On the definition of multi-Koszul modules

In [8] we introduced the notion of multi-Koszul algebra: it is an extension of the definition of generalized Koszul algebra given by R. Berger in [1] for homogeneous algebras (see also [7]) that can be applied to any nonnegatively graded connected algebra over a field k. The goal of this article is on the one hand to properly extend the notion of multi-Koszul algebra to the case where the base ring K is a product of copies of a field k, which a priori allows us to treat quiver algebras, and on the other hand to introduce the notion of multi-Koszul module such that it extends the usual definition of generalized Koszul module over a generalized Koszul algebra. We show eventually that multi-Koszul algebras and multi-Koszul modules are strongly linked via the notion of one-point extensions, as in the case of generalized Koszul algebras. Moreover, we describe the complete structure of right -module on over , where M is a multi-Koszul module over a multi-Koszul algebra A, extending a result in [9]. As a corollary, we obtain that the underlying right module structure of over is generated by the component of cohomological degree zero, as in the case of generalized Koszul modules over generalized Koszul algebras.