Noetherianity and Ext

Let R=R0circled plusR1circled plusR2circled pluscdots, three dots, centered be a graded algebra over a field K such that R0 is a finite product of copies of K and each Ri is finite dimensional over K. Set J=R1circled plusR2circled pluscdots, three dots, centered and image. We study the properties of the categories of graded R-modules and S-modules that relate to the noetherianity of R. We pay particular attention to the case when R is a Koszul algebra and S is the Koszul dual to R.