Relative assembly maps and the K-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 <i>K</i>-theory of twisted group rings of a group <i>G</i> with coefficients in a regular ring <i>R</i> 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 <i>p</i> for which <i>G</i> contains a non-trivial finite <i>p</i>-group and <i>p</i> is not invertible in <i>R</i>. The key ingredient is the detection of Nil-terms of a twisted group ring of a finite group <i>F</i> after localizing at <i>p</i> in terms of the <i>p</i>-subgroups of <i>F</i> 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 <i>K</i>-theory of the Hecke algebras of subgroups of reductive <i>p</i>-adic groups in prime characteristic.