Finite polynomial cohomology for general varieties

Nekovar and Niziol have introduced in [arxiv:1309.7620] 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 the first author 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 Nekovar and Niziol, modified by replacing 1 - Phi (where Phi is the Frobenius map) with other polynomials in Phi. This is the analogue for general varieties of the finite-polynomial cohomology defined by the first author for varieties with good reduction. We use this cohomology theory to give formulae for p-adic regulator maps on curves or products of curves, without imposing any good reduction hypotheses.

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).
We define a filtered (ϕ, N, G K )-module over K to be the data of a finite-dimensional 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 = pϕN ; a K nr 0 -semilinear action of G K ; and a decreasing filtration of the K-vector space D K = (D ⊗ K nr 0 K) GK by K-vector subspaces Fil i D K . Such objects form an abelian category in the obvious way (with morphisms required to be strictly compatible with the filtration on D K ).
Theorem 1.1 (Colmez-Fontaine). The subcategory of "weakly admissible" filtered (ϕ, N )-modules is equivalent to the category of semistable Q p -linear representations of G K , via the functor V → D pst (V ) = Remark 1.3. 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.

Cup products.
Definition 1.4 (cf. [Bes00a,Definition 4.1]). If P, Q ∈ 1 + T L[T ], then we define P ⋆ Q ∈ 1 + T L[T ] as the polynomial with roots {α i β j }, where {α i } and {β j } are the roots of P and Q respectively. 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 cupproducts in the various degrees are given by the following table: One verifies easily that changing the value of λ, or the polynomials a and b, changes the product by a chain homotopy.
1.4. Convenient modules. Definition 1.6. Let D be a filtered (ϕ, N, G K )-module. We say D is convenient (for some choice of L and P ) if N = 0 and P (Φ) and P (qΦ) are bijective as endomorphisms of D st,L .
Note that D is convenient if and only if D * (1) is convenient. If D is convenient, then the inclusion 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 (qΦ)x since (w, x, y) is a cocycle, but by assumption N = 0 and P (qΦ) is bijective, so in fact we have x = 0. Note that this map commutes with change of P and of L (where defined). The following special case will be of importance below: Proposition 1.7. 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 1.8. Suppose that D is crystalline (i.e. N = 0 on D st and G K acts trivially), and that D is convenient. Then for any λ ∈ Fil 0 D * (1) K = (D K / Fil 0 ) * , there is a polynomial Q such that λ ∈ H 0 st,K,P (D * (1)) and (P ⋆ Q)(1) = 0, (P ⋆ Q)(q −1 ) = 0; and if P is such a polynomial, then we have a commutative diagram where the right-hand vertical map is the cup-product of the previous section.
Remark 1.9. If we take λ = 1 in the formulae for the cup-product pairing, we see that the cup product C 1 st,K,P × C 0 st,K,P → C 1 st,K,P restricted to the direct summands D K ⊆ C 1 st,K,P and Fil 0 D K ⊆ C 0 st,K,P is just the natural tensor product.
Remark 1.10. The utility of this remark 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. Note that even if P is some very simple specific polynomial such as 1 − T , we still need to be able to choose Q freely for this to work.
2. Finite polynomial cohomology for general varieties 2.1. Summary of the theory of Nekovář-Nizio l. We briefly summarize some of the main results of the paper [NN13]. Let Var(K) be the category of varieties over K (i.e. reduced separated K-schemes of finite type).
The cohomology theory X → RΓ syn (X h , r) has pushforward maps for projective morphisms, and has a functorial map from Voevodsky's motivic cohomology, compatible with theétale realization map via ρ syn .
Theorem 2.2. There is a "syntomic descent spectral sequence" ) from Theorem 1.2 assemble into a morphism from the syntomic spectral sequence to the Hochschild-Serre spectral sequence H i (K, H j et (X K,ét , Q p (r))) ⇒ H i+j et (X K,ét , Q p (r)), compatible with the period morphism ρ syn on the abutment.

2.2.
Definition of P -syntomic cohomology. We now develop a theory which very closely imitates that of [NN13], 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 + T L[T ].
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 RΓ 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 There is a K 0 -semilinear Frobenius ϕ on RΓ cr (U, U ) Q , and we let Φ = ϕ f , which is L 0 -linear and thus extends to an L-linear operator on RΓ cr (U, Remark 2.3. Curiously, the complex RΓ 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 Definition 2.4. We define where the brackets signify mapping fibre, in the ∞-derived category of abelian groups).
Remark 2.5. Note that the use P (Φ r ), rather than P (Φ); 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 RΓ syn,L,P (U, U , r) and the finite-polynomial cohomology of [Bes00a].
The complexes RΓ 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 r,P RΓ syn,L,P (U, U , r) is an E ∞ -algebra over L).
Remark 2.6. Here P ⋆ Q is the convolution of the polynomials P and Q. We give an explicit formula for the cup product in Proposition 2.17 below.
The following properties are immediate: Proposition 2.7.
(2) (Change of P ) If P, Q are two polynomials, there is a map RΓ syn,L,P (U, U , r) Q → RΓ syn,L,P Q (U, U , r) Q functorial in (U, U ) and compatible with cup-products, which is given by the diagram (3) (Change of L) Let L ′ /L be a finite extension, and suppose that we have functorial in (U, U ) and compatible with cup-products. If L ′ /L is an unramified extension, this is an isomorphism. Proof.
where the vertical maps are the natural maps induced from the inclusion L ⊂ L ′ .
the vertical maps are isomorphisms.
We will be particularly interested in a special case of this: Proposition 2.8. If P (1) = 0 then there is a morphism of complexes RΓ syn (U, U , r) Q → RΓ syn,L,P (U, U , r), 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-Nizio l's cohomology RΓ 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 RΓ syn,Qp,1−T (U, U , r) → RΓ syn,Qp,P (T f ) (U, U , r) Qp .
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 : RΓ B HK (U, U ) K ∼ ✲ RΓ dR (U, U K ) which is defined at the level of complexes and does not depend on making a choice of uniformizer of K.
Proposition 2.9. For each uniformizer π of V , there is a quasi-isomorphism RΓ syn,L,P (U, U , r) Remark 2.10. 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 (Φ) 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 ) ≥ −nA for n ≫ 0. This implies the convergence of the series a n Φ n on I ⊗ W (k) H i HK (U, U ) L , which gives an inverse of P (Φ).
Continuing to follow [NN13], 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 2.12. We are principally interested in the case L = K, but it seems to be easier to prove basechange 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 RΓ syn,P (U, U , r) K ∼ ✲ RΓ(G, RΓ syn,P (T, T , r) K ′ ), but this does not seem to be so easy to prove, and it is not needed for the calculations below.
Proposition 2.13. For any arithmetic pair (U, U ) that is fine, log-smooth over V × and of Cartier type, the canonical maps RΓ syn,L,P (U, U , r) → RΓ syn,L,P (U h , r) are quasi-isomorphisms.
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 2.14. There is a "P -syntomic descent spectral sequence" where D j (r) = D pst (H j (X K,ét , Q p (r))) as above. This spectral sequence is compatible with extension of L (where defined) and change of P . Moreover, it is compatible with cup-products, where the cup-product on the terms E ij 2 is given by the construction of §1.3. 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 §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 Lvector 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): Definition 2.15. 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 2.16. It seems natural to conjecture that the "knight's move" maps H 0 st,L,P (D j (X h )(r)) → H 2 st,L,P (D j−1 (X h )(r)) should be zero for smooth proper X, extending the conjecture for syntomic cohomology formulated in Remark 4.10 of [NN13]. 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.
2.4. Explicit cup product formulae. We now use Proposition 2.9 to give an explicit formula for the cup product H i syn,L,P (X h , r) × H j syn,L,Q (X h , s) ✲ H i+j syn,L,P ⋆Q (X h , r + s), 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 du = 0, dv = 0, 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 ′ .
Proposition 2.17. The cup-product is given by Here, we write e.g.
where Φ 1 and Φ 2 are the q-power Frobenius maps of the two factors.
Proof. This is simply a translation of the formulae of §1.3 into the setting of complexes.
2.5. Pushforward maps and compact supports. We now extend to our setup the constructions of Déglise's Appendix B to [NN13]. Proof. Each of the sheaves S L,P (r) is h-local, by definition. They are also A 1 -local: this is virtually immediate from the corresponding result for the underlying Hyodo-Kato and de Rham cohomology theories, as in Prop 5.4 of op.cit.. The same methods also yield the projective bundle theorem.
Hence we can consider the direct sum of h-sheaves given by S L,P := r S L,P (r).
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 Ω-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 2.19. There exist compactly-supported cohomology complexes RΓ 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 RΓ syn,L,P,c (X h , r) → RΓ syn,L,P (X h , r) which is an isomorphism for proper X.
The compactly-supported cohomology has a descent spectral sequence Finally, we have a projection formula: Theorem 2.20. There are cup-products RΓ syn,L,P,c (X h , r) × RΓ syn,L,Q (X h , s) → RΓ syn,L,P ⋆Q,c (X h , r + s), and for f a smooth proper morphism, we have the projection formula Proof. This follows from the construction of the pushforward map, cf. [DM12].

Application to computation of regulators
We'll now apply the constructions above to give formulae describing the syntomic regulator map , 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 finitepolynomial cohomology (in [Bes00c], [Bes12], and [Bes00a] respectively); and these have been applied in proving explicit reciprocity laws for Euler systems (in [BD13,BDR12,DR14] 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 §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.) 3.1. Syntomic regulators. Now let us consider the following general setting: X is a smooth connected d-dimensional affine K-variety, and z ∈ H d+1 mot (X, j) for some j. We have an etale realization map ré t : H d+1 mot (X, j) ✲ H d+1 et (X K,ét , j). The Hochschild-Serre descent spectral sequence gives a map H d+1 (X K,ét , Q p (j)) → H 0 (K, H d+1 (X K,ét , Q p (j)); but the latter group is zero (because an affine variety of dimension d hasétale cohomological dimension d) and thus we obtain a map H d+1 (X K,ét , Q p (j)) → H 1 (K, H d (X K,ét , Q p (j))).
Theorem B of [NN13] shows that we have ré t = ρ syn • r syn , where where D d (X h ) = D pst H d (X K,ét , Q p ) , the vertical maps are induced by the syntomic and Hochschild-Serre descent spectral sequences, and the bottom horizontal map is the generalized Bloch-Kato exponential for the de Rham G K -representation H d (X K,ét , Q p (j)). Now let D be a quotient of D d (X h )(j) (in the category of weakly-admissible (ϕ, N, G K )-modules) which is crystalline and convenient, in the sense of §1.4 above. Then we have isomorphisms is the Tate dual of D, which is a submodule of D d c (X h )(d + 1 − j). As we saw above, for any class η ∈ Fil 0 D * (1) K , we may choose a polynomial P such that η ∈ H 0 st,K,P (D * (1)) and P (1) = 0, P (q −1 ) = 0; and the natural perfect pairing D K / Fil 0 × Fil 0 D * (1) K → K coincides with the cupproduct H 1 st,K (D) × H 0 st,K,P (D * (1)) → H 1 st,K,P (Q p (1)) ∼ = K. We now relate this to cup-products in P -syntomic cohomology.
Using the above formula together with the projection formula for cup-products (Theorem 2.20), we obtain the following consequences (in which j = 2 and 1 ≤ d ≤ 3). For η ∈ H d dR,c (X/K) satisfying the hypotheses of Theorem 4.2, we write λ η for the map H d+1 syn (X h , j) → K given by the composition of pr D and pairing with η.
Proposition 3.3. If d = 3 and z = i n i [Z i ] ∈ H 4 mot (X, 2) is the class of a codimension 2 cycle with smooth components, and η is as in Theorem 3.2, then we have λ η (r syn (z)) = i n i tr Zi,syn,K,P (ι * i (η)), where ι i is the inclusion of Z i into X.
Since the maps tr X,syn,K,P are compatible with pushforward (as is clear by comparison with their de Rham analogues), the result is now immediate from Theorem 2.20.
Proposition 3.4. If d = 2 and z = i (Z i , u i ) ∈ H 3 mot (X, 2), where Z i are codimension 1 cycles and u i ∈ O(Z i ) × , and η is as in Theorem 3.2, then we have λ η (r syn (z)) = i n i tr Zi,syn,K,P (r syn,K (u i ) ∪ ι * i (η)), where ι i is the inclusion of Z i into X. Here, r syn is the composition O(Z i ) × = H 1 mot (Z i , 1) ✲ H 1 syn (Z i , 1).