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 PoincarĂ© 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.


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.)