Periodic resolutions over exterior algebras

In this paper we study modules with periodic free resolutions (that is, periodic modules) over an exterior algebra. We show that any module with bounded Betti numbers (that is, a module whose syzygy modules have a bounded number of generators) must have periodic free resolution of period less than or equal to 2, and that for graded modules the period is 1. We also show that any module with a linear Tate resolution is periodic. We give a criterion of exactness for periodic complexes and a parameterization of the set of periodic modules. (C) 2002 Elsevier Science (USA). All rights reserved.