Continuous Cohomology of Permutation Groups on Profinite Modules

ABSTRACT

Model theorists have made use of low-dimensional continuous cohomology of infinite permutation groups on profinite modules, see Ahlbrandt and Ziegler (1991 Ahlbrandt , G. , Ziegler , M. ( 1991 ). What's so special about (?/4?)^ω? Archive for Math. Logic 31 : 115 – 132 . CSA] Crossref] , Google Scholar]), Evans (1997b Evans , D. M. ( 1997b ). Computation of first cohomology groups of finite covers . J. Algebra 193 : 214 – 238 . CSA] CROSSREF] Crossref], Web of Science ®] , Google Scholar]), Evans et al. (1997 Evans , D. M. , Ivanov , A. A. , Macpherson , H. D. ( 1997 ). Finite covers . In: Evans , D. M. , ed. Model Theory of Groups and Automorphism Groups . London Mathematical Society Lecture Notes 244 . Cambridge : Cambridge Univ Press , pp. 1 – 72 .Crossref] , Google Scholar]), and Hodges and Pillay (1994 Hodges , W. , Pillay , A. ( 1994 ). Cohomology of structures and some problems of Ahlbrandt and Ziegler . J. London Math. Soc. 50 ( 2 ): 1 – 16 . CSA] Crossref] , Google Scholar]), for example. We expand the module category in order to widen the cohomological toolkit. For an important class of groups we use these tools to establish criteria for finiteness of cohomology.