On p-adic absolute Hodge cohomology and syntomic coefficients. I

We interpret syntomic cohomology defined in [50] as a p-adic absolute Hodge cohomology. This is analogous to the interpretation of Deligne-Beilinson cohomology as an absolute Hodge cohomology by Beilinson [8] and generalizes the results of Bannai [6] and Chiarellotto, Ciccioni, Mazzari [15] in the good reduction case. This interpretation yields a simple construction of the syntomic descent spectral sequence and its degeneration for projective and smooth varieties. We introduce syntomic coefficients and show that in dimension zero they form a full triangulated subcategory of the derived category of potentially semistable Galois representations. Along the way, we obtain p-adic realizations of mixed motives including p-adic comparison isomorphisms. We apply this to the motivic fundamental group generalizing results of Olsson and Vologodsky [56, 71].