Periodic resolutions over exterior algebras.
The author analyzes the structure of periodic modules over the exterior algebra \(E=\bigwedge V\) where \(V\) is a finite dimensional vector space over a field \(K\). A module \(M\) is called periodic if it is isomorphic to one of its syzygy modules \(\Omega_i(M)\) and the smallest index \(i\geq 1\) for which isomorphism occurs is called the period. It is shown that every module with bounded Betti numbers is periodic of period \(\leq 2\), and that the period is \(1\) if the module is graded. Moreover, a module with a linear Tate resolution is periodic of period \(1\). In the last part of the paper the author parametrizes the family of periodic modules over algebraically closed fields in terms of their representation matrices. A particularly nice description is obtained for graded periodic modules: In a sense, they are close to direct sums of quotients \(E/(a)\) where \(a\) is a linear form.
scientific article; zbMATH DE number 1868092