and an open neighbourhood , then we can find a smooth function whose support is compact and lies in and Then we can write and define Since in then is clearly well-defined and smooth; we note that which is indeed well-defined and smooth, therefore the same is true for . Now we show that on . since is holomorphic in . applying the generalised Cauchy formula to we find since , but then on . Since was arbitrary, the lemma is now proved. Proof of Dolbeault–Grothendieck lemma Now are ready to prove the Dolbeault–Grothendieck lemma; the proof presented here is due to Grothendieck. We denote with the open polydisc centered in with radius . Lemma (Dolbeault–Grothendieck): Let where open and such that , then there exists which satisfies: on Before starting the proof we note that any -form can be written as for multi-indices , therefore we can reduce the proof to the case . Proof. Let be the smallest index such that in the sheaf of -modules, we proceed by induction on . For we have since ; next we suppose that if then there exists such that on . Then suppose and observe that we can write Since is -closed it follows that are holomorphic in variables and smooth in the remaining ones on the polydisc . Moreover we can apply the -Poincaré lemma to the smooth functions on the open ball , hence there exist a family of smooth functions which satisfy are also holomorphic in . Define then therefore we can apply the induction hypothesis to it, there exists such that and ends the induction step. QED The previous lemma can be generalised by admitting polydiscs with for some of the components of the polyradius. Lemma (extended Dolbeault-Grothendieck). If is an open polydisc with and , then Proof. We consider two cases: and . Case 1. Let , and we cover with polydiscs , then by the Dolbeault–Grothendieck lemma we can find forms of bidegree on open such that ; we want to show that We proceed by induction on : the case when holds by the previous lemma. Let the claim be true for and take with Then we find a -form defined in an open neighbourhood of such that . Let be an open neighbourhood of then on and we can apply again the Dolbeault-Grothendieck lemma to find a -form such that on .