Noetherianity and Ext

Let R=R0⊕R1⊕R2⊕⋯ 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=R1⊕R2⊕⋯ and S=⊕n≥0ExtRn(R/J,R/J). 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.