The Exponential Homomorphism for the Second Syntomic Cohomology

Let OK be a complete discrete valuation ring with perfect residue field F of characteristic p ≥ 3 and M a free filtered Dieudonné module. The second relative syntomic cohomology H2((OK, F), (M, 2)) with coefficients in M is an object related to arithmetic geometry, such as the albanese kernel. In this article, we study H2((OK, F), (M, 2)) by constructing the exponential homomorphism. We determine the structure of H2((OK, F), (M, 2)), under the assumption that M is of Hodge–Witt type and that the absolute ramification index of OK is prime to p.