The main objective of this paper is to give a new construction of syntomic cohomology
for smooth schemes over the ring of integers of a *p*-adic
field and to construct regulators from K-theory into this cohomology. Our
construction is better behaved than previous constructions: the resulting cohmology
is always finite dimensional, and it maps to most other constructions.

We also define a new cohmology theory, "modified syntomic cohomology" which is better behaved in explicit computations yet is isomorphic to syntomic cohomology in most cases of interest.

The techniques introduced in this paper form the basis for
A generalization of Coleman's
*p*-adic integration theory