On the tubes of exterior algebras

In this paper, we study the Koszul modules of complexity I of an exterior algebra of a vector space V over an algebraically closed field. We prove that there are P(V)-orthogonal families of such modules, each of which consists of Koszul modules filtered by the cyclic ones corresponding to the same point in P(V). We also prove that they are invariant under syzygy, so they are all on the homogeneous tubes.