and we let = ϕ f , which is L 0 -linear and thus extends to an L-linear operator on R cr (U, U ) L . For r ≥ 0 we write r = q −r = ( p −r ϕ) f .

Remark 2.2.1
Curiously, the complex R cr (U, U ) Q is already a complex of K -vector spaces, but this K -linear structure interacts very badly with the Frobenius, and passing to Frobenius eigenspaces "kills off" all contributions from K \K 0 . So we must "add the K -linear structure a second time" in order to obtain a K -linearized theory of syntomic cohomology.
For any r ≥ 0, the quasi-isomorphism γ −1 Note that the use P( r ), rather than P( ); this choice gives somewhat cleaner formulations of some results (e.g. the pushforward maps and the syntomic descent spectral sequence), but has the disadvantage of introducing a notational discrepancy between R syn,L ,P (U, U , r ) and the finite-polynomial cohomology of [4].
The complexes R syn,L ,P (U, U , r ) are complexes of L-vector spaces, functorial in pairs (U, U ), equipped with L-linear cup-products which are associative and graded-commutative, up to coherent homotopy (i.e. the direct sum is an E ∞ -algebra over L). These cup-products are given explicitly by the same recipe as in 1.3.2 above.
The following properties are immediate:  Proof (1) and (2) are obvious by construction. For (3), note that we have r = ( p −r ϕ) f and r = ( p −r ϕ) f , so P( r ) = P ( r ). Hence we have a commutative diagram where the vertical maps are the natural maps induced from the inclusion L ⊂ L .
If L /L is unramified, so L = L L 0 , then we have a canonical isomorphism of complexes of L 0 -vector spaces so the vertical maps are isomorphisms.
We will be particularly interested in a special case of this:

functorial in (U, U ) and compatible with cup-products.
Proof This is built up by combining all three parts of the above proposition. Firstly, (1) identifies Nekovář-Nizioł's cohomology R syn (U, U , r ) Q with the special case P(T ) = 1 − T, L = Q p of our construction. If P is any polynomial with P(1) = 0, then P(T f ) is divisible by 1 − T , so (2) gives a morphism Finally (3) gives a morphism R syn,Q p ,P(T f ) (U, U , r ) → R syn,L ,P (U, U , r ).

All of these are visibly functorial in pairs (U, U ).
We now show a relation analogous to Proposition 3.7 of op.cit.. Suppose that (U, U ) is of Cartier type, so Hyodo-Kato cohomology is defined. We use Beilinson's variant of Hyodo-Kato cohomology, which has the advantage of having a comparison map ι B dR : where q = p f .

Remark 2.2.7
This map does actually depend on the choice of a uniformizer π, although its source and target are independent of any such choice.
Proof We follow exactly the same argument as the proof given in op.cit.. It suffices to check that P( ) is invertible on I ⊗ W (k) H i HK (U, U ) L , for any P ∈ 1 + T L[T ], where I is the ideal in a divided power series ring over W (k) considered in op.cit.. We note that since P is monic the formal Laurent series 1/P(T ) = n≥0 a n T n has positive radius of convergence, so there is some A such that ord p (a n ) ≥ −n A for n 0. This implies the convergence of the series a n n on I ⊗ W (k) H i HK (U, U ) L , which gives an inverse of P( ). Continuing to follow [10], we have Proof This follows by exactly the same proof as in op.cit.: the proof proceeds by constructing compatible maps between the various variants of crystalline, de Rham, and Hyodo-Kato cohomology, and these all remain compatible after extending scalars to L. Remark 2.2. 9 We are principally interested in the case L = K , but it seems to be easier to prove base-change compatibility in K for a fixed L. It seems eminently natural that if K /K is totally ramified and both fields are finite over Q p then we should get a quasi-isomorphism but this does not seem to be so easy to prove, and it is not needed for the calculations below.

h-sheafification
We now sheafify in the h-topology. We write S L ,P (r ) for the sheafification of (U, U ) → R syn,L ,P (U, U , r ), and we define R syn,L ,P (X h , r ) = R (X h , S L ,P (r )).

Proposition 2.3.1 For any arithmetic pair (U, U ) that is fine, log-smooth over V × and of Cartier type, the canonical maps
Proof This is exactly the generalization to our setting of Proposition 3.16 of op.cit.. The long and highly technical proof fortunately carries over verbatim to general P.

Proposition 2.3.2
There is a "P-syntomic descent spectral sequence" the (ϕ, N , G  Proof For the existence of the spectral sequence, see Proposition 3.17 of op.cit.. The compatibility with cup-products is immediate from the definition of the cup-product on P-syntomic cohomology (see Sect. 2.4 below for explicit formulae).
Note that the spectral sequence implies that the cohomology groups H i syn,L ,P (X h , r ) are zero if i > 2 dim X + 2, and if K /Q p is a finite extension, then the cohomology groups are finite-dimensional L-vector spaces (with dimension bounded independently of P). Moreover, the spectral sequence obviously degenerates at E 3 , and this gives a 3-step filtration on the groups H i syn,L ,P (X h , r ): 3 We write Fil m H i syn,L ,P (X h , r ) for the 3-step decreasing filtration on H i syn,L ,P (X h , r ) induced by the P-syntomic spectral sequence. Concretely, we have ,

Remark 2.3.4
It seems natural to conjecture that the "knight's move" maps should be zero for smooth proper X , extending the conjecture for syntomic cohomology formulated in Remark 4.10 of [10]. We do not know if this conjecture holds in general, but the applications below will all concern cases where either the source or the target of this map is zero.

Explicit cup product formulae
We now use Proposition 2.2.6 to give an explicit formula for the cup product which generalizes the description of the cup-product on the complexes C • st,L ,P (D) given in the previous section.
A class η ∈ H i syn,P (X h , r ) is represented by the following data: which satisfy the relations Let η = [u , v ; w , x , y ; z ] be a corresponding representation of η ∈ H j syn,L ,Q (X h , s). We want to find explicit formulae for the class η = η ∪ η .
As before, fix polynomials a(t, s) and b(t, s) such that

Proposition 2.4.1 The cup-product is given by
Here, we write e.g. a( , Proof This is simply a translation of the formulae of Sect. 1.3 into the setting of complexes.

Pushforward maps and compact supports
We now extend to our setup the constructions of Déglise's Appendix B to [10]. Cup-product gives us maps S L (1)⊗S L ,P (r ) → S L ,P (r +1), where S L (1) is defined using the polynomial 1 − T ; this gives the direct sum S L ,P the structure of a Tate -spectrum.
The same argument as in op.cit. now gives pushforward maps (and the projection formula for these holds by construction). It is clear that this argument is compatible with change of L and of P.
The same argument also gives compactly-supported cohomology complexes: Theorem 2.5.2 There exist compactly-supported cohomology complexes R syn,L ,P,c (X h , r ), contravariantly functorial with respect to proper morphisms and covariantly functorial ( with a degree shift) with respect to smooth morphisms; and there is a functorial morphism which is an isomorphism for proper X . The compactly-supported cohomology has a descent spectral sequence Finally, we have a projection formula: and for f a smooth proper morphism, we have the projection formula Proof This follows from the construction of the pushforward map, cf. [9].

Application to computation of regulators
We'll now apply the constructions above to give formulae describing the syntomic regulator map H i mot (X, j) → H i syn (X h , j), where X is a product of copies of an affine curve Y . More specifically, we will give formulae in the following cases: • X = Y is a single curve and we are given a class in H 2 mot (X, 2) that is the cup product of two units on Y ; • X = Y 2 is the product of two curves, and we are given a class in H 3 mot (X, 2) that is the pushforward of a unit along the diagonal embedding Y → Y 2 ; • X = Y 3 is the product of three curves and we consider the class in H 4 mot (X, 2) given by the cycle class of the diagonal Y → Y 3 .
In the case where Y has a smooth model over Z p these regulators have been described using finite-polynomial cohomology (in [4,6,7] respectively); and these have been applied in proving explicit reciprocity laws for Euler systems (in [2,3,8] respectively). We use the generalization of finite-polynomial cohomology to arbitrary varieties described in the preceding section to extend this description to the case of arbitrary smooth curves over K . However, our results are less complete, in that we only obtain a full description of the image of the regulator in a "convenient" quotient of the cohomology of X in the sense of Sect. 1.4. (In other words, we allow X to have bad reduction, but we project to a quotient of its cohomology which looks like the cohomology of a variety with good reduction.) Definition 3.2.2 Under the above assumptions, we define the triple symbol [η; ω 1 , ω 2 ] ∈ K by the formula [η; ω 1 , ω 2 ] = tr Y,syn,P 0 P 1 P 2 (η ∪ω 1 ∪ω 2 ) whereη ∈ H 1 syn,K ,P 0 ,c (Y K ,h , 0) andω i ∈ H 1 syn,K ,P i (Y K ,h , 1) are liftings of η and the ω i . Proposition 3.2.3 The above quantity is independent of the choice of liftings and of the polynomials P i .
Proof Firstly, we note that the natural map H 1 syn,K ,P 0 ,c (Y K ,h , 0) → H 1 dR,c (Y /K ) is an isomorphism, since the degree 0 compactly-supported cohomology is zero. Thus the classη is uniquely defined.
Ifω 1 is a class in H 1 syn,K ,P 1 lifting ω 1 , and [u, v; w, x, y; z] is a representative ofω 1 in R 1 syn,K ,P 1 (with z = 0, necessarily, for degree reasons), then any other choice of lifting can be represented by [u, v; w + λ, x + μ, y; z] for some constants λ, μ ∈ K (with μ = 0 unless P 1 (1) = 0). From the definition of the cup-product, one sees that varying μ has no effect onη ∪ω 1 ∪ω 2 ; while varying λ changes the cup product by a multiple of η ∪ ω 2 , which is zero by assumption. Similarly, the assumption that η ∪ ω 1 = 0 implies that the cup-product is independent of the choice of lifting of ω 2 .
So [η; ω 1 , ω 2 ] is well-defined for a fixed choice of polynomials P i . However, both the cup-product and the map tr Y,syn,P 0 P 1 P 2 are compatible with change of the polynomials P i , so the symbol [η; ω 1 , ω 2 ] is also independent of these choices.

Remark 3.2.4
We may also carry out the same construction if Y is projective, rather than affine, if we add the assumption that P 0 (1) = 0 and P 0 (q) = 0; this assumption implies that there is still a unique lifting of η to H 1 syn,K ,P 0 ,c (Y K ,h , 0) = H 1 syn,K ,P 0 (Y K ,h , 0). In particular, this holds if P 0 is pure of weight 1.

Products of curves
We now use the theory of the previous section to reformulate special cases of Propositions 3.1.4-3.1.6 in terms of the triple symbol. We continue to assume that Y is a connected smooth affine curve over K .   Let z ∈ H 2 mot (X, 2) be the cup-product of the classes u, v ∈ H 1 mot (Y, 1) ∼ = O(Y ) × . Then λ η (r syn (z)) = [η; dlog u, dlog v].

An alternative description of the triple symbol
We conclude by giving an alternative, equivalent description of the symbol [η; ω 1 , ω 2 ], which we hope may be useful in relating our results to p-adic modular forms (as in the calculations of [8] in the good-reduction case).

Then we have
Proof From the definition of the cup product, we see that it respects the 3-step filtration on P-syntomic cohomology, and the cup-products induced on the graded pieces coincide with the usual de Rham cup products. Sinceω 1 ∪ω 2 lies in Fil 1 , and the trace isomorphism factors through Fil 1 / Fil 2 , we obtain the above compatibility using (2).