Syntomic cohomology as a p -adic absolute Hodge cohomology

Abstract

The purpose of this paper is to interpret rigid syntomic cohomology, defined by Amnon Besser [Bes], as a p-adic absolute Hodge cohomology. This is a p-adic analogue of a work of Beilinson [Be1] which interprets Beilinson-Deligne cohomology in terms of absolute Hodge cohomology. In the process, we will define a theory of p-adic absolute Hodge cohomology with coefficients, which may be interpreted as a generalization of rigid syntomic cohomology to the case with coefficients. (2000):14F30, 14G20 Mathematics Subject Classification IntroductionIn the paper "Notes on absolute Hodge cohomology" [Be1], Beilinson gave an interpretation of Beilinson-Deligne cohomology as an absolute Hodge cohomology; in other words, as extension groups in the derived category of the category of mixed Hodge structures. It is widely believed that the p-adic analogue of Beilinson-Deligne cohomology is syntomic cohomology. The purpose of this paper is to give the p-adic analogue of the result of Beilinson mentioned above. Namely, we will show that rigid syntomic cohomology, first defined by M. Gros [G] then fully developed by Amnon Besser [Bes], is expressed as extension groups in the derived category of weakly admissible filtered Frobenius modules MF f K . In the process, we will define a theory of p-adic absolute Hodge cohomology with coefficients. This may be interpreted as a generalization to the case with coefficients of rigid syntomic cohomology. This is also a generalization of absolute syntomic cohomology with coefficients defined in our previous paper [Ba1] to the case when the base field is ramified over Q p . We remark that a cohomology theory with coefficients for proper and smooth 444 K. Bannai schemes has been considered by W. Nizioĺ [Ni]. The philosophy of this paper played an important role in the papers [Ba1] [Ba2].Let K be a finite extension of Q p with ring of integers O K and residue field k. We let K 0 be the maximum unramified extension of Q p in K and W its ring of integers. Let σ be the lifting to K 0 of the Frobenius automorphism on k.The Philosophy. For a scheme X smooth and of finite type over O K , there should exist a Tannakian category MHM p (X) of the p-adic mixed Hodge modules. This category should be the p-adic analogue of the category of mixed Hodge modules defined by Morihiko Saito [Sa]. In particular, on the derived category D b (X) of MHM p (X), we should have the formalism of six Grothendieck functors.Although the category MHM p (X) still has not been constructed, in the special case X = Spec O K , the category MHM p (Spec O K ), which we denote by MHM p (O K ), should be the category MF f K of weakly admissible filtered modules defined by Fontaine ([Fon1] 4.1.4), consisting of objects (M 0 , ϕ, F • ) where (i) M 0 is a finite dimensional K 0 -vector space with a σ-linear isomorphism ϕ : M 0 → M 0 , which we call the Frobenius automorphism. (ii) F • is a descending exhaustive separated filtration on M = M 0 ⊗ K, which we call the Hodge filtration. (iii) M with the above structure is weakly admissible.

Syntomic cohomology and p-adic regulators for varieties over p-adic fields

“…The guiding idea here was that just as Selmer groups classify extensions in certain categories of "geometric" Galois representations their higher dimensional analogs -syntomic cohomology groups -should classify extensions in a category of "p-adic motivic sheaves". This was shown to be the case for H 1 by Bannai [1] who has also shown that Besser's (rigid) syntomic cohomology is a p-adic analog of Beilinson's absolute Hodge cohomology [2].…”

“…Moreover, S is a Tate Ω-spectrum because each S (r) is h-local and A 1 -local, and the map obtained by adjunction from σ r is a quasi-isomorphism because of the projective bundle theorem for P 1 (an easy case of Proposition 5.2). Now, by definition of DM h (K, Q p ) and because of property (DM1) above, for any variety X, and any integers (i, r), we get: For points (1) and (2) Remark B.5. Note that the construction of the syntomic ring spectrum S in DM h (K, Q p ) automatically yields the general projective bundle theorem (already obtained in Prop.…”

“…These quasi-isomorphisms h-sheafify well: for a variety Xover K, they induce the following quasiisomorphisms [4] We obtain syntomic complexes S (r) on X K,h by h-sheafifying the complexes Y → RΓ syn (Y, r) and (geometric) syntomic cohomology by setting RΓ syn (X K,h , r) := RΓ(X K,h , S (r)). They fit into an analog of the exact sequence (1) and, by the above, generalize log-syntomic cohomology.…”

“…Unless otherwise stated, we work in the (stable) ∞-category D(A), i.e., stable ∞-category whose objects are (left-bounded) chain complexes of projective objects of A. For a readable introduction to such categories the reader may consult [37], [45,1]. The ∞-derived category is essential to us for two reasons: first, it allows us to work simply with the Beilinson-Hyodo-Kato complexes; second, it supplies functorial homotopy limits.…”

“…It was generalized to log-syntomic cohomology for semistable varieties by Kato [41]. For a log-smooth scheme X over a complete discrete valuation ring V of mixed characteristic (0, p) and a perfect residue field, and for any r ≥ 0, rational log-syntomic cohomology of X can be defined as the "filtered Frobenius eigenspace" in log-crystalline cohomology, i.e., as the following mapping fiber (1) RΓ syn (X , r) := Cone RΓ cr (X , J where RΓ cr (·, J [r] ) denotes the absolute rational log-crystalline cohomology (i.e., over Z p ) of the r'th Hodge filtration sheaf J [r] and ϕ r is the crystalline Frobenius divided by p r . This definition suggested that the log-syntomic cohomology could be the sought for p-adic analog of Deligne-Beilinson cohomology.…”