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

Abstract

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].