Denote byRthe power series ring in countably many variables over a fieldK; thenŔis the smallest sub-algebra ofRthat contains all homogeneous elements. It is a fact that a homogeneous, finitely generated idealJinŔhave an initial ideal gr(J), with respect to an arbitrary admissible order, that is locally finitely generated in the sense thatfor all total degreesd. Furthermore, gr(J) is locally finitely generated even under the weaker hypothesis thatJis homogeneous and locally finitely generated.