Want to define syntomic in terms of generic fiber. Use spectra, either via six functors formalism, or more specifically via gluing??? It might be the case that syntomic cohomology is the absolute theory associated to the geometric theory which is rigid cohomology. For the latter, I think generic fiber theorems are known, based on vague impressions from Illusie in Motives 1 and Le Stum. Is there a way we can express the absolute/geometric dichotomy in terms of motivic stable homotopy theory, and somehow conclude that syntomic cohomology depends only on the generic fiber?? For example, maybe there is Hochschild-Serre spectral sequence from rigid cohomology.
Bannai: 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.
Kato and Messing: Syntomic cohomology and -adic étale cohomology. Tohoku 1992.
MR1703301 (2000g:11062) Somekawa, Mutsuro(J-TOIT) Log-syntomic regulators and -adic polylogarithms. (English summary) -Theory 17 (1999), no. 3, 265–294. (Uses, and perhaps defines, syntomic cohomology for simplicial schemes)
arXiv: Experimental full text search
AG (Algebraic geometry)
Charp
There should be a paper by Nekovar with a useful appendix, see http://www.math.uiuc.edu/K-theory/0212
A useful thesis
Look also at the thesis of Hannu
See Galois Cohomology of Fontaine rings, in Fontaine theory folder
Things by Besser, see his webpage.
Articles on syntomic regulators:
MR1031903 (91e:11070) Gros, Michel(F-RENNB-isomorphism) Régulateurs syntomiques et valeurs de fonctions -adiques. I. (French) [Syntomic regulators and values of -adic -functions. I] With an appendix by Masato Kurihara. Invent. Math. 99 (1990), no. 2, 293–320.
Also part II by Gros, in Inventiones.
MR1632798 (99f:11151) Kolster, Manfred(3-MMAS); Nguyen Quang Do, Thong(F-FRAN-M) Syntomic regulators and special values of -adic -functions. (English summary)
MR1728549 (2001d:11070) Besser, Amnon(IL-BGUN); Deninger, Christopher(D-MUNS-isomorphism) -adic Mahler measures. J. Reine Angew. Math. 517 (1999), 19–50.
MR1909217 (2003e:11070) Bannai, Kenichi(J-TOKYOGM) On the -adic realization of elliptic polylogarithms for CM-elliptic curves. (English summary) Duke Math. J. 113 (2002), no. 2, 193–236.
Geisser: Over a discrete valuation ring of mixed characteristic (0,p), we construct a map from motivic cohomology to syntomic cohomology, which is a quasi-isomorphism provided the Bloch-Kato conjecture holds.
arXiv:1109.1635 A question on image of syntomic regulator on CH^2(X,1) from arXiv Front: math.AG by Masanori Asakura We give explicit examples on image of syntomic regulator on CH^2(X,1). Taking account of the results, we propose a new question.
The syntomic regulator for K-theory of fields, by Amnon Besser and Rob de Jeu: http://www.math.uiuc.edu/K-theory/0523
arXiv:0910.4436 Cohomologie syntomique: liens avec les cohomologies étale et rigide from arXiv Front: math.AG by Jean-Yves Etesse Syntomic cohomology here defined yields a link between rigid cohomology and etale cohomology, viewing the last one as the fixed points under Frobenius of the former one. Let V be a complete discrete valuation ring, with perfect residue field k = V/m of characteristic p > 0 and fraction field K of characteristic 0. Having defined syntomic cohomology with compact supports of an abelian sheaf G on a k-scheme X, we show that it coincides with etale cohomology with compact supports when G is a lisse sheaf. If moreover the convergent F-isocrystal associated to G comes from an overconvergent isocrystal E, then the rigid cohomology of E expresses as a limit of syntomic cohomologies: then the etale cohomology with compact supports of G is the fixed points of Frobenius acting on the rigid cohomology of E.
arXiv:1003.2810 Cyclotomic complexes from arXiv Front: math.AT by D. Kaledin We construct a triangulated category of cyclotomic complexes, a homological counterpart of cyclotomic spectra of Bokstedt and Madsen. We also construct a version of the Topological Cyclic Homology functor TC for cyclotomic complexes, and an equivariant homology functor from cycloctomic spectra to cyclotomic complexes which commutes with TC. Then on the other hand, we prove that the category of cyclotomic complexes is essentially a twisted 2-periodic derived category of the category of filtered Dieudonne modules of Fontaine and Lafaille. We also show that under some mild conditions, the functor TC on cyclotomic complexes is the syntomic cohomology functor.
In http://www.ams.org/mathscinet-getitem?mr=2275605 (review of Niziol ICM talk) it says that syntomic cohomology has a motivic description, due to work of Geisser, and assuming the BK conjecture.
From Hannu’s thesis, section 2.2.4: “Syntomic cohomology is an “arithmetic” version of crystalline cohomology, whose purpose in p-adic Hodge theory is to bridge the gap between etale and crystalline cohomology. It was originally defined by Fontaine and Messing in [FM87] using “syntomic topology”, but soon thereafter an easier approach, based on “syntomic complexes”, was found by Kato [Kat87]. Our definition, taken from [Tsu00], follows this approach.”
arXiv:1211.5065 The rigid syntomic ring spectrum from arXiv Front: math.NT by Frédéric Déglise, Nicola Mazzari The aim of this paper is to show that Besser syntomic cohomology is representable by a rational ring spectrum in the motivic homotopical sense. In fact, extending previous constructions, we exhibit a simple representability criterion and we apply it to several cohomologies in order to get our central result. This theorem gives new results for syntomic cohomology such as h-descent and the compatibility of cycle classes with Gysin morphisms. Along the way, we prove that motivic ring spectra induces a complete Bloch-Ogus cohomological formalism and even more. Finally, following a general motivic homotopical philosophy, we exhibit a natural notion of syntomic coefficients.
nLab page on Syntomic cohomology