Relative assembly maps and the -theory of Hecke algebras in prime characteristic.

We investigate the relative assembly map from the family of finite subgroups to the family of virtually cyclic subgroups for the algebraic -theory of twisted group rings of a group with coefficients in a regular ring or, more generally, with coefficients in a regular additive category. They are known to be isomorphisms rationally. We show that it suffices to invert only those primes for which contains a non-trivial finite -group and is not invertible in . The key ingredient is the detection of Nil-terms of a twisted group ring of a finite group after localizing at in terms of the -subgroups of using Verschiebungs and Frobenius operators. We construct and exploit the structure of a module over the ring of big Witt vectors on the Nil-terms. We analyze the algebraic -theory of the Hecke algebras of subgroups of reductive -adic groups in prime characteristic.