## A generalization of Coleman’s $$p$$-adic integration theory.(English)Zbl 1053.14020

The autor introduces a new cohomology theory $$H_{fp}$$, called finite polynomial cohomology, for smooth schemes over a $$p$$-adic ring so that syntomic cohomology $$H_{syn}$$ is embedded functorially into $$H_{fp}$$. For a proper scheme, this new cohomology has Poincaré duality and hence Gysin maps and cycle class maps. For zero-cycles, the cycle class map is shown to be given by Coleman’s $$p$$-adic integration, and $$H_{fp}$$ is therefore interpreted as a generalization of Coleman’s integration theory to higher-dimensional cycles. The main result is an explicit description of the syntomic Abel-Jacobi map in terms of this generaized Coleman integration.

### MSC:

 14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.)

### Keywords:

syntomic Abel-Jacobi map; finite polynomial cohomology